AP Seminar!

AP Seminar, by the book, is a foundational course that engages students in cross-curricular conversations that explore the complexities of academic and real world topics and issues by analyzing divergent perspectives. We will read and analyze articles, research studies and foundational, literary and philosophical texts and we watch and listen to speeches and broadcasts and personal accounts and we experience works and performances in this inquiry based course framework. We will synthesize information from multiple sources, develop our own perspectives and deliver oral presentations. Ultimately, this course aims to equip students with the power to analyze and evaluate information with accuracy and precision in order to craft and communicate evidence based arguments.

I like to look at our course as an exploration of communication. And at the very core lies the concepts of argument construction and rhetorical analysis through modes of persuasion. We have been trying to convince people to take our point of view for ages. We want to stay up later when we are kids and we want people to see our favorite film, as was the original intension of this blog, or dine in our favorite restaurant. We try to convince people to read our screenplay and we try to convince Cornell to accept us. And it is through the concepts within this course that we build our case for ourselves or our favorite pizza place.

I’ll have a go a the latter. In the fall of 2018 a brand new pizza restaurant opened in the Old North End of Burlington. With limited seating and pies nearing $40, it certainly doesn’t seem as if it is place that could find success. However, it is by far the best pizza, tossed by an eccentric in the air of high volume Grateful Dead tunes that values authenticity over any other aspect of his business, some people might say over service. He uses naturally leavened dough, fresh everyday, and true to the motherland San Marzano tomatoes. He sells square pies and traditional pies and, in fact, one of his most amazing feats was to rework his Neapolitan dough during the past couple years into the traditional crust to accommodate the carry out craze! It is my special occasion establishment and it should be yours too.

Some people would instantly balk at the price without even coming within a mile of Riverside. However, you do not have to go every night? The ingredients are superior and come at greater cost so he is basically breaking even. Some people want to be greeted, seated and waited on hand over feet. This will not happen at Ida. You will simply make a new friend a leave as you escape under the hypnosis of the most wonderful pizza.

AP Seminar Assessment

Performance Task 1

In this project, three to five students collaborate as a team to identify a real world problem or issue (local, national or global, academic, theoretical or philosophical). Each team develops a team research question and conducts preliminary research. They identify approaches, perspectives, or lenses and divide responsibilities for individual research that will address the team’s research question.

Individual Research Report

Individually, you investigate your assigned approach, perspective or lens on the issue or topic of the team research question. The IRR is essentially a well written, 1200 word, lite literature review to present to your team that does the following.

(a) Identifies the area of investigation and its relationship to the overall problem or issue.

(b) Summarizes, explains, analyzes and evaluates the main ideas and reasoning of the chosen sources. This report essentially gets your sources talking to one another.

(c) Identifies, compares and interprets a range of perspectives within your chosen lens about the problem or issue.

(d) Cites all sources used and includes a list of works cited in either MLA or APA format. Either is acceptable but choose one or the other.

Team Multimedia Presentation

Now, working collaboratively, your team considers all of the research and analyses presented in the IRRs for the purpose of proposing one or more solutions or resolutions to the problem or issue and does the following.

(a) Collaboratively synthesizes and evaluates individual findings and perspectives to create a collective understanding of different approaches to the problem or issue.

(b) Considers potential solutions or resolutions and conducts additional research in order to evaluate different solutions within the context of the problem or issue.

(c) Proposes one or more solutions or resolutions and prepares an argument to support your proposal.

The team will then develop an 8-10 minute presentation that presents a convincing argument for your proposed solutions or resolutions. We should be sure the claims made are supported by evidence which should be attributed. We should ensure you have considered different perspectives and the limitations and implications of their proposed solutions or resolutions.

Oral Defense

After your presentation, I will ask each member of your team one question. The questions are scripted and designed to prompt reflection on your experiences with group collaboration. You should be able to answer questions about each aspect of the presentation.

Performance Task 2

Individual Written Argument

Each year, our friends at the College Board will release cross curricular stimulus material representing a range of perspectives connected through range of themes with one or two emerging overarching themes. You will read and analyze these stimulus materials to identify these thematic connections and possible areas of inquiry. Your inquiry must be based on a thematic connection between at least two of the stimulus material. It really can be anything and you should make sure it is something that is of interest!!

(a) You then compose your own research question.

(b) Conduct research, analyze, evaluate and select evidence to develop an argument!!

(c) Present and defend your conclusions and the final paper must integrate at least one of the stimulus materials as part of the actual response.

We must have at least 30 school days with the stimulus materials to complete your research, compose your essays and develop your presentations and the presentations will be scheduled outside of the 30 day window. Once the stimulus materials are revealed we will explore the connections in class and discuss possible topics through different texts and artifacts like documentary films.

For the IWA, you will read and analyze the stimulus material to identify the thematic connections as stated above. Once you discover your areas of inquiry and research question prompted by the stimulus materials. Once you gather additional information through research, as the majority of your paper will be derived from sources outside of the stimulus relevant to your area of inquiry, you will analyze and select evidence to develop a logical and well reasoned argument of 2000 words. Mind your sources and plagiarism.

Individual Multimedia Presentation

In this component of Performance Task 2, you will develop a 6-8 minute presentation to convey your perspective and present your conclusions from your IWA. You can assess your topic of inquiry through different lenses and different perspectives within the lens to make your argument. You will present your evidence to support your claims and plant your perspective in a larger context instead of simply summarizing your research. This is called argumentative synthesis! The presentation and media will consider audience, context and purpose.

The Oral Defense

In conclusion, after the presentation, I will ask you two questions. This is designed to assess your response to, and understanding of, your reflection on the research process and your extension of the argumentation through effective questioning and inquiry.

Dates

On May 1, 2023, your IWA and IRR are to be submitted via the Digital Portfolio on the College Board website for them to score. It is recommended that you stay ahead and submit earlier than the last minute to avoid any glitches and deliver a fantastic paper on both tasks!! The presentations are expertly scored by Darren and recorded to be submitted to the College Board. The TMP will take place in early to mid March with the IMP taking place the week after spring break

In early May, we will be taking the End of Course Exam for AP Seminar. The exam consists of four items. Three short-answer questions which assess analysis of an argument in a single source or document. The essay question assesses students’ skills in synthesizing and creating an evidence-based argument.

There will be five sources included in each exam. Sources and readings on the exam will represent a range of disciplines and perspectives. If we have a look at the components of the exam we can share how they are assessed!

In Part A, there is one source provided and you are asked to analyze an argument using evidence.

(1) Identify the author’s argument, main idea, or thesis.

(2) Explain the author’s lines of reasoning by identifying the claims used to build the argument and the connections between them.

(3) Evaluate the effectiveness of the evidence the author uses to support the claims made in the argument.

This will count as 30% of 45% for the exam. The suggested time for this section is 30 minutes.

In Part B, you are asked to build your own argument using at least two of the four sources that are provided. Each the the four sources will explore a common theme through a different lens allowing multiple entry points for student’s to approach the topic. We suggest 90 minutes for this section.

The instructions will ask you to read the four sources, focusing on a theme or issue that connects them and the different perspective each represents. Then, you will write a logically organized, well-reasoned and well written argument that presents your own perspective on the theme or issue you identified. You must incorporate at least two of the sources provided and link the claims in your argument to supporting evidence. You may also use the other provided sources or draw upon your own knowledge. In your response, refer to the provided sources as Source A through D or by the author’s names.

AP Lang!!

An exciting aspect of the AP Seminar course is that many of the concepts integral to the course are mirrored through AP Language and Composition. It is therefore encouraged to sit for the AP Language exam!!

AP Lang aims to test our understanding of the literary concepts covered in the course units, as well as our ability to analyze texts and develop written arguments on our interpretations. It is a long exam, at 3 hours and 15 minutes, and comprised of two sections.

In Section 1, we have 1 hour to do 45 questions and this makes up 45% of the score. Excepts from nonfiction texts accompany the questions. In 23-25 of the questions, you’ll be asked to read and analyze nonfiction texts. In 20-22, the writing questions, you’ll be asked to read like a writer and consider revisions to the text! This is the reading comprehension portion of the program

In Section 2, the free response section, there are 3 questions that make up 55% of your score and you are given 2 hours and 15 minutes to complete this section which includes a 15 minute reading period. In the free response section, you’ll respond to three questions with written answers. This section tests your skill in composition in three areas!!

(1) Synthesis: after reading 6-7 texts about a topic, including visual and quantitative sources, you will compose an argument that combines and cites at least three of the sources to support your thesis.

(2) Rhetorical analysis: you will read a nonfiction text and analyze how the writer’s language choices contribute to the intended meaning and purpose of the text. We can do this now!!

(3) Argument: you will create an evidence based argument that responds to a given topic.

Course Essentials!!

Throughout the course we are going to learning about arguments, lines of reasoning, claims, synthesis, rhetoric, irony, logical fallacies, rhetorical strategies and evidence through interaction with what we call artifacts or texts. This interaction can be through (a) reading articles and research studies (b) reading foundational, literary and philosophical texts (c) viewing and listening to speeches, broadcasts and personal accounts and (d) experiencing artistic works and performances.

Argument

Argument is probably the most important aspect of this course and broadens your portfolio in communication and reasoning. The argument that may first come to mind is that of an exchange of diverging views, typically a very heated one. Like a quarrel or row. In literary terms, an argument is a reason or set of reasons given with the aim of persuading others that an action or idea is right or wrong. Take a great look at the this classic clip from Monty Python featuring the great John Cleese!

In academic writing, the kind of writing central to Sem, an argument is usually a main idea, often referred to as a claim or thesis statement backed up with evidence that supports the idea. In the majority of all the writing you will do in your academic life, you will need to make a claim and use evidence to support it. This profound ability will separate your work from students that view assignments as mere accumulations of facts and details. Therefore, we don’t take a topic and simply write anything anymore. We must take a position and prove why it’s a good position for a thoughtful person to hold (writingcenter.unc.edu).

(a) USWNT deserves to be paid equally.

(b) Rogue One is best Star Wars film outside of Empire.

(c) Cinnamon makes everything taste better.

In each case, the rest of my paper will detail the evidence that has led me to believe that my position is the best. That is, “What is my argument?”

As we have defined and reiterated, argument is a reason or set of reasons given with the aim of persuading others that an action or idea is right or wrong. Let’s make sure our thesis takes a clear stand!! We want to always avoid expository writing or writing to convey factual information. It is the language of learning and understanding the world around us like a how-to article or a textbook where it is simply the facts. This type of writing simply offers the reader information and is encapsulated in the following sentence.

“Steven Parker says that language is a way of understanding human nature.”

Developing an Argument!

As we develop our arguments, we must remember that arguments use reason and evidence to convey perspective, point of view or some version of the truth that is stated or implied in the thesis or conclusion. Arguments are supported and unified by carefully chosen and connected claims, reasons and evidence. An argument may acknowledge other arguments and respond to them with counterarguments. The line of reasoning is a clear, logical and sequential path leading the audience through the reasons toward a conclusion. This is sometimes referred to as the train of thought and is organized based on the argument’s purpose.

We always want to connect claims and evidence during our stream of thought!! When writing an argument, it is important that you organize the points you want to make in a clear sequence that makes the overall argument easy to follow. The other important part of developing an argument is the commentary which is the explanation of how the evidence relates to the claims being made and how that supports the overall thesis and argument! Evidence should be sufficient, typical, relevant, current and credible to support the conclusion.

In particular, one way to signal to the reader relationships between ideas is through the use of transitions. Transitions are words or phrases that provide sign posts to help the reader understand the point you are trying to make. Let’s look at a few examples. This will also be very prevalent in our AP Lang prep!

(a) If the relationship between ideas is example or illustration; “For example, thus, therefore, for instance, to illustrate, in particular, specifically, namely, such as.”

(b) Contrast or exception; “However, nonetheless, in spite of, in contrast to, conversely, while this may be true, on the contrary, on the other hand, rather.”

(c) Addition; “In addition, furthermore, moreover, besides, likewise, similarly, as well as, next, equally important.”

(d) Cause or result; “Accordingly, so, hence, therefore, consequently, as a result, for this reason, in other words, due to.”

(e) Summary; “In short, in brief, finally, in conclusion, on the whole, to sum up.”

Whenever you’re reading a text, always ask yourself what the author is doing. Again, the main idea of an argument, which can be its conclusion or thesis, is the main point the author is trying to make the reader accept and the line of reasoning is the reasons or claims the author uses to support their main idea. Sometime the argument can be explicit and it can be difficult to discern the claims into the premises and the conclusion or thesis. Is this the main point or is this a reason to believe another claim? Luckily, this is where our transition words can really come into play and we can think of it like since (premise) versus hence (conclusion)!!

Synthesis

The writing process for composing a good synthesis essay requires curiosity, research and original thought to argue a point. Let me reiterate that curiosity and original thought are integral to writing as you cannot simply list data points or summarize a text. The subject must be interesting to you and allow for exploration of an original idea. Synthesis essay writing involves a great deal of intellectual savvy, but knowing how to compose a compelling written discussion of a topic can give you an edge in so many fields from social sciences to screen writing.

A synthesis essay gathers information from a variety of sources to form a new idea, question or argumentative thesis. Writers composing a synthesis essay will discuss ideas, data and evidence from a series of sources to either explain or argue something original. There are two different types of synthesis essay.

(a) Explanatory synthesis uses different sources to explain a particular point of view. These synthesis papers objectively examine the similarities and differences between ideas but does not necessarily choose a side or attempt to sway the reader in a certain direction.

(b) Argumentative synthesis follows the same structure as a typical argumentative essay where the thesis will argue one specific point.

Writing Synthesis

A great synthesis essay walks readers through a series of ideas and sources to prove or explain a larger point. Here are some tips!

(1) Choose a topic you are curious about! Brainstorm a few ideas for your synthesis essay topic prioritizing subjects you feel passionate about.

(2) Do your research! This will help develop a point of view that is backed up by concrete information. Use credible sources that are unbiased and objective.

(3) Be aware of your point! Your entire essay should focus on either explaining a certain perspective or making an argument. Discuss how each of your sources relate to your theme and support your idea.

(4) Write your introduction! Present the general premise of your paper providing any necessary background information and capture the reader’s attention and contains a strong thesis statement or conclusion. This is where you will state the point of view that you’re explaining or the argument you are making.

(5) Include body paragraphs! A good body paragraph contains a topic sentence, relevant supporting sentences and a transition sentence. The structure keeps your paragraphs focused on the main Idea, providing clear and concise information. Each paragraph should discuss different elements and supporting arguments of your thesis, along with evidence. It is here where you would explain the common theme between your resources and synthesize your argument and include counterarguments as well as how source material can discredit those claims and support your own ideas!

(6) Write a great conclusion!! A strong conclusion is the final piece of your research paper, essay or article that summarizes the entire piece. This paragraph of your synthesis essay will restate your thesis, summarize the key supporting ideas you discussed throughout and offer your final impression of the central idea.

(7) Proofread!! This is extremely important!! I cannot tell you how many pieces were turned in with obvious grammatical errors spotted through Docs!! This ensures grammar, syntax and flow are as accurate and clear as possible. This is the final impression on your reader as a credible source!

Rhetoric

Rhetoric, as defined by the Oxford English Dictionary, is the art of using language so as to persuade or influence others; the body of rules to be observed by a speaker or writer in order that they may express themselves with eloquence. Rhetoric is speech or writing intended to be effective and influential. In the past, rhetoric was reckoned to be one of the seven “liberal arts” being comprised of grammar, logic, arithmetic, astronomy, music and geometry. Logical arguments are those which determine whether a particular statement is true or false. Rhetorical arguments are those which attempt to persuade a person or audience that a particular statement is true or false, regardless of whether it actually is true or false.

The art of persuasion, along with grammar and logic is one of the three ancient arts of discourse. Discourse is simply an orderly and formal expression of thought on a subject. It is somewhat of a linguistic unit larger than a sentence.

Rhetorical Appeals!!

If rhetoric is regarded as the art of persuasive language and speaking, it reasons that there would be modes of persuasion. These were characterized by Aristotle as the three artistic proofs. The first depends on the personal character of the speaker; the second on putting the audience into a certain frame of mind; the third on the proof, or apparent proof, provided by the words of the speech itself. Ultimately, these are strategies of rhetoric that classify a speaker’s or writer’s appeal to their audience. This is the rhetorical triangle!!

(1) Ethos; appealing to ethics, morals, and character.

(2) Pathos; Appealing to emotion.

(3) Logos; Appealing to logic, reason, arrangement, facts and evidence.

The initial persuasive appeal, ethos, has reference to the speaker and leads to the entire notion of the spokesperson. Ethos is Greek for “ethics” which is inherent to character and has been made to represent the credibility of the person making the communication, the speaker. Ethos is established through a variety of factors including status, awareness, professionalism, celebrity endorsement, research and so forth. We build ethos to make our audience trust what we are saying!

The second persuasive appeal, pathos, references the audience and is the Greek term for “emotion”. This persuasive appeal has been made to represent how an audience feels or experiences a message. The appeal of pathos makes us feel excited, sad, angry, motivated, jealous or any other emotions that persuade us to act based on the speakers words.

The final persuasive appeal that we will consider is logos which is the Greek term for “logic”, logically, and references the message. It has been made to represent the facts, research and other message elements that provide proof or evidence to a claim. We use logos to convince our audience that what they are hearing or reading or seeing is well researched, well built or otherwise worth their time and investment.

The Many Faces of Rhetoric

Rhetoric comes in all shapes and sizes and we actually use the term quite frequently in our daily lives. Rhetoric is omnipresent but I believe we can sum up its many faces in the following few sentences.

Rhetorical questions guide the reader in a certain direction. Rhetorical or logical fallacies distract the reader with various appeals instead of sound reasoning. Rhetorical devices use words in specific ways to strengthen an argument. Rhetorical analysis is close reading that examines the interplay between author, text and audience. Rhetoric, as we have stated before, is the art of persuasion.

A rhetorical question is a question that either doesn’t require an answer and is posed simply to make a point. It can also be a question where the answer is so obvious that it doesn’t need to be stated or doesn’t have an answer. Simply, a rhetorical question is where the asker is not expecting an answer or the question has an extremely obvious answer that is does not need to be stated.

Recently, during the epic final episode of the Season 4 of Stranger Things, Eddie asks Dustin if he is ready for the most metal concert in the history of the world and Dustin responds, “Is that a rhetorical question?” It’s a brilliant line in a brilliant scene. Famously, George Carlin once posed, “Isn’t a bit unnerving that doctor’s call what they do “practice?” George is simply making a humorous point or observation and not actually expecting someone in his audience to answer him. Think about this Geico Ad campaign or simply ask someone, “Can you believe this is happening?”

Rhetorical or logical fallacies are flaws in reasoning that appear as trick or illusions of thought. And those who can wield them to their advantage are truly magicians, or just bombastic liars. They are flawed, deceptive or false arguments that can be proven wrong with reason. Arguments are an important part of academic discourse but not every argument is perfect. Fallacies can be very persuasive and it’s important for us to understand these types of appeals so we can make sure we are making sound and logical arguments. In this video and this text we can become familiar with over 20 common logical fallacies and we can see how prevalent they are in our rhetoric. Let’s look at a couple of popular ones.

False Equivalence

This fallacy involves drawing an equivalence between two subjects because they share some characteristics but ignores the significant differences between them. This is often called the apple to oranges fallacy.

Apples and oranges must tastes the same. After all, they are both round fruits!

Strawman

This is a common fallacy that misrepresents an opposing argument, showing it to be more extreme than it actually is so that it is easier to attack. If someone wants to legalize cannabis, their argument can be construed to include the legalization of all drugs. Or the ACA makes healthcare a government run entity.

Ad Populum or Bandwagon

This is a fallacy extremely common in advertising as the assumption that the opinion of the majority has to be valid. Everyone believes it, so should you! This is represented in fashion trends, weight loss trends and sports and social media challenges. “Cousin Greg wore Crocs to the Emmy’s so I just bought mine today.”

Rhetorical Analysis

Rhetorical analysis is the cornerstone of a liberal arts education. Rhetorical studies prepares students to thrive intellectually in whichever situations they might encounter, while also expanding their capacities to reflect on the human condition and serve the public good. I

A singular rhetorical analysis attempts to break down an artifact into its components and then continues on to offer an explanation as to how each part of the piece work to combine to create a certain effect. Artifacts or texts can be in classical essay or article form but can also present as cartoon, advertisement, interview or speech and the role of the analysis is to decipher whether the text is intended to persuade, entertain or inform.

We want to explore the artifacts of discourse which includes words, phrases, images, gestures, style and performance that speakers, directors, teachers, YouTubers and TikTok stars use to communicate. This may seem like a daunting task as we must develop observational and analytical skills. As we write our rhetorical analysis, we are not agreeing or disagreeing with the argument but we are discussing hoe the speaker makes the argument and whether or not this approach facilitated success! What are the speaker’s goals, techniques used and are those techniques effective. Think about what the movie preview is trying to accomplish!!

There are several methods of analysis and SPACECAT is my favorite for obvious reasons!! This link offers a purposeful guide to the vital ability to break down a text!

Rhetorical Devices

Rhetorical devices are the playful side of language as the author writes through creativity and imagination. These linguistic tools use a specific type of sentence structure, sound or pattern of meaning in order to elicit a particular reaction from the audience. As we have been saying, any interaction in which you try to inform or persuade an audience or argue with someone, you engage in rhetoric. This is where we find all of our good friends such as metaphor, juxtaposition, oxymoron, alliteration and hyperbole and a list can be found in this article. We could also have a look at this list which has a link to common rhetorical devices in pop songs!! One of these devices was commonly used by Yoda as hyperbaton inverts the idiomatic order of a phrase like, “enjoy ice cream, do you? or “run to the market, will we?” It’s very Irish as well!!

It is important to realize that there are many literary devices and it is pretty difficult to memorize all of them. The easiest process to apply is to ponder what the author is actually doing in the text. Are they beginning each sentence with the same phrase? If so, just mention that in your analysis as you recognized the manifestation of the tool. If you remember that it’s anaphora that’s great but tying up your analysis with the Greek terminology, besides the commonest gadgets like metaphor, may be slightly distracting.

A metaphor, as defined by the Oxford Dictionary is a figure of speech in which a word or phrase is applied to an object or action to which it is not literally applicable. Or simply poetically calling something something else. Or as defined by Grammarly a metaphor is a figure of speech that describes an object or action in a way that isn’t literally true, but helps explain an idea or make a comparison. Are really a couch potato? A night owl? An early bird? A spark plug?

Metaphors are part of figurative language where we tap our reader’s imagination by taking a flat or factual statement and inject it with life color, humor to make it more interesting.

Irony

The 1994 hit film Reality Bites, starring Ethan Hawke and Winona Ryder, captures a generation that, according to the Atlantic, preferred to be overlooked. Troy (Hawke) famously defines irony to Lelaina (Ryder) as “when the actual meaning is the complete opposite of literal meaning.” According to StudioBinder, this fundamental difference can present in language in the form of what we say v. what we actually mean or circumstance in the form of what we expect to happen v. what actually happens.

The three different types of irony are (a) dramatic irony, (b) situational irony and (c) verbal irony. Irony can be sad or tragic and funny and satirical which renders the technique limitless. It creates suspense, conflict and complexity which are the pillars of good storytelling.

Masterclass talks about dramatic irony as when the writer lets their reader or viewer know something that the characters don’t. The obvious example is Romeo & Juliet as this is also regarded as tragic irony. This fills the audience with suspense and anticipation and relief and is a classic tool in horror films! Also, see The Matrix!!

Next we have the dog chews up the “Dog Training for Dummies” book! Situational irony is at play when the expected outcome is subverted like the classic story of the Gift of the Magi or Remy as a master chef in Ratatouille!

Lastly, verbal irony is when the speakers mean something very different than what he or she is saying. My favorite example of this is this clip when Donkey asks if he can stay with Shrek!! This commercial also exemplifies verbal irony as everything he says is completely opposite to what would be expected after each unfortunate circumstance!!

The Written Assessments Refined!

IRR

(1) A Performance Task 1 assessment of 1200 words with a 10% cushion and focuses on one (1) lens of a research question and typically at least two perspectives.

(a) A lens is something that facilitates and influences perception, comprehension or evaluation; a way of looking at a problem.

(b) A perspective is a particular way of considering something defined as a point of view conveyed through an argument. Perspectives are easiest to identify by grouping sources by common theme or reason for agreement or disagreement.

(2) We are investigating a self-selected topic with a group of our peers.

(3) Again, this is considered a lite literature review that explains existing information that addresses the Research Question through the lens of our choice. In the IRR, no argument is made as it is only a report of existing research.

A quality IRR identifies and explains a complex problem with a wide variety of appropriate sources and connects the problem to a larger context. It utilizes appropriate source material using strategic selection of evidence in support of claims. It explicitly addresses the credibility of the sources and organizes source material into clear perspectives and addresses the line of reasoning of the chosen source material. Remember the line of reasoning is how the author gets from A to B using claims and evidence referred to as the train of thought.

Let’s talk about a fantastic Research Question!!

We need to develop a wonderful research question. Without this, we can end up picking a question that is too broad to be properly researched or does not allow for debate between perspectives within a lens. Good research questions require judgment or evaluation to made and they are researchable with possible relevant and credible sources lined up. The question involves genuine points of ongoing debate, invites engagement with alternative perspectives and are simple without containing multiple or nested questions. (Seminar Workbook)

A good research question pinpoints exactly what you are trying to find giving your work clarity and purpose. As conceptualized in this Scribbr article, we want our research question to be focused on a single issue, researchable using primary and secondary sources, feasible to answer within practical restraints, specific, complex enough to develop the answer within a thesis and relevant to society or your work.

The IRR adheres to all standard rules of grammar and syntax and accurately attributes information to sources using in text citations and a full bibliography. The rule of thumb is about 200 words filtered from each source and therefore we should have at least seven sources.

IWA

(1) A Performance Task 2 component of 2000 words also with a 10% cushion and allows for multiple lenses and perspectives.

(2) Your paper must be aligned with the a stimulus theme from the college board stimulus material that connects at least two of the artifacts.

(3) The IWA is a well-supported and aligned argument based a round a research question which requires a position to be taken and argued for and the line of reasoning and the quality of evidence is the biggest component of the paper!

Let’s look at the IWA!!

The IWA requires development of a well-reasoned, well-supported argument or a broad theme based on stimulus material from the College Board. For example, the material contains literary, artistic, scientific and mathematical resources all related to the idea of work. At least two of these sources must be used as an integral or central piece of evidence or context of the argument.

In the IWA we must have a line of reasoning that progresses from a main claim through evidence based sub claims to a logical conclusion. Continuing the work example, Jolene defines the Puritan work ethic and ties this top the idea of overwork, connects the idea of overwork to increased mental illness and depression and concludes the Puritan ideals around work are detrimental to a healthy life.

The IWA requires a logical conclusion or solutions to the topic being explained. The argument addresses the problem. Jolene can argue for changes in work regulations to reduce overwork or cultural shifts in understanding a solid work life balance. She could just as easily defended her conclusion by addressing counter claims.

A quality IWA does the following:

(1) Demonstrates understanding of the stimulus material theme and sources by utilizing at least two sources as a keystone component.

(2) Contextualizes and situates the research question and argument as part of a larger idea.

(3) Utilizes multiple perspectives and clearly explains how they are related to one another.

(4) Identifies limitations, connects claims or implications of the chosen perspectives.

(5) Provides a logical, well-reasoned and well-supported argument by applying lines of reasoning.

(6) Tightly connects claims and evidence which leads to a sound and logical conclusion.

(7) Utilizes evidence from appropriate and credible sources (RAVEN and CRAAP) although explicit and extensive evaluation of credibility is not needed.

(8) Adheres to all standard rules of grammar and syntax and accurately attributes information to sources using in text citations and a full biography.

Report v. Argument!

For the IRR in PT1, we need to write an analytical report presenting the most important perspectives they have discovered in their research into a single lens of the team’s problem or issue of the day. In contrast, the IWA in PT2 requires students to present their own perspective in a well-reasoned argument.

The IRR draws on scholarly and credible sources. The different perspectives within a lens should be in conversation with one another to convey the most important aspect of the topic and what experts have to say about them. This involves examining details of different perspectives to explain points of overlap, agreement, contention or conflict that help a reader understand the complexity of the problem or issue.

In terms of our voice, we should NOT make an argument. Our report presents an analysis of the. Ian issue and perspectives in our chosen topic that will contribute to the team’s overall research question. The only argument present in the IRR is an implicit one that what is in the report is the most important issue for that topic and the most important perspectives. The IRR is a small scale literature review on one lens of the problem or issue. It is a scaffolded step in analyzing and synthesizing what experts say prior to the team forming their own argument for the TMP.

On the other hand, the IWA should draw on scholarly and credible sources as well. The different perspectives should be in conversation with each other to draw conclusions that support our argument. This involves examining the details of the perspective and drawing on them to support the individual claims and the overall argument we are making.

Our voice must be driven by our argument. The argument needs to be organized in such a way that our perspective should be accepted. These reason’s and claims should be supported by credible and relevant evidence that is clearly explained so the audience understands how that evidence supports our overall argument. Our argument needs to reach a plausible conclusion that either answers our question or converts our thesis!!

Look at Lenses!!

Lenses are particular ways to look at a problem or issue and in order to sift through the different lenses, we can ask a few questions. How is your research connected to the understanding of the group topic? What type of professional or academic might study the subject or teach about the subject? The College Major Paradox. Please do not write, “my lens is (blank) in the IRR” and always remember that not all lenses are logically connected to your theme, issue or topic. The eight lenses generally considered for the College Board are as follows:

Cultural and Social; how do people interact? Explain how daily issues have an impact on our daily lives, relationships, customs or beliefs.

Artistic and Philosophical; explore how artwork expenses ideas about the issue and what philosophers think.

Ethical; explain an issue at a moral level considering human and animal rights and laws.

Political and Historical; explore an issue’s effect on government decisions and consider the background.

Futuristic; explore the impact of this issue on the future.

Environmental; explore the effects on the environment and surroundings or how the environment effects the issue.

Economic; explore the economic impact of the issue from economists.

Scientific; explore the issue based on or characterized by the methods and principals of science

With the AP Seminar course, we will make cross cultural connections within a theme by looking at the theme through different lenses. A lens is ultimately a working tool to help generate issues for further investigation. Also, when searching for texts, we want to gather sources that view the theme through some of the different lenses for the cross curricular approach.

Finally, we have a way to examine our sources. With Tier 1 sources being peer-reviewed journals and Tier 5 sources including Wikipedia and blogs, we need a way to distinguish between them. RAVEN is a tool to evaluate how evidence needs to be evaluated, presented and considered.

Reputation; do past actions indicate unreliability? Is the author in a position of authority?

Ability to Observe; Is the author allowed access to reliable evidence? Did the author actually observe the event?

Vested Interest; does the author have personal stake in the topic or event? Would the author gain anything by lying?

Expertise; does the author have specialized knowledge on the topic?

Neutrality; is the author neutral or bias?Is the source of the evidence bias?

Oscillations!!

It may seem that if no one is in the room, there is no motion around you but can you sit absolutely still? Guess again!! It may seem as if no one is in the room, there is no motion around you but your heart is pulsing at a rate of about a cycle per second. Our eardrums vibrate softly in response to the sound of our breathing. Also, electrons move back and forth about 60 times per second as they supply energy to our computers or tea kettles. This is a special kind of motion called an oscillation, in which an object moves back and forth around a position of equilibrium. The equilibrium position is a point at which the object experiences zero net force.

(a) The wings of a hummingbird oscillate up and down or back and forth during hovering about 50 flaps every second!

(b) The time keeper at the heart of most wristwatches is a tiny piece of quartz that oscillates 32,768 times every second!

(c) An MRI is made by mapping how protons oscillate with molecules in the body in response to radio waves!

An object swinging back and forth on the end of a string is in equilibrium when it’s hanging straight down (FT = Fg). The few examples above describe oscillations in nature and technology. Oscillations have a cycle that repeats. The time for one complete cycle of oscillation is called the period of oscillation.

Object’s can only be in oscillation if they experience a restoring force. The restoring force is a force that always pushes or pulls the object towards the equilibrium point. Oscillations are particularly important and simple in the natural world if the restoring force is directly proportional to how far the object is displaced from equilibrium. This concept approximates many types of oscillations!

The rise and fall of our lungs and the beating of our hearts are oscillations! Compare the frequency of your breaths with the frequency of your heartbeats whilst sitting quietly for a minute. Does there seem to be a relationship. What are some questions whose pursuit could reveal a cause of the relationship if one does exist or talk about the isolation of these events if there is no relationship.

(a) We can control our breathing rate by voluntary muscular contraction. However, contractions of the heart are controlled by the Autonomic Nervous System (ANS). Can the period of the heartbeat become a multiple of the heart rate. That is, is there a relationship? It turns out, sometimes, a factor of 8 is commonly seen!

(i) Does the awareness of your heartbeat control your breath rate, and if so, is there a voluntary control of your heartbeat?

(ii) Does the awareness of your breath rate control your heart rate, and if so, is there a path of communication between breath rate and heart beat?

(iii) If the heart rate is controlled by the ANS, is there signaling between thought and the ANS, such as the release of a downstream signaling molecule?

Some other everyday examples of oscillations include strings, reeds, air columns or drum heads and their musical implications. In nature, we marvel at the oscillations in the surface of the sea or ocean and we breathlessly watch the swinging back and forth of a tall pine tree. We should always think of how the periodic behavior makes the system oscillate and identify the restoring force.

In the ENSO cycle, the cold and warm phases alternate but the period of oscillation is not regular. The mechanisms causing the effect are not understood completely. On Earth, periods of day follow periods of night with regularity. The cycle of seasons periodically repeats. The mechanisms that cause these affects are understood. Why is it easier to understand the mechanisms that cause periodic events than it is to understand the causes of seemingly random events such as the period of the warm phase of the ENSO cycle?

(i) The mechanisms that cause periodic events such as day following night, can sometimes be found by finding other processes with the same period. In this case, Earth rotates once on its axis each day, turning toward or away from the sun. It is more difficult to explain events that are seemingly random because they may involve several causes, and although each mechanism may be periodic, when they appear together, they seem random.

Oscillations

Oscillations are the result of the interplay between a restoring force and inertia. The classic object attached to a horizontal spring and is free to slide back and forth in the absence of friction!

(a) The object is in equilibrium, that is the spring is neither stretched or compressed. The distance from the equilibrium point is zero, x = 0.

(b) When the spring is stretched, the object is at x > 0. The spring force then pulls the object back towards equilibrium (F < 0).

(c) When the spring is compressed the object is at x < 0. The spring force pushes the object back toward equilibrium (F > 0).

The force exerted by the spring is the restoring force. It does not matter which way the object is displaced from equilibrium, the spring force on the object toward equilibrium.

Is a bouncing ball an example of SHM? Nope!! The force of gravity is always in the same direction and not a restoring force directed toward equilibrium.

F = – kx

Always remember that an ideal spring is a spring that abides by Hooke’s Law, that is, the greater the displacement, x, the great the force. Therefore, period and frequency are characteristics of the spring, not the amplitude by which a random variable may stretch or compress the spring.

Oscillation Period and Frequency

The period of oscillation for a stiff spring, which will push and pull the block with more force and more rapidly through a cycle, will be shorter.

Lonnie has a resting pulse of 50 bpm. When Lonnie sprints down the track, her pulse increases to 150 bpm. Compared to when she is at rest, What is the he oscillation period of her heart rate whilst sprinting?

The number of beats per unit time is the frequency of oscillation. The oscillation period is the reciprocal of frequency, so when frequency increases by a factor of 3, the period decreases by a factor of three to 1/3 its original value. f’ = (3)f and therefore T’ = (1/3)f.

During “Simple” on Night 13 of the Baker’s Dozen in New York City, Paige played the A4 note on the piano (440 Hz). What was the oscillation period of my eardrum whilst Melissa danced along?

f = 1/T = 1/440 = 2 x 10-3 s

Oscillation Amplitude

We will consider only restoring forces that stretch a certain distance as when they compress the same distance. That is, the object will move as far to left of equilibrium as it does to the right. This distance is given the value of amplitude, A, of oscillation. It’s equal to the magnitude of the maximum displacement of the object from equilibrium. The amplitude is always positive and is related to the equilibrium point not maximum stretch or compression possible.

(1) Oscillations depend on inertia to carry objects past the equilibrium point, even though the net force is zero at this point.

(2) An object in Simple Harmonic Motion has momentum at equilibrium.

Hooke’s Law involves the vectors of force and displacement.

The vector form of Hooke’s Law has a (-) sign which indicates the spring force is in the opposite direction of displacement. The spring constant is a scalar and always positive. Any negative relates to direction!

The simplest form of oscillation occurs when the restoring force obeys Hooke’s Law. The negative sign in Hooke’s Law tells us that if the spring is stretched, x > 0, it exerts a force in the negative direction. And if the spring is compressed so that x < 0, it exerts a force in the positive direction.

Uniform Circular Motion and Hooke’s Law

How does UCM relate to oscillations? Imagine that you view the circular path of an object edge-on so that you can see the (x) component of its motion. From this vantage point, you’ll see the object oscillating back and forth along a straight-line path. With some imagination, this is like our little jumpers flying around the Thriller record. However, this crash course video is your best friend when it comes to understanding SHM and how it relates to UCM!! Let’s see if we can describe it with words.

(1) An object moves at constant speed around a circle of radius A. It completes one trip around the circle in one period of revolution, T.

(2) The angular velocity or angular frequency, w = 2(pi)/T. The angle (theta) increases at a rate w as (theta) = wt.

(3) If you view the motion edge-on, the object oscillates back and forth between, -A < x < A, momentarily coming to rest at the endpoint and moving the fastest in the middle.

The (x) component of UCM looks like straight line motion oscillating around an equilibrium position at x = 0.

Position

Now, the (x) coordinate of an object as it moves around a circle is:

x = A cos (theta); (theta) = wt and x = A cos (wt)

(1) The vector from the center of the circle to the object has a length A and an angle (theta) from the +(x).

(2) The (x) coordinate of the object of the object is x = A cos (theta) = A cos (wt)

(3) The object’s velocity vector has a magnitude has a magnitude v = wA

(4) The (x) component of the object’s velocity is vx = -wA sin (wt)

(5) The object’s acceleration vector is a = w2A and is 180o from the (x) axis.

(6) The (x) component of the object’s acceleration is a = -w2A cos (wt).

Simple Harmonic Motion: Angular Frequency, Period and Frequency!

We’ve seen that the period, T, of the oscillation is the same as the T that it takes the object in UCM to travel once around the circle. The angular frequency is w = 2(pi)/T and therefore T = 2(pi)/w.

Formulas!!

Angular frequency = Angular velocity = w = sqrt(k/m) = 2(pi)/T = 2(pi)f

Period = T = 2(pi)/w = 2(pi) sqrt(m/k)

frequency = f = 1/T = 1/2(pi) sqrt(k/m) = w/2(pi)

(i) In the first equation, we’ve introduced a new quantity called angular frequency. This is the angular velocity, the back and forth motion of a block/spring is equivalent to the (x) component of UCM.

(ii) The Period of Oscillation is directly proportional to the square root of the mass, m, and inversely proportional to the square root of the spring constant, k. Therefore, increasing the mass, m, makes the period longer and increasing the spring constant, k, makes the period shorter. Angular frequency, period and frequency of an oscillation are independent of amplitude if it is an ideal spring. This is the harmonic property.

Since doubling the amplitude means that the oscillating object has to cover twice as much distance during an oscillation and so it seems it should make the period longer. However, doubling the amplitude also means that we have doubled the force, according to Hooke’s Law. This causes a greater acceleration back toward equilibrium and the object moves faster through an oscillation.

The Harmonic Property holds only for Ideal Springs!

The word harmonic invokes a certain musical vibe, right? Indeed, the harmonic property is important for musical instrumentation. We know “pitch” of a musical sound is determined by frequency of oscillation, while loudness is determined by amplitude. If changing the amplitude also changed the frequency, playing the same key on the piano would make a completely different pitch if you pressed the key soft or hard. This would render the entire musical scale completely useless and arbitrary!

The Harmonic Property is the result of the restoring force being directly proportional to the displacement from equilibrium. For this reason, the kind of oscillation that results from a Hooke’s Law restoring force is called Simple Harmonic Motion (SHM).

Mechanical Energy is Conserved in SHM!

We have discussed how the spring can provide the restoring force for oscillation to occur. However, springs can also be used to store potential energy. Let’s look at the kangaroo!! A kangaroo’s tendons acts as springs to store Us which is transferred to K when the ‘room hops and then they stretch again when the ‘roo lands. As hopping is repeated, the efficiency of movement is amplified!

Dolphins use a similar mechanism when swimming so effortlessly through the ocean waters. Even when a dolphin increases its speed by beating its tail faster, there is hardly any change in the rate of oxygen consumption. The reason is because the tissue in the tail of the dolphin acts much like an ideal spring! Let’s see, spring potential energy as the tail flips up and down and then transferring Us into K. As a result, the dolphin swims extremely efficiently at high speeds!! Incredible!!

Us = 1/2 kx2

This equation says the Us varies during an oscillation. At x = 0, Us = 0 J. When 0 < x < 0, the Us is going to be greater than zero. The Us is greatest at maximum extension, x = +A or maximum compression, x =-A. At both of these endpoints of oscillation where Us = 1/2 kA2.

Kinetic Energy in SHM!

The K of the oscillating block, K = 1/2 mv2 also varies during an oscillation. The block has a maximum K when passing through equilibrium!

At any other point, the block is moving more slowly than when it passes through equilibrium, so K < Kmax @ x = 0! The K has a minimum value, K = 0 @ A, when the block momentarily comes to rest at the endpoints, @ x = -A and x = A.

(i) The K for a block on an ideal spring is maximum when Us is minimus and K = ) when Us is maximum at x = -A and X = +A. If there are no non-conservative forces, the total energy, (ET), will remain constant and the energy will be transferred back and forth between its K and Us forms during an oscillation. This lands us right where we want to be! This concept allows us to find the speed or position of an object at any point along the oscillation.

Total Mechanical Energy of an Oscillating Object Spring System

ET = K + Us = 1/2 mv2 + 1/2 kx2 = 1/2 kA2

Owing to the fact that both k and A are constant for a specific oscillation, the total energy of an oscillating system is constant! This is another statement of the Conservation of Energy! The ET transforms between K and Us but total ME remains constant!

K = ET – Us = 1/2 kA2 – 1/2 kx2 = 1/2 mv2

Note that Us is greatest at the two endpoints and zero at equilibrium. K is zero at the two endpoints and greatest at equilibrium.

When x = 0, ET = 100% K and at x = -A or x = +A, ET = 100% Us.

Like we said before, this is extremely useful in our analysis of oscillations!

In vertical springs, you can “ignore gravity”. We do not need to add Fg with N2L or Ug for energy considerations, even if Fg is exerted. The component of Fg in the direction of displacement of the spring will change the equilibrium but will NOT impact the resulting harmonic motion of the object on the spring around this new equilibrium!

Pendulums!!

It turns that Miley Cyrus and Edgar Allen Poe have more in common than originally thought; they are both great American writers who wrote iconically about the movement of a pendulum! We’ve all been on the swings in our childhoods and these playground staples, like wrecking balls, are really great examples of the wonder and limitations of pendular motion. Wilma swinging on the playground is analogous to the motion of an object on the end of a string. When Wilma or any object is displaced from equilibrium, released, and allowed to move freely, the force of gravity (Fg) pulls it back toward equilibrium. It is here where the defining characteristic of pendular motion shines; the Fg is the restoring force.

As with object-spring systems, Inertia causes Wilma to overshoot past equilibrium, resulting in SHM and oscillations. A pendulum is defined as objects that oscillate back and forth sue to the restoring force of gravity.

Since pedal rotate around point, we can use rotational motion as a model. Instead of relating a restoring force to the acceleration it produces, we will describe pendular motion in terms of restoring torque that produces angular acceleration.

A simple pendulum refers to the entire mass of a system is concentrated at a single point. This is an idealized version of the bob at the end of a string where the string has a length l and zero mass and we can use the object model for whatever is tired to the end of the string, since gravity is outside of our system.

In describing pendular motion we are always referring to Small Amplitude Oscillations (SAOs), so the angle (theta) is small and will obey Hooke’s Law with the angular frequency given by w = sqrt(g/l). Think about when Wilma flies a little too high on the swing and she free falls for a slight amount of time before the swing becomes taut again and resumes pendular motion This bit of free fall disobeys Hooke’s Law!!

As in springs, the period, T, is equal to 2(pi)/w and the frequency, f, is equal to 1/T or w/2(pi). Therefore we have a collection of relationships that mimic an object-spring system with g analogous to k and l analogous to m. That is, in our system or environment, we are given k and g and they are characteristics that cannot change for the system. Jeanine, our scientist, can change the length of the string or the mass oof the object on the spring!

Pendular Formulas in SHM

w = sqrt(g/l)

Period, T = 2(pi)*sqrt(l/g)

frequency, f = 1/2(pi)*sqrt(g/l)

These equations do NOT show a dependence on the mass of the bob. This is because the restoring force is provided by the Fg. Doubling the mass, doubles the rotational inertia which would slow down the oscillation (rotation) but since the restoring force is mg, we double the restoring force. Also, with SAOs, the amplitude of the oscillations does not make a difference.

Conceptual Problems

(i) Barbara constructs a simple pendulum by hanging a .06 kg marble from a light thread of negligible mass. With the marble attached, the thread if .4 m long. (a) What is the period when the marble is pulled into SAO? (b) We replace the pendulum with a .260 marble and increase the string length to .5 m long, what is the new period?

(a) T = 2(pi)*sqrt(l/g) = 2(pi)*sqrt(.4)/(10) = 1.26 s

(b) T = 2(pi)*sqrt(.5)/(10) = 1.40 s

(ii) A grandfather clock, which keeps time on Earth by a simple pendulum, is taken to the moon, where acceleration due to gravity is 1/6 of what it is on Earth. (a) If the clock is operated on the moon in the same manner, will the clock run fast or slow? (b) Calculate the time that passes on Earth while the hands on the clock on the moon move through 12 hours.

(a) As seen in the formula, T and g are inversely proprtional. The period on the moon will be longer than the period on Earth, So each of the swing of the pendulum records 1 second on Earth and more than 1 second on the moon. The clock will run slower, recording less time per second.

(b) Proportionate Reasoning!! T(moon) = 2(pi)*sqrt(l/g) = 1(1)*sqrt((1)/(1/6)) = 1*sqrt(6) = 2.44 T(Earth)

T(moon) = 2.44 T(Earth) = 2.44(12) = 29.3 hrs

(iii)If the period of a simple pendulum is T and we increase its length so that it’s 4 times longer, what will the new period be?

T(new) = 2(pi)*sqrt(l/g) = 1*sqrt((4)/(1)) = 2 T(old)

(iv) A simple pendulum on the surface of the earth is 1.24 m long. What is the angular frequency of the oscillation?

T = 2(pi)*sqrt(l/g) = 2(pi)*((1.24)/(10)) = 2.2 s

(v) Rearrange T = 2(pi)*sqrt(l/g) to solve for (g).

T2 = 4(pi)2 *((l)/(g)) —> g = 4(pi)2*(l)/T2

Momentum!!

When two objects interact with each other, they exert forces of equal magnitude and opposite direction on each other. This idea reconsiders N3L much the same way Kinetic Energy and the Work Energy Theorem got us thinking about N2L! An important way to express this concept is in terms of the momentum, p, of an object which is the product of the object’s mass times its velocity!

Since velocity is a vector, so is momentum. We can describe the behavior of two interacting objects or systems, like a baseball and a bat or a squid ejecting water as a source of motility. Each object in a pair undergoes a change in velocity and, hence, a change in momentum and momentum changes are equal in magnitude and opposite and direction. This leads us down the glorious path towards another Great Conservation Law, The Law of Conservation of Momentum! This Great Law is essential in describing what happens when two objects or systems interact. Interestingly, this law happens to harbor the fundamental physical principle from which Newton’s Three Laws of Motion arise! Momentum can also be used in more general situations than those in which we are able to apply Newton’s Laws, including cases when mass changes instead of velocity!

To discuss forces on systems and the Work Energy Theorem, we had to understand the notion of the CoM. The object model is founded on whether every point in the system we are considering could be approximated to move the same displacement as the CoM. In the case of a system of objects, this special point moves as though all of the mass of the system were focused there. Now, we can see that systems behave as if all forces on the system are exerted at this special point and we will finally expand our idea of the CoM by expanding our understanding of momentum!

Let’s take the example of Paula jumping off her skateboard. Paula is standing atop a stationary board and then jumps to the left off the board and the board rolls off to the right. Obviously, the skateboard flies off at a much faster speed than Paula. But why? Our first introduction to the Impulse-Momentum Theorem gets us pointed in the right direction, pun intended!

F(delta)t = (delta)p

This theorem will apply to both the skateboard and Paula and N3L tells us that the forces will be equal in magnitude and opposite in direction. Also, the time of each interaction will clearly be the same. Ergo, we are able to set the two impulses equal to each other and, by definition, their changes in momentum become equal!

m(sb)v(sb) = – m(p)v(p)

(1) The minus sign tells us that the the skateboard and Paula are moving in opposite directions. To push the board to right, she must move to the left.

(2) If we look at the magnitudes on both sides of the equation, the speed of the skateboard is equal to Paula’s speed multiplied by the ratio of the masses of Paula and the skateboard, m(p)/m(s). Since Paula is more massive than the skateboard, the board moves off at a much faster pace.

(3) If Paula pushes off the board with greater force, both Paula and the skateboard will fly off at faster speeds. But this force doesn’t appear in any equation and therefore, indicating, the ratio of the speed’s will be the same without regard to how hard Paula pushes off the board. An object’s mass multiplied by its velocity is called the objects linear momentum.

p = mv

We use a lowercase p for momentum, as per the Latin, petere.

The units for momentum, p, are simply the units of mass times the units for velocity or kg(m/s) or N(s). Unfortunately, there is no special name for this unit, so we just have to roll with it. Now we can easily say that the formula for Paula and the Skateboard is as below.

p(paula) = -p(skateboard)

A block of mass M is moving such that its kinetic energy has a value of K1 and its momentum, p1. The speed of the block is increased such that its kinetic energy K2 = 3K1. In terms of p1, what is the new magnitude of the momentum of the block? Use your proportionate reasoning to find the ratio of the velocities! (p2 = sqrt(3)p1)

Pro Tip

The magnitude of momentum for an object is mass times the magnitude of the object’s velocity. At any instant, the magnitude of the object’s velocity is its speed. Since Kinetic Energy depends on speed, we can use this relationship to find the Kinetic Energy in terms of the magnitude of momentum!

K = 1/2 mv2 = 1/2 m(p/m)2 = p2/2m

Let’s jump back to the equations that say that the momentum object acquires during the push off the skateboard is equal to the Fnet multiplied by the time over which the force is exerted. N3L says that Paula and the skateboard exert forces of equal magnitude on each other and that when one is present, the other has to be present as well. Therefore, both forces are exerted for the same time and Paula and the skateboard have equal magnitudes of momentum, p.

The kinetic energy each object acquires is equal to the work done on that object during the push off and work is equal to force times displacement of the object in the direction of the force. So, we treat Paula and the skateboard as individual objects, the forces Paula and the skateboard exert on each other are external forces and can therefore do work. The skateboard moves faster than Paula during the push-off, so it travels a greater distance than Paula and so has more work done on it by the same amount of force! Therefore the skateboard ends up with more kinetic energy! The momenta of the two objects is the same so therefore the lighter of the two must have the greater speed. The kinetic energy of the object is 1/2 mv2 = 1/2 pv, so for the same mass, the object with the greater speed has the greater magnitude of momentum.

The total momentum of a system is always conserved and it is constant for closed and isolated systems. In the case of the skateboard, the total momentum is equal to the vector sum of the momentum of Paula and the skateboard. During the interaction of Paula pushing off the skateboard, nothing else was interacting with Paula or the skateboard with anywhere near a significant force. Fg and FN balance and Ff can be considered negligible in comparison, therefore the system of Paula and the skateboard can be considered closed and isolated. The total momentum of the system just before and just after the interaction is constant. Just like the total energy of the system is always conserved but will be constant under certain circumstances.

Forces

(1) Forces exerted by other members of the system are considered internal forces.

(2) Forces exerted by objects outside the system are considered external forces.

Can we rewrite N2L in terms of momentum? Of course we can!

F = ma —> a = (delta)v/(delta)t —> F = m(delta)v/(delta)t —> F = (delta)p/t —> F (delta)t = (delta)p

For two billiard balls, the internal forces are the ones that the ball exerts on another when they collide. External forces are the FN and Ff exerted by the table and Fg exerted by Earth and the force you exert on the cue. Internal forces can either pull the components of a system together or push them apart and can allow energy to convert from one form to another, while keeping the total energy constant. In order for two true objects to interact, one of them would already have to be moving, since objects only have kinetic energy! The internal forces that one object in our system exerts on another and the details of these forces is irrelevant. We can calculate the effects of a collision with knowing the details.

Impulse Momentum Theorem

sum F(ext) (delta)t = (delta)p

This is truly remarkable! Only the external forces exerted on a system can affect the system’s total momentum. The internal forces of one object on another allow momentum to be transferred between objects (the cue ball strikes the eight-ball at rest and sends the eight-ball flying) but they don’t affect the total momentum. This parallels our results that internal forces can allow energy to be converted or transferred between objects in a system but cannot change the total energy of the system. This is the most general expression of the conservation of momentum. Just like the Work Energy Theorem told us that any change in energy of the system must equal the transfer of energy into or out of the system, any change in momentum of a system must equal the transfer of momentum, F(ext)(delta)t. Just as F(ext)(delta)x is called Work, F(ext)(delta)t is called Impulse.

The Law of Conservation of Momentum for the special case of no external forces can also be used in all collisions and proves to be so remarkably useful!

pi = pf

Tennis anyone?

One morning, I was running on the treadmill whilst watching the Australian Open on the telly! I looked up to see Sasha Zverev clock a serve at 217 km/h. Now, we are all taught to follow through when swinging anything, including a tennis serve. Let’s analyze this phenomena in terms of the Impulse Momentum Theorem! (See Google Classroom)

If we treat two objects of systems A and B as a single system, the interaction between A and B is most easily described as equal and opposite internal forces. Recall external forces can do work, changing the total energy but the internal forces can only rearrange energy! The Impulse Momentum Theorem reckons that to change the momentum of an object, a certain amount of impulse is required. The left side of the equations, F (delta)t, is actually referred to as the “impulse” that is exerted on the system. This can be a result of either a large external force over a short time or a smaller external force over a longer time. The Impulse Momentum Theorem is very much a statement of conservation as any change in momentum of a system has to be due to a transfer of momentum into or out of the system by an external force. If momentum changes due to a collision, the net external force on an object is predominantly due to the other object with which it is colliding. Any other forces (friction) are weak in comparison.

In F (delta)t = (delta)p, the (delta)t is representative of the contact time or the amount of time that the colliding objects interact. If a car comes to stop suddenly, the momentum of its occupants changes from a large initial value to a final value of zero. So the momentum change is very large and, in turn, the Impulse (F (delta)t) is also large. If there is no air bag, the passengers have a hard collision with the dash or steering wheel and come to a sudden stop. In this case, (delta)t is very short and the force of the collision is again very large. Therefore, the structure of the car exerts a tremendous force on the passengers. On the other hand, if the car is equipped with air bags, the air bags compress much more than the dash or the steering wheel as the driver strikes it so the pair are in contact for a greater amount of time. The force of the collision, represented as the air bag on the driver, is greatly reduced and injury can be avoided!! By the same accord, when you jump down from an elevation, or ski a set of moguls, you bend your knees to “absorb” the force. Flexing your legs during the collision between your feet and the ground maximizes the contact time that it takes your CoM to come to rest!!

The Conservation of Momentum!!

If the net external forces on an object or system of objects is zero, the total momentum of the system is constant. That is, the total momentum of the system at the beginning of a time interval is the same as the total momentum of the given system at the end of the time interval. It is so important to remember that the “Impulse”, or left side of the Impulse-Momentum Theorem, is equal to the change in momentum, not the total momentum. So, F(delta)t = (delta)p shows that of the net external force is equal to zero, then the change in momentum is equal to zero. This is what is meant by momentum is “conserved”. Momentum is not constant (it changes) if there is a net external force on the system. Although it is not generally constant when a system is not closed and isolated, momentum is always conserved, i.e., you can always account for the changes.

ptotal(i) = ptotal(f)

m1v1(i) + m2v2(i) = m1v1(f) + m2v2(f)

Collisions

The law of conservation of momentum for closed, isolated systems states that, if the internal forces during an interaction are much greater in magnitude than external forces, the total momentum of the interacting objects has the same value just before and just after the interaction.

When hockey players collide on the ice, the internal forces are much greater than the friction forces the ice exerts on the players. Similarly, the force between a tennis ball and a tennis racket distort the balls configuration so the external forces of gravity and Sasha’s hand on the racket are feeble in comparison. The total momentum of a system of colliding objects just after the collision is the same as before. The definition of a collision, in physics, is any brief interaction where internal forces dominate, like jumping off a skateboard. Similarly, when cars collide, the net vertical forces are zero and we can ignore the frictional forces as they pale in comparison to the enormous internal forces.

In a collision, momentum is constant only during the collision! Once the collision is over, the large internal forces are no longer exerted. If cars collide, the total momentum just before is equal to the total momentum just after the collision. Once this instant is over, the friction of the racetrack and other external forces, cause the car to come to a stop, transferring all the momentum to the Earth. With the mass of the Earth being so great these effects are unobservable!

The law of conservation of momentum is a vector equation. If the momentum vectors are equal before and after the collision, their (x) and (y) components are also equal in magnitude!

The Format of Vintage Collision Problems!!

An object of mass ma = 9.0 kg slides to the right at 10.0 m/s on a surface of negligible friction. It collides with a second object that moves left at 8.0 m/s before the collision but after the collision moves to the right at 12.25 m/s. What the m2 if m1 is still moving right at 3.25 m/s after the collision? (m1 = 3.0 kg, also see hockey collision example)

A stationary object explodes into two pieces. Piece A has a mass mA and piece B has a mass mB = 2mA

(a) Immediately after the explosion, which piece has the larger magnitude of momentum?

(b) Which piece is moving faster?

(c) Which piece has more kinetic energy?

Answer: (a) We know the momenta will be the same because momentum is zero before the explosion. Therefore the momentum after the explosion must be equal and opposite. In (b) the object with the smaller mass must move faster. And, finally, in (c) the smaller object must also have the kinetic energy because speed is squared in the kinetic energy formula.

A 2.0 kg object is moving east at 4.0 m/s when it collides with a 6.0 kg object at rest. After the collision, the larger object moves east at 2.0 m/s. What is the final magnitude and direction of the smaller object after the collision, assuming external forces are negligible? (v = 2 m/s west)

Two ice skaters, Twyla and Ani, face each other while stationary and push against each other’s hands. Twyla’s mass is two times larger that Ani’s. Calculate their relative speeds after they lose contact assuming friction forces are negligible.

mvtotal(i) = mtvt + mava

0 = mtvt + mava

mtvt = mava

2vt = 1va

2vt = va

In this case, we are looking for speeds only but we obviously know that the directions will be opposite!

Inelastic Collisions Dissipate Mechanical Energy

Are both mechanical energy and momentum constant throughout collisions? Sometimes!

We learned that mechanical energy of a system is constant only if conservative forces (external) do work and there are only conservative interactions inside the system. An example may be a system made up of two small systems that behave like ideal springs. They collide and compress, converting kinetic energy to spring potential and then relax, converting spring potential back to kinetic. A collision of this kind, in which internal forces between colliding objects are conservative, is an elastic collision. In an elastic collision, both total mechanical energy AND total momentum are constant. During the interaction, all the kinetic energy not required to conserve momentum is converted to potential energy. After, all the potential energy is restored to kinetic. If the initial momentum of a system is not zero, then not all the kinetic energy can be stored because enough kinetic energy must remain to keep momentum constant.

Elastic collisions are happening all around us. When oxygen and nitrogen in the air collide, the collisions are always elastic. The molecules compress slightly and undergo a reversible change in configuration, storing energy as potential. As we know, in the observable universe, the collision between billiard balls on a pool table is very nearly elastic!

Something very different happens when autos collide. The bodies of the autos undergo irreversible changes in configuration. These nonconservative forces, like heat and sound, dissipate mechanical energy. In a collision in racing, this is by design! By converting mechanical energy to internal energy as it deforms, the structure of the race car prevents all that energy from being used to do potentially harmful work on the driver. A collision in which mechanical energy is not constant is called an inelastic collision. (Even simply pushing off the skateboard, you convert internal energy to kinetic.)

In inelastic collisions, during the interaction, all the initial kinetic energy not required to conserve momentum is converted to internal forms but not all is converted to potential energy. Some is directly converted into internal energy. After the interaction, any any energy converted to potential energy is restored to kinetic but this internal energy is considered dissipated.

If we know the masses of colliding objects and their velocities before and after the collision, its straight forward to determine whether the collision is elastic inelastic. If the kinetic energy has the same value before and after the collision, the collision is elastic. If there is less total kinetic energy after the collision, the collision is inelastic and is the more common result.

A 20.0 g bee bee is fired into a 1.50 kg wood block initially at rest. The speed of the bee bee is 200 m/s just before it strikes the wooden block.

(a) Find the speed fo the block – bee bee system just after the collsion.

(b) Calculate the amount of mechanical energy dissipated in the collision.

(a) mbvb = mwvw = (mb + mw)vf —> .020(200) + 0 = 1.52vf —> 4 = 1.52vf —> vf = 2.63 m/s

(b) K(before) = 1/2mv2 = 1/2(.020)(200)2 = 400 J; K(after) = 1/2mv2 = 1/2(1.52)(2.63)2 = 5.23 J

400 J -5.23 J = 395 J

Completely Inelastic Collisions

The type of collision where the most mechanical energy is dissipated is a completely inelastic collision in which two objects stick together after they collide. A collision between two cars is completely inelastic if the two cars lock together and do not separate. An odd example is the fertilized egg!

In a completely inelastic collision, none of the initial kinetic energy is stored as potential energy. The amount of kinetic energy needed to keep the momentum constant remains, but all the rest of the initial kinetic energy is converted to internal energy. When an object of mass ma and velocity va(i) undergoes a completely inelastic collision with a second object of mass mb and velocity vb(i), we can regard what remain after the collision as a single object of mass (ma + mb). Momentum conservation can then give us an equation for the velocity vf of the combined object!

mava(i) + mbvb(i) = (ma + mb)vf

Because all of the mass of the systems travels off together after a completely inelastic collision, the final velocity in such a collision is velocity of the CoM of the system!

A bit about Newton’s Cradle

The cascade of events, collisions, back and forth continue for quite some time!

(a) Explain why the system behaves in such a manner? For each collision, momentum is conserved and mechanical energy is constant. No energy is transferred out of the system and the rigid metal balls do not acquire internal energy.

(b) Consider Newton’s Cradle consisting of only two balls. One ball is raised and released. Justify the claim that the height after the collision of the ball initially at rest can be no greater than the height from which the first ball was released. No energy is being transferred into the system, so total mechanical energy cannot increase. At the beginning of the swing all the energy is Ug, so at the end, the swing cannot be greater!

(c) Given this conclusion, justify the claim that the velocity of the ball colliding with the ball initially at rest cannot be negative after the collision. If momentum after the collision of the raised ball were negative, then the momentum of the ball initially at rest would have to increase by this amount. This would violate the conservation of mechanical energy.

(d) Given the conclusion in (b) and (c), justify the claim that all of the momentum from the raised ball must be transferred to the ball initially at rest. The ball that is initially raised cannot have a velocity after the collision. Because momentum is conserved, the momentum of the system after the collision must therefore be the momentum of the ball that was initially at rest!

(e) Predict what happens when two balls are raised and released in a Newton’s Cradle with at least three balls. You can raise and release two balls simultaneously and both the momentum and mechanical energy of the system are constant. When two balls are initially raised, two balls must be raised finally, otherwise mechanical energy would not be conserved.

There are three special cases of Completely Inelastic Collisions. Let’s look at the general scenario where Object A moves toward Object B, initially at rest, and they stick together.

mava(i) + mbvb(i) = (ma + mb)vf

vf = mava(i)/(ma + mb)

(1) When object A moving has a much great mass than object B resting! Whenever one quantity in an expression is significantly larger than another, it’s a good approximation to ignore the smaller quantity. So we can replace the denominator in our equation with just ma.

vf = mava(i)/ma ~ va(i)

(2) The two objects have the same mass, ma = mb.

vf = va(i)/2

This states that when a moving object collides and sticks to a stationary object of the same mass, the combined system moves at half the initial speed.

(3) The moving object has much less mass that the object at rest. In this case, the mass of object A is much less than the mass of object B and the numerator is much less than the denominator. Object B hardly moves when struck by object A.

ma/(ma + mb) ~ 0

The Center of Mass

If a system is rotating, like a gymnast flipping through the air or a barrel rolling down a hill or something changing shape, the object model no longer holds. So the quantity “a” in a = F/m refers to the acceleration of a certain point, the Center of Mass (CoM) of the system. The CoM moves as though all of the object’s mass is squeezed into a tiny blob at that point and all the external forces are exerted on the blob.

The position of a system’s CoM is kind of an average that takes into account the masses and positions of all the objects that make up that system. In a weighted average, each value affects the results to a greater or lesser extent depending on where it appears in that set. In this case, weight doesn’t refer to Fg but refers to the amount that each value contributes to the result. If you hold a lunch tray with only one hand, you will place your hand under the CoM and that position will be closer to the heaviest food or drink!

xcm = (m1(x1))/mT + (m2(x2))/mT + … (mn(xn))/mT

The more massive an object, the greater the ratio mn/MT and the greater the importance of that single object’s position. On the exam, you will not be expected to calculate the CoM of a collection of masses! But you will be expected to understand the concept of CoM, and you should be able to locate the approximate position qualitatively!

Two students are sitting on a seesaw, one at each end, that is pivoted about a point at its center. Claire, on the left end, has a mass of 60 kg and Ben, on the right end, has a mass of 80 kg. If the seesaw has a length of 2.40 m, how far from Ben is the CoM of the system of the students. That is, where do you put the pivot so they balance? We will designate x = 0 where Claire sits.

xcm = (60(0))/140 + (80(2.4))/140 = 1.37 m from Claire or 1.03 m from Ben!

The CoM is the geometrical center of a symmetrical system.

Let’s justify the use of the idea that the CoM moves as though all of the mass of the system were concentrated there. The total momentum of a system equals the vector sum of the momentum of all the objects in the system and also equals the total mass of the system multiplied by the velocity of the CoM. In other words, the total momentum of a system of objects of total mass, MT, is the same as if all of the objects were squeezed into a single blob moving at the velocity of the CoM. This means that the total linear momentum of a football doesn’t depend on how fast the football is rotating but only on the velocity of the CoM.

(sum)F(ext) = mTacm

The CoM of a system of objects moves exactly as if all the mass were concentrated there and all of the external forces on the system were exerted there. Only external forces affect the motion of the CoM. Forces on various points of the system can affect how those parts move relative to each other but have no effect on the motion of the CoM. This is the case of the complicated movements of a ballet dancer where her CoM follows a simple arc!

Frank launches a canon shell over level ground at a target 200 m away. Frank is perfectly aimed to hit the target, and air resistance can be neglected. At the highest point of the shell’s trajectory, the shell explodes into two identical halves, both of which hit the ground at the same time. One half falls vertically downward from the point of explosion. Where does the other half land?

(a) on the target (b) 50 m short of the target (c) 50 m beyond the target (d) 100 beyond the target (e) 150 m beyond the target

(d) If the shell did not explode, the CoM would hit the target. The force that blows the shell apart is internal to the shell, so it does not affect the motion of the CoM. This means that the CoM must still hit the target. This explosion happens when the shell is exactly halfway along its trajectory, so the half that fall’s vertically lands 100 m short of the target (half the horizontal). Since the CoM of a system of two equal masses is halfway between the masses, the other mass must land 100 m on the other side of the target!!

Grinnell stands on one end of a long wooden plank with a length L that rests on an icy surface. The ice exerts a negligible friction force on the plank. The plank and Grinnell have the same mass, m. Grinnell’s mom stands to the side of the plank, marking the CoM.

(a) Describe the CoM of the Grinnell/Plank system relative to the end of the plank where Grinnell is standing. The CoM of the system is L/4 from Grinnell. It is so important to remember that the CoM of a uniform plank or rod is its geometrical center. Therefore the point halfway between the end, where Grinnell stands, and the geometrical center of the plank is 1/4 the distance of the plank from where Grinnell is standing. We could also use the formula we discovered but like we said, we only need to know the intuition regarding the CoM!

xcm = mg(xg)/mT + mp(xp)/mT —> the plank and Grinnell have the same mass, m —> xcm = 0 + m(1/2)L/2m = L/4

(b) Grinnell runs to the other end of the plank with a speed vo while Mom maintains her position on the ice. Predict how the CoM of the system moves while the child runs. The CoM doesn’t change because there are no external forces!

(c) Predict the location of the middle of the plank and the location of Grinnell relative to Mom when Grinnell has reached the other end of the plank. Relative to Mom, Grinnell has moved to a position that is L/2 on the other side from where Grinnell started, whereas the center of the plank has also moved L/2 in the opposite direction.

(d) From Grinnell’s POV, she has been displaced by a distance L. Explain why her displacement from Mom’s POV is NOT equal to L. From Grinnell’s POV, she has moved a distance L (in the coordinate frame of the plank, the displacement is L.) But in the coordinate frame fixed on Mom, she has only moved L/2 because the plank has moved a distance L/2 opposite to the direction of Grinnell’s motion!

A great example of the manipulation of the CoM is the Fosbury Flop that redefined the high jump and thereby the Olympics forever! Have a look at this great video!

Bowery, Kimchi and the CoM!!

Bowery and her dog, Kimchi, sit at opposite ends of a 3 m long boat on which there are negligible horizontal forces exerted on the flat bottom boat by the surface of the still pond. The boat is pointed toward the shore with Kimchi in the bow at the front of the boat and Bowery in the stern at the rear of the boat. The bow is 6 m from the shore. Bowery has a mass of 52 kg and Kimchi has a mass of 20 kg. The mass of the boat is 135 kg.

(a) Discuss the location of the CoM relative to the geometrical center of the boat. In the reference frame with the origin at the center of the boat, Kimchi is 1.5 m from the center (m = 20 kg) and Bowery is 1.5 m from the center of the boat in the opposite direction (m = 52 kg). So the CoM must be closer to Bowery than Kimchi. Therefore, the CoM is between the center of the boat and Bowery, but closer to the center of the boat!

(Quantitatively xcm = (mbxb + mkxk + mvxv)/mT = ((52(0)) + 20(3) + 135(1.5))/207 = 1.26 m from Bowery and therefore, the CoM is 0.24 m from the center of the boat.)

Kimchi suddenly walks rapidly rapidly towards Bowery!

(b) Justify the claim that, in the reference frame that moves with the boat, momentum is not constant. Initially, the total momentum is zero. In the reference frame fixed on the boat, the boat does not move and Bowery remains seated. However, Kimchi is moving in the reference frame of the boat and the total momentum becomes the momentum of Kimchi and is no longer zero!

(c) Justify the claim that in the reference frame fixed on the shore, momentum is constant. Momentum of a the Bowery-Kimchi-boat system is constant in the absence of a net external force. Since there are negligible forces in the plane of the surface of the water exerted on the boat by the water, the momentum of the Bowery-Kimchi-boat system must be constant. From this reference frame, as Kimchi moves backwards, Bowery moves forwards and the CoM does not move!

(d) Predict the direction that the boat moves as Kimchi is moving. Kimchi moves in a direction away from the shore. The CoM of the system must remain constant in the absence of an external force, so the boat and Bowery must move toward the shore. In the reference frame fixed on the shore, Kimchi moves in a direction away from the shore. The CoM of the system must remain constant in the absence of an external force, so the boat must move toward the shore, although the CoM of the system does not move.

Por lo General

In a system of masses, the ratio of the distances from the CoM is inversely proportional to the ratio of the masses.

Work and Energy!!

We live in a complex world! To describe and explain phenomena, it is often necessary to simplify real objects, systems and processes. These simplifications are models and they are tested by using them to predict how new phenomena may occur. In some cases, a simplified model may give results that are “good enough”, like predicting where the salt shaker will land after your cat knocks it off the table using kinematics. In other cases, we must invoke complexity to analyze Clayton Kershaw’s slider, which behaves in a very imaginative manner!!

Models represent large aspects of our course. The “object” model means that we can ignore anything going inside or any internal structure and we just treat it as a point moving through space. This is used extensively in our course to mean you are able to neglect internal structure, like a particle. In contrast, we use the “system” model when internal structure cannot be ignored. This distinction is important because it determines the manner in which we can solve the problem.

For example, according to the problem to be solved, or the process being described, it may be appropriate to describe an atom as a system, paying attention to the protons, electrons and neutrons inside. But other times, we don’t really care what’s happening inside the atom, or a school bus perhaps, it would be entirely logical to use the object model and ignore internal structure.

Consider a ball. Whilst a ball is simple, it does have internal structure. A ball can be squeezed or spun. If the question you’re trying to answer depends on the fact not all parts of the ball move the same distance when you do something to it, we must refer to it as a system. If, however, the properties we are trying to study do not depend on the internal structure of the ball, such as predicting how far a soft toss may go, we can simply use the object model!!

A system can be complex, like the solar system. If we need to know how the sun or the planets move within the solar system, then we must treat it as a system. However, even with something large and complex as our solar system, if we look at how the solar system moves throughout the galaxy, we could simply use the object model. Living things, like grizzly bears and humans or flowers, are some of the most complex systems around. In order be able to chase you in the woods, the grizzly needs to convert that salmon she ate from the river into energy. Similarly, humans use bagels to give them energy to dance in the mornings. These are very intense systems and processes, but sometimes we are only concerned with how far we jump. In this case, we get a good approximation by treating the jumper as an object once we know the initial velocity and launch angle of the Center of Mass!

So, when we ignore internal structure of a system, we model it as an object. Choosing to model something as an object or a system is fundamental to determining how to describe and analyze a physical situation.

In our class, we use the object model to imply that you can neglect the internal structure of the system. To determine if you can use the object model, we must make sure we don’t need to attribute anything to it that requires internal structure. We can use the object model as long as all points on the system move together in exactly the same way. In other words, we treat it as an object if you can completely describe the system by a single point in space and its motion with the motion of that point. When we cannot neglect internal structure, like if the system’s shape changes or it spins, we use the system model!!

An object model can be defined when it is not necessary to consider internal structure. When internal structure plays a role in behavior we must use the system model defined in terms of the composing objects.

In one or more sentences, justify your claim that an object model is or is not appropriate for the bold word in each of the following contexts: when the system model is needed, reflect on the roles played by internal structure!

(a) A description of the motion of Earth as it orbits the sun. (The object model. The motion is directed simply by the path of a single point.)

(b) An explanation for the motion of Earth as it orbits the Sun. (Since both the Earth and Sun were highlighted, we are asked to describe this motion in terms of the interaction between them, so we use the system model.)

(c) A description of the motion of Filomena on a bike coasting down the hill to the baseball field. (If Filomena’s interaction with the bike is important, we need the system description, but in many cases this interaction does not need to be taken into account. Such is the case presented, Filomena coasting indicates that she is not interacting with the bike so she and the bike can be taken as a single object.)

(d) The mass of the water in Fred the Goldfish’s bowl. (The property of mass requires no structure to understand so the object model can be used.)

(e) Water poured into Fred the goldfish’s bowl, assuming the shape of the bowl. (The explanation of the shape of the water requires the description of the interaction between the water and the bowl. We need the system model.)

(f) Janet the Ice Skater gliding in a straight line across the ice. (We can use the object model since Janet’s position can be taken as a moving point.

(g) Janet spinning on the ice. (All points on Janet are not moving in the same direction at the same speed during this motion, so Janet is a system!!)

In these cases, if you must include the interaction between objects, which affect the behavior of the system, or the internal structure of the system is demonstrated because not all points move in exactly the same way, we must use the system model. When the behavior we are interested in can be sufficiently described by the position of a single point in space, the object model is appropriate.

The distinction between an object and a system is foundational to an understanding of Energy!!

We’ve described the notion of an object. We’ve explained how to use the concept of force to describe an interaction between objects and systems. Now, let’s begin applying one of the most fundamental ideas in all of Physics: Conservation!!

The changes that occur as a result of interactions between objects and systems are constrained by conservation laws. Conservation is often poorly understood because of its connotation in everyday life. That is, if I want to conserve butter because the grocery store is closed and I’m making grilled cheeses, I’m going to use less butter. Or I’m not going to make popcorn so I can use it all for the cheeses. Or determining whether to make popcorn or cupcakes, depending on which requires less butter.

In physics, we however mean something much more profound than simply “using less” when talking about conservation. In physics, a conserved quantity is a quantity that can be transferred between objects or systems, or converted from one type to to another but is neither created nor destroyed. When quantities are neither created nor destroyed, the amount of that quantity does not change. This simple concept gives rise to some of the most powerful and fundamental laws in all of physics: The Conservation Laws!!

Conservation laws constrain the possible motions of the objects in a system. Or the outcome of the interaction or process. A conservation law is a statement that a measurable physical quantity of a system does not change as the system evolves over time. This physical quantity can be used to characterize a system.

We use the term Energy almost everyday, mostly when talking about how sleepy tired we are. In physics, Energy is defined by a scalar quantity used to measure the state or motion of an object or system. Energy is always conserved but not all energy is equally as useful. So when we say we want to conserve Energy for the big game, we really mean we want to to not waste the energy that is most useful to us!

So, since conservation also considers the transfer of a quantity, we need to define our system to know how to apply a conservation law.

We will define a closed, isolated system as one where no energy or matter is transferred to, or from, the system and there are no interactions between objects in the system and objects outside the system.

A force is also a way to describe the interaction between two objects, so another way to define an isolated system is a system for which no forces are exerted on objects inside the system by objects outside the system. We will use forces to transfer energy. The total amount of energy in a system cannot change and all interactions and processes in the system are constrained by this fact!

In an open system, energy can cross the boundary of the system in which case conservation no longer means the energy in the system is constant. It means that changes in energy in the system are equal to the transfer of energy into and out of the system by interactions with other systems or processes.

The Conservation of Energy

This is literally the most pervasive conservation law across all of Physics.

Energy is used in every living and breathing process (moving). Within these processes, there are different types of Energy including kinetic, potential and internal. The Law of Conservation of Energy states that energy can be converted from one type to another but never destroyed. Energy is a scalar quantity used to measure the state or motion of an object or system. Ok, so what in the hay does that even mean??

It’s easier to begin our discussion of Energy by considering one of the ways to “transfer” energy. To delve into how energy is transferred, it is only logical to first explore the concept of Work!! Again, work, like energy, has many meaning in everyday life.

In Physics, work is defined as the transfer of energy.

Specifically, work is defined as the transfer of Energy from one object or system to another through a mechanical process that happens when a force is exerted on an object or system along the direction of motion as the object or system moves!! An example is lifting a clean pan up to the shelf above the stove or when Harley pushes her bobsled across the snowfield. In each of these cases, the point of contact, where the force is exerted on the object, moves.

The definitions of Work and Energy reference each other. If work is the transfer of energy, we will see that, if an object or system has energy then that object or system has the ability to do work!! (It’s kinda like when Ben Gates decides to steal the Declaration of Independence in order to protect it!)

One type of energy is Kinetic Energy (K) which is the energy an object has due to its motion. An object we Kinetic Energy has the ability to do work, that is a moving ball has the ability to displace objects in its path. For anything for which we use the object model, the only type of energy that it can have is kinetic energy. This is because by the definition of an object, we cannot change its shape, or the way its internal components are moving, since an object has no internal structure. Conversely, Systems, since they have internal structure, can have internal and potential energy!

An open system can exchange both energy and matter with its surroundings. A closed system cannot exchange matter with its surroundings. An isolated system cannot exchange matter or energy with its surroundings. Categorize each of the following systems as (i) open (ii) closed or (iii) isolated and describe evidence from your own experience to support your categorization.

(a) 🌎 Earth 🌎! Open. The system is open because energy and matter are constantly entering our atmosphere. We see energy in the form of light from the Sun, stars and matter as shooting stars burning up in the atmosphere.

(b) You!! Open. You take in matter and energy with the pancakes 🥞 you ate for breakfast, emit energy through heating and you breathe in and out exchanging gassed with the environment.

(c) a Yeti cooler with the lid shut! Closed. The ice eventually melts so we know energy enters the system but until the icy 🥶 water is removed, it’s relatively closed!

(Q) Describe how you know that winds blowing across a field of tall grass have energy and that work is done by the wind 💨 on the grass. ((a) The wind has energy of motion (K). This gives the grass energy of motion (K). Work is defined by this transfer of energy!)

☎️ The work done by a constant force exerted on a moving object depends on the magnitude of the force and the distance the object moves through in the direction of the force! 🎙

Loretta pushes a crate of Kimball Brook whole milk up a ramp. The amount of work Loretta does depends not only on how hard she pushes on the crate, that is, the magnitude of the force she exerts but also on the distance over which she moves the milk. Or the displacement of the crate.

Similarly, if Max, in detention, has to push desks around the gym, he will be more exhausted if Principal Carol asks him to push the desks all the way across the gym rather than only to the first foul line. That is, if he had to exert the force over a longer distance. On our mini pool table, K gets transferred to one pool ball from another as two balls collide without friction or deformation. The K of the desk does not continue to increase as Max pushes because of the large resistance to motion provided by friction between the floor and the desk.

These examples suggest how we should define the work done on an object or system by a force exerted on an object or system. If a constant force on an object as it moves through a straight-line displacement, d, and the force, F, is in the same direction as the displacement, the work done by the force is equal to the product of the magnitude of the force and the magnitude of the displacement over which the point of contact where the force is exerted on the object moves!!

W = Fd cos(theta)

Theta is the angle between the force and the displacement!

I am doing work on the snowball!!

This is force OVER a distance.

It is important to keep track of what object exerts a given force and on what object that force is exerted. It is equally important to keep track of both the object that exerts a force and the object on which the force does work. For example, when Loretta pushes the milk crate up the ramp, the object exerting the force is Loretta and the object on which the force is exerted and on which work is done, is the milk crate.

Just like a force must be exerted by something external to the object or system, work is done on an object or system by an external force. Work is one way in which we transfer energy!

Rule Of Thumb!!

Work is a scalar that can be positive or negative. (a) Positive work adds energy to a system and speeds things up and (b) negative work removes energy from a system and slows things down!!

Now, if the force used to pull your puppy in a wheelbarrow is not exerted in the direction of the objects motion, the force, F, that you exert on the wheelbarrow is angled with respect to the displacement, d, of the wheelbarrow. So, there is only one component of the force in the direction of the displacement. This is the horizontal component or F cos(theta)!

W = Fd cos(theta)

The work done by a constant force, F, angled at (theta) from the direction of the object’s displacement, d.

So, let’s 👀 at some special cases!

(a) if the angle between F and d is greater than 0 (zero) but less that 90, then the cos (theta) is less than 1 but still positive. Therefore W is less than Fd but still positive. (0 < (theta) < 90).

(b) if F is perpendicular to motion, cos(90) = 0. In this case, work is equal to zero! This is the work done by the puppy, W(puppy) = 0, since the force of the puppy is mg directed downward and motion is ——->. The angle between F and d is 90 and therefore the adorable does no work to ride along!!

(c) if the angle, theta, between F and d is greater than 90, the value of cos(theta) becomes negative. Therefore, work done by this force is negative, W < 0. This is when Diana tries to slow down a rolling cart of potatoes!! The force, F, that Diana exerts on the potatoes is directed opposite the cart’s displacement, d, so the angle is 180 and the cos(180) = -1. This means that Diana does negative work (-W)!!

Alas, remember the rule?? When force, and therefore acceleration, is directed opposite the object’s velocity and displacement, the object slows down!!

🙌 Therefore, negative work slows an object down and positive work speeds an object up!! 🏇🏿

This makes complete sense as Work is the transfer of energy, you make a positive transfer of energy, you increased the energy of the system receiving it. If you make a negative energy transfer, you remove energy from the system.

With d to the right, F(D on 🥔 ) <————> F(🥔 on D) = -F(D on 🥔)

(a) as Diana tries to make the cart of potatoes slow down, the cart and her hands move together to the right. They have the same displacement since her hands and the cart are in contact.

(b) the F of Diana on the cart is opposite to the cart’s displacement. Hence (theta) = 180, cos (theta) = -1, and Diana does negative work on the cart of potatoes.

(c) per N3L, the cart exerts an equally strong force on Diana but in the opposite direction, that is, in the same direction of the displacement of Diana’s hands. So the cart does positive work on Diana.

For objects that are in contact, if object A does negative work on object B, then object B does an equal amount of positive work on object A!

🎱 Think about pool balls!! 🎱

Always remember…negative work removes energy from a system and positive work adds energy to a system!

🏓 The Work-Energy Theorem 🏑

The Work-Energy Theorem is a general relationship between the total amount of work done and the change in the object’s speed. Remember that for something to be modeled as an object, all points on that object must move the same distance in any motion. So the point of contact of the force and the center of mass (CoM) have the same displacement.

An object can only have Kinetic Energy!!

K = 1/2 mv^2

Again, units are Nm or Joules.

The Work-Energy Theorem states that the Net Work done on an object is equal to the change in kinetic energy.

W(net) = K(f) – K(i)

W(net) = 1/2 mv(f)^2 – 1/2 mv(i)^2

When an object undergoes a displacement, the Work done on it by the Net Force equals the change in the object’s kinetic energy, that is, the object’s final kinetic energy at the end of displacement minus its initial kinetic energy at the beginning of displacement.

This is the very essence of the Work-Energy Theorem!!

It is valid as long as the object model is appropriate, such as when a system is rigid, that is, when it doesn’t deform. The Work-Energy Theorem is also valid if the object follows a curved path and the forces vary, not just straight line motion with constant forces. Remember it’s the New Work done which means the sun of all the work done by all the forces!!

What does the Work-Energy Theorem mean?

If Carl gives a cart a push along a frictionless surface, the Net Force on the cart equals the F that Carl exerts (because Fn and Fg cancel and there is not friction), so the work Carl does equals the W(net) by the Net Force. The cart starts at rest, so v(i) = 0 and the cart’s K(i) = 1/2 mv^2 = 0. After the push, the cart has a final speed v(f) = v and the K = 1/2 mv^2.

W(Carl on cart) = K(f) – K(i) = 1/2mv^2 – 0 = 1/2 mv^2

In words, the cart’s Kinetic Energy equals the Work done to accelerate it from rest to its present speed. Now imagine you have pushed the cart over to your friend Lyra to stock the potatoes and she brings it to a halt. The cart’s K(f) = 0 and K(i) = 1/2 mv^2. The Net Force is the force exerted by Lyra so the work she does equals the Net Work on the cart.

W(Lyra on cart) = K(f) – K(i) = 0 – 1/2 mv^2 = – 1/2 mv^2

Therefore, our second notion of Kinetic Energy is an object’s K equals the amount of work it can do in the process of coming to rest from its current speed. Remember, energy is the ability to do work!!

You know this, which is why you duck when Phil throws a nicely packed snowball at your face!!

Potential Energy

A system defined as a single object has no potential energy or internal forces, but external forces can be exerted on it. If a system has potential energy, it is due to internal interactions, so a system with potential energy must be made up of more than one object or be something for which the object models fails (it has internal structure or it can be deformed).

For anything for which the object model can be used cannot have potential energy. Potential Energy is always associated with two or more objects within a system that are interacting via conservative forces, such as gravity, or with the systems that do have internal structure, like springs! Practice saying things like, the ball-Earth system has potential energy! 🏀 – 🌍 system!

For a force to be conservative, (a) it must depend on a reversible change in configuration such as the change in length of a spring or a change in separation between the Earth and an object and (b) no energy must be dissipated to exert it. Potential energy is associated with reversible changes in a system’s configuration!!

Quantifying Potential Energy

An object near the Earth’s surface gains K when it is dropped. Let’s use an emphatic “mic drop” as an example. When the microphone is held out at shoulder’s length, the 🎤 has no K because it isn’t moving. When DJ Stir Fry drops the mic, it falls to the stage, gaining K as it does and possibly leaving a small indentation on the floor by exerting a large downward force on the floor over a small distance. If we treat the 🎤 as an object, then the Earth exerts an external force of gravity that does work on the mic as it falls to the stage, increasing the mic’s K. Since Fg is conservative, we can instead think of a potential energy associated with this change in location of the mic. Potential Energy has come from the configuration of the system. The shape of the mic doesn’t change but the distance between the mic and the Earth’s surface does change.

If we choose our system to be the mic and the Earth, we can no longer use work done by the force of gravity (Fg) because Earth is no longer outside of the system!! Therefore, Fg is no longer an external force! Instead we talk about Gravitational Potential Energy (Ug) of the system.

How we choose our system completely determines whether we must use Work or Potential Energy (Ug) to describe effects of conservative interactions.

There are two ways to determine the amount of potential energy stored in the Earth-mic system.

(1) Compare the potential energy due to a reversible change in the configuration of a system to the work that would be done if the object was isolated. If the system is just the 🎤 and it has a mass m and is initially a height h above the floor (we consider the floor to be h=0) as it falls, Earth (which is not in the same system, so exerts an external Fg on the mic in its direction of motion) would do an amount of work on the mic.

Wg = Fd cos(theta) = Fgh cos(theta) = mgh

From the Work- Energy Theorem, this is equal to the K that the dropped mic has gained just before it hits the stage. So we say that the gravitational potential energy (Ug) of the Earth-mic system before the ball was dropped was Ug = mgh and this potential energy was converted to potential energy as the mic fell. So the change in gravitational potential energy is the negative change in K, (change) Ug = -mgh.

This is why changes in energy are quite advantageous in problem solving!

(2) Consider when the initial potential energy came from the Earth-mic system. To see this, consider when the weightlifter raises the mic from the floor to height h. During the lift, the mic begins with 0 K and ends up with 0 K (at rest at shoulder level). The net change in K = 0 and Fg could do no work!

Since Fg in internal to the Earth-mic system, the positive work that DJ Stir Fry did to raise the mic must provide the potential energy, mgh, associated with the mic when it reaches height h (assuming the floor is h=0). The potential energy stored in the configuration of a system because of a gravitational interaction is called gravitational potential energy, mgh. DJ Stir Fry converted internal energy into the work done on the mic. Anytime energy comes from changing the configuration of a chemical of a chemical in food, some of the internal energy goes into warming up the DJ. In general, if an object of mass m is at a vertical coordinate y, the Ug of the Earth-mic system is mgy.

Ug = mg (change)y

When, The DJ raises the mic, Ug of the Earth-mic system increases!! When the DJ lowers the mic, (change)y decreases and the Ug decreases.

As the mic falls, Fg does work on it and this work goes into changing the object’s K. The total work done,Wg, by Fg along the curved path is the sum of the Wg for each segment along the path.

In words, the work done by gravity on an object if we choose our system to be the isolated object is equal to the negative of the change in gravitational potential energy of the Earth-object system if we choose our system to be the Earth and the object.

If an object descends, we can think of it as the downward Fg doing (+) work on the object or as the gravitational potential energy of the Earth-object system decreasing (change is (-)). If an object rises, we describe it as the downward Fg doing negative work on it or the gravitational potential energy of the Earth-object system increasing (change is (+)). If an object begins and ends its motion at the same height we can describe it as Fg doing no work on it or as there being zero net change in the Earth-object system’s gravitational potential energy (Ug)

Finally, remember, we must define our system, so we must only choose one of these interpretations.

(1) Either Earth is in the system and we use potential energy.

(2) Or Earth is out of the system and we use Work.

However the system is defined, we will get the same results!!

Spring Potential Energy!!

If we take a tennis ball 🎾 and chuck it at the wall, we see that the point of contact of the force, F, does not move as it is compressed against the wall but we do see that the Center of Mass, CoM, of the ball gets further away from the wall as it expands. Remember, work is determined by the displacement of the point of contact while the force doing work is being exerted. However, the Work-Energy Theorem describes the motion of the CoM for the object model to be valid. This means the displacement of the CoM while the force is being exerted describes the change in K of the system.

Now, given what we know, our first reaction could be that the wall has done work on the ball. After all, it has lost all of its K at the instant it is fully compressed against the wall and changes direction.

And it looks like an object, am I right??!!!

The wall does no work, however. Our definition of work done on an object states that the force must be exerted on the object as the point of contact of the force on the object moves. We see the point of contact of the force does not move so the force of the wall on the ball does no work!!

Have a think!!! I could lean against the wall all day and I’m not going to start moving!!

Therefore, we must consider the use of the the system model in the case of our tennis ball. For a system, the displacement of the CoM and the point of contact where the force is exerted can be different and, therefore, other types of energy other than K can change!!

So, the wall is exerting a Fn on the ball but the point of contact does not move so the total work done by the wall is zero. The wall transfers no energy to the ball. However, the CoM of the ball does change whilst the wall is exerting this force on the ball. So the ball’s K does change. Now let’s write the full Work Energy Theorem!!

W = (change) E = (change) K + (change) U + E (thermal)

(a) (change) K of the system is equal to the force x displacement of the CoM.

(b) (change) U due to reversible changes in its configuration.

These ideas about displacement of the point of contact relative to the CoM direct us into the object or system model which then determine whether or not we use the Work Energy Theorem or the Conservation of Energy!! This is also relevant to rigid systems where the PoC and the CoM are in different spots but move relative to one another, that is, in the same direction and the same distance.

Work Done by Varying Force

Often times, a force of variable magnitude does work on an object or system. Par example, you must do work to stretch a spring. That force you exert on a spring to stretch it is NOT constant. The further you stretch a spring, the greater the magnitude F you must exert.

Circular Motion

Mario “holding” on to the turn!! Notice the “banking” angle of the track!

The Space Shuttle, including Astronaut Sandra Magnus and all its inhabitants (animate and inanimate) follows a nearly circular path as it orbits Earth. The shuttle is constantly changing direction and therefore she is always accelerating. This acceleration is proportional to the square of the velocity and inversely proportional to the distance from the center of the earth.

Newton’s 2nd Law (N2L) tells Sandra and her crew that for an object to be accelerating, there must be a Net Force exerted on it. What could be this force?? Well, it is our good friend the force of gravity (Fg) and this force vector points directly toward the center of Earth. This certainly means that everything aboard the shuttle is constantly accelerating toward the center of the Earth because that is the direction of the net force!!! Everything, therefore, is in Free Fall!!

Now, since the famous astronaut ice cream is falling at the same rate as the astronauts, they stay in the same position relative to each other. It seems as if they are floating, or that they are in a state of weightlessness. Weightlessness is NOT a lack of gravity!! The Fg due to an object is proportional to the inverse square of the distance to the object’s centre. Because the shuttle and other “satellites” are considered to be “Low-Earth Orbiting” , the acceleration due to gravity is only about 10% less than that on the Earth’s surface. And normally, we will consider this distance negligible in the vastness of space.

🎤 Important🎤

If the shuttle is constantly feeling a tug toward the Earth, why doesn’t it come crashing down to the surface? Well, the shuttle is always falling but, because of its velocity tangential to the orbital path, it’s always missing the Earth!!

This was first postulated by one of Isaac Newton’s greatest thought experiments!!

He thought a cannon could fire a ball with enough velocity that it’s trajectory would match the curvature of Earth. It would therefore remain the same height above the Earth but fall around, not toward, Earth! We would essentially cross the boundaries of projectile motion and into circular motion!

Science fiction has become reality! We launched the Voyager Probe that is still telling us about how weird interstellar space is 43 years later!! We can trace the origins of space flight to Isaac Newton’s Law of Universal Gravitation; the idea that all objects in the universe attract each other through the force of gravity. This is the gravitational force exerted by an object with mass on any other object with mass, and so it goes!! These gravitational components are even responsible for holding Earth together!!

Now we all know, from skiers to show dogs, that in order to make a high speed, sharp turn we must have a larger acceleration than when making a high speed gentle turn. This is specifically evident it the “racing line” that expert drivers try to find when maximizing the radius of turns on a race track. This is because acceleration also represents the change in direction of the velocity vector. So, what is this relationship between an objects acceleration, it’s speed and the radius of the circular path?

🏎 Let’s Delve!! 🛵

An object traveling in a circular path at constant speed is experiencing Uniform Circular Motion (UCM). Because the object’s speed is constant, it’s acceleration has no component tangent to its trajectory!! This tells us that the entire acceleration vector must point directly to the center of curvature!! We call this centripetal acceleration taken from the Latin for “center-seeking”

Now, it is of immense importance in the AP world and beyond to understand that in order to move with a constant velocity in a circular path, we must accelerate!! Why? It’s because our velocity vector is always changing direction. This is in direct contrast to linear motion with constant velocity where acceleration is zero (0).

The Great Hornbill!!

Let’s consider the motion of the great hornbill in flight throughout one-quarter of a revolution. If she is initially moving North such that after turning 90 degrees, it’s moving at the same speed, only now to the East. It’s velocity vector originally had a zero (0) x-component but ends with an entire eastward velocity. The opposite occurs in the vertical direction. The loss of speed by one component is perfectly offset by a gain in speed in the other component so that the overall speed remains constant!! Finally, we have a formula to calculate the centripetal acceleration of an object following a circular path.

a(c) = v^2/r

This is read as, the magnitude of the centripetal acceleration is proportional to the square of the speed and inversely proportional to its radius. Therefore, there is greater a(c) the greater the hornbill’s speed and smaller the radius (the tighter the turn). This equation also applies to an object traveling at a constant speed around a portion of a circle. An object moving in non-uniform circular motion has a component along the direction of motion as well as perpendicular. No matter what, the hornbill and other objects moving in a circle have an acceleration directed toward the center of the circular path!!

Now is a good time to introduce the concepts of frequency, f, and period, T, of revolution. Frequency is a rate of any per second. It could be beats or flaps per second. The Period is the time is takes for one revolution. These two values are inversely related!! The unit for frequency is simply a placeholder and doesn’t appear in the Period.

Since velocity, as always, is distance divided by time, and the distance around a circle is its circumference. We have a formula for tangential velocity!!

v = 2(pi)r/T

Now, reconnecting with Sandra and the astronauts, the feeling of weightlessness means that any surface Sandra and the other astronauts touch has this same acceleration as Sandra so that it is impossible to exert a Normal Force on her.

This helps dispel the common misconception that an object in orbit is beyond the pull of the Earth’s gravity. But if gravity weren’t exerted by Earth on an orbiting satellite, it would have zero (0) acceleration and couldn’t stay in circular orbit. It would maintain constant velocity and fly off into space in a straight line. F(g) provides a(c)!! Sandra seems to be weightless because she is really a satellite of her own following the same orbit as the Shuttle and there is nothing pushing her toward any of the walls!

🏇🏿 Centripetal Force

…centripetal force, that is!!

Think of Jill on her Vespa making a sharp turn. She follows a curved path, so the direction of the velocity vector is always changing. Because the velocity is always changing, the Vespa is always accelerating, with a component of the acceleration vector directed toward the center of the circle or curve. We know that this acceleration depends on the radius of the curve and the Vespa’s velocity. And our good friend Isaac Newton tells us that an object accelerates because of a net external force exerted on it and this F(net) must point in the direction of the acceleration!!

What forces provide the acceleration needed to make an object follow a curved path?

1. The Force of Static Friction!! This could be between a track athlete’s shoes and the track or the tyres of a race car and the track. An example would be Jill on the Vespa.

2. The Force of Tension!! This could be the tension in a string swinging keys around a circle on a lanyard.

3. The Force of Gravity!!! This is the case of our orbiting satellites, moons and everything else in the universe!!

4. The Normal Force (or the Lift Force in case of an airplane)!! This could be a loop on a rollercoaster!!

Since the acceleration is always toward the center of the circle, N2L says that at any instant the F(net) exerted on the Vespa must also be directed toward the center of the circle. Furthermore, the magnitude of F(net) must equal mass times the centripetal acceleration!!

F(c) = ma(c)

This equation says that in UCM, the magnitude of the F(net) exerted on the Vespa is equal to the Vespa’s mass times centripetal acceleration. This leads to our equation for centripetal force.

F(c) = mv^2/r

Banking and Lift and Conical Motion!!

The Fn that we feel when we stand in the ground is analogous to the F(l), or the lift force that a plane feels whilst in flight! So let’s look at how an airplane makes a turn and in the process, discuss the concepts of banking and multiple components contributing to the Fc.

An airplane banks (dips its wings) to one side to turn in that direction. By banking the plane, the lift force, a force exerted perpendicular to the direction of flight due to the motion of air over the airplane’s wings, ends up with both a vertical and horizontal component. The horizontal component of this force provides the centripetal acceleration needed to make the plane move around in a circle.

(a) If an airplane of mass m is traveling at speed v and is banked by an angle (theta), derive an expression for the radius of the turn.

(b) Perry the Pilot has a mass of m(p) and a weight Fg = m(p)*g. If the pilot is sitting on a scale as t he plane makes a banked turn, what does the scale read?

After we draw our FBDs similar to our conical motion lab with the pigs or a car rounding a banked turn, we see that, besides Fg, a second force has a vertical component balancing Fg so the airplane doesn’t accelerate up and down and a horizontal component that provides the a(c). In our problem, want to find the radius of the turn and the Fn that the scale exerts on Perry. This is referred to as Perry’s apparent weight or how much she feels she weighs throughout the turn.

Obviously, we write N2L I component form. The (x) component involves the radius of the turn but we don’t know F(l). We find F(l) from the (y) component equation and then we substitute the value of F(l) into the (x) equation and solve for r. Obviously the airplane’s acceleration is directed toward the center of the turn and is therefore a(c).

(x): F(l) sin(theta) = mv^2/r

(y): F(l) cos(theta) = mg

Ultimately (see derivation), we end up with

r = v^2/g tan(theta)

The component form of N2L for Perry is very similar to that for the plane with the F(l) replaced by the normal force, F(n).

Fn = m(p)g/cos(theta)

Now let’s analyze our results! Part (a) says that the faster the plane’s speed for a given bank angle (theta), the larger the radius of the turn. To make a right turn at high speed (large v and small r), the quantity tan(theta) has to be as large as possible. So Perry needs a steep bank to make a tight turn. The greater the bank angle (theta), the greater the value of tan(theta). Again, notice the disappearance of mass from relevance. A bird and a jet observe the same physics no matter their difference in mass!!

Perry could pass out!!!

Our result in Part (b) shows the danger of too steep a bank angle. As (theta) increases, cos (theta) decreases, so the reading on the scale, Fn, becomes larger and larger. Perry’s apparent weight increases and she feels heavier than normal. The same is true for every part of Perry’s body, including her blood!! If the bank angle is too steep, her heart can’t pump this heavy blood to her brain. As a result, pilots will lose consciousness when their weight is about 4-5 times mg!!

Gravitation!!

Newton’s Law of Gravitation begins to explain the orbits of planets and satellites. The simplest type of orbits to analyze is a circular orbit an many 🌍 satellites, including the International Space Station (ISS) and GPS navigation satellites are in circular or nearly circular orbits.

Isaac Newton knew what was required to put a satellite in orbit three centuries before the first 🌎 satellite was set in orbit. It was his thought experiment that is a ball is thrown at just the right speed, the surface of the Earth would fall away below the ball so that the ball remains the same height above the surface. It is 🌎’s gravitation that causes the ball to accelerate toward 🌎’s center. If the speed is just right, the result is UCM as it keeps falling and missing 🌎!!

We can find the speed v required for circular orbit of radius r from a = v^2/r. For a satellite of mass m orbiting 🌎, the acceleration is provided by 🌎’s gravitational force from N2L!!

F(Earth on satellite) = ma(c)

F(g) = Gm(1)m(2)/r^2 = m(1)v^2/r

Solve for v

v = sqrt(Gm/r)

where G = 6.67 x 10^-11 Nm^2/kg^2

Many low Earth orbiting satellites (LEOS) have a height of a few hundred kilometers which is only a short distance compared to the Earth’s radius of 6370 km. So we can find the speed of LEOS by replacing r with Earth’s radius.

The result is v = 7.91 x 10^3 m/s

(17,700 mph or 28,500 kph)

Form this equation, we see that increasing orbits radius decreases the speed of the circular orbit!! The 🌚 orbits the 🌎 at 3.48 x 10^8 m, about 60 times the Earth’s radius and its orbital speed is 1020 m/s. You can now understand why speed decreases with increasing orbital radius by whirling a ball on the edge of a string and increasing and decreasing the radius. A satellite that orbits close to 🌍 experiences a substantial gravitational pull and moves at a higher speed while a planet with a larger radius of orbit experiences less gravitational force and moves at lower speed!!

Another way to describe how rapidly a satellite moves around its circular orbit is in terms of orbital period, T, which is the time required to complete one orbit. In UCM the speed v is constant so the orbital period, T, is the circumference 2(pi)r of the orbit divided by the speed.

T = 2(pi)r/v

This results in a formula for period by substituting orbital velocity.

T = 2(pi)r(sqrt(r/Gm(Earth))

Conveniently squaring both sides to remove the radical we have a proportional formula!!

T^2 = 4(pi)^2r^3/Gm(Earth)

This equation yields the period for LEOS as T = 5.06 x 10^3 s

84.4 minutes.

In words, the square of the period is directly proportional to the cube of the radius.

From v = sqrt(Gm/r) we also know that orbital speed is inversely proportional to the square root of the radius.

Consider a satellite that orbits at r = 4(r(Earth)) from 🌎’s center. Since r is “4^3” and equals 64 times greater. So, T^2 is also 4 times greater than for an object in LEOS which means that the T = sqrt(64) = 8 times larger!!

The orbital speed is proportional to 1/sqrt(r) which is 1/sqrt(4) = 1/2 as great for a satellite in LEO.

The proportionality tell us that for a satellite with r = 4r(Earth), the orbital period is 8 x 84.4 minutes = 675 minutes (11.2 hours) and the orbital speed is 7910/2 = 3950 m/s!!

Dynamics!!

What force causes the acceleration of Yoshi’s friends!?!

Newton’s Laws!!

When one object or system interacts externally with another, we describe that interaction as a force. When two objects touch, they push or pull on one another. A push or pull is a common way to describe a force. Forces allow us to predict the future motion of objects or systems!! A force is a vector, which means it has magnitude and direction.

Newton’s First Law (N1L)!!

The Law of Inertia refers to an object’s tendency to resist changes in its motion. That is, an object at rest will remain at rest unless acted upon by an external force. Also, an object in motion will tend to stay in motion with the same speed until acted upon by an external force. We quantify the inertia of an object by the object’s mass!!

Mind the differences between mass, weight and inertia. Mass is the quantity of material in an object; no matter where in the universe you take the object, it’s mass remains the same. Weight is the magnitude of the gravitational force (Fg) on an object. It is proportional to mass but depends on the the value of ‘g’ at the object’s location. Inertia is the tendency of all objects to maintain the same motion!

#dyk that N1L contradicts the older theories of motion developed by ancient Greeks like Aristotle?

Newton’s Second Law!!

F = ma

The most commonly used notion to describe the relationship between force and motion. In words; if a net external force is exerted on an object, the object’s accelerates in the direction of the force. The net external force is equal to the product of the object’s mass and acceleration.

Remember that acceleration is due to the sum of all the forces exerted on an object. It sometimes beneficial to think of N2L in terms of acceleration.

a = F/m

In words; the acceleration of an object is proportional to and in the same direction as the net external force exerted on the object and inversely proportional to the mass of the object.

So, a net external force on an object causes the object to accelerate in the same direction. If you double the net external force on an the object the acceleration also doubles. If you apply the same net force to an object with double the mass the acceleration is half as great.

The SI unit of force is the Newton (N). A net external force of 1 N applied to an object with a mass of 1 kg gives the object an acceleration of 1 m/s^2. A Newton is also equal to .225 lb, so about 1/4 pound.

Ultimately, N2L tells us that a net external force is required to cause an object to accelerate, that is, change it’s velocity. If we push the paper football so that the football’s velocity is constant, it simply means that the force exerted is balanced by the force of kinetic friction the table exerts on the football. What happens if you stop applying the force on the football?

a: the net external force now points backwards and therefore acceleration is now pointing backwards and is now opposite motion. The football slows down!!

We say that an object is in equilibrium if the net external force on an object is zero. A hanging Halloween spider decoration is on equilibrium just the same as Mario moving along at a constant velocity!!

If Mario drives at constant velocity, he is in equilibrium as the NET external force is equal to zero.

Newton’s Third Law (N3L)!!

This is the law of action/reaction pairs. Or, every action has an equal and opposite reaction. This relates the forces that two object’s exert on each other. You can feel the slap of the basketball against your hand as you push down on it!

So, if object A exerts a force on object B, object B exerts a force on object A that has the same magnitude but opposite direction.

You need the right recipe to make those perfect chocolate chip cookies. Now, The key that unlocks physics problems involving forces and the recipe that allows you to navigate these problems in the Free Body Diagram (FBD)!! The FBD is a graphical representation of all the external forces exerted ON an object. It’s useful because N2L and N1L both involve the sum of all the external forces on an object. The term “free body” mans that we draw only the object on which the forces are exerted and not the other objects that exert those forces!!

In the same sense, the Center of Mass (CoM) of a system moves as though all of the system’s mass were concentrated there. If every object in a system moves with the same acceleration as the CoM, then the entire system can be modeled as an object.

FBDs help connect the “real world” with “pen and paper” problems. They are an essential tool to solve most N2L problems. The FBD represents each external force on an object by a vector that originates on the object and points in the direction of the force. Remember to draw a simple representation of the object of interest. Here are the rules of drawing FBDs!!

1.) Sketch a simplified version of the object on which the forces are exerted. A dot, a small circle or a square is perfect. Something where you can easily place your arrow tails for your vectors. Remember we are representing the CoM so shape does not make a difference!!

2.) Identify what other objects exert a force on this object. This includes Earth, which exerts the force of gravity (Fg) and anything else that touches the object.

3.) For each force exerted on the object, draw a force vector with its tail at the center of the object. Don’t include forces the object may exert in other objects!

4.) Label each force properly with its symbol. Be sure to identify what other object exerts each force. Be sure it’s a real force.

5.) Choose, draw and label the directions of the positive (x) and (y) axes. Choose the axes so that as many force vectors as possible lie along one of the axes. That is, it might be advantageous to try your axes for objects on an incline!!

Look here for an simple example of the FBDs for the small block on a table! Remember to be able to draw the force vectors to scale to estimate the size of forces, particularly relative to one another. We also unambiguously label the vectors using a capital F and a subscript describing the force, that is, Fg.

We can also apply N2L to inclines!!

Ramps and Inclines

Like Josh Neuman cruising down the Swiss Alps we always manipulate the rules of nature and apply Newton’s Laws in practice. If we analyze any sort of motion down a hill the most important aspect is to, of course, draw an FBD.

When we draw an FBD for an object accelerating down a hill there are a few things to keep in mind.

(1) Use a straight edge to draw a diagonal line from the horizontal.

(2) Label the angle between the incline and the horizontal, theta.

(3) Draw the Normal Force perpendicular to the contact surface.

(4) Draw the Force of Gravity pointing straight down!!

(5) Tilt your coordinate plane so that the x-axis is parallel to the ramp and the y-axis is parallel to the Normal Force. This means that Fg is the only force not directed along one of the axes.

Motion!!

I want to go fast!!

We are constantly in motion and we are constantly surrounded by motion. Whether it’s speeding cars, scampering kittens and even the breeze! Our world is constantly moving even when we sitting down, we are spinning on Earth which is spinning around a sun in a solar system spinning around in a galaxy that is spinning around the universe!

The study of motion and all its varieties is crucial to the study of physics and is mostly considering under the umbrella of kinematics. In order to understand complex motion, we need to understand it’s simplest form which is straight line or linear motion. Let’s think about a rocket launching, blood flowing through a capillary or the trotting horse

How fast is Bowser going? How fast is he going relative to Mario!?!

Let’s talk about reference frames!!

Any measurement of position, distance or speed must be made with respect to a frame of reference.

Ron is traveling on a train which is traveling at 80 kph. Suppose Harry walks past Ron toward the front of the train at 5 kph. This value, 5 kph, is Harry’s speed with respect to the train’s frame of reference. However, with respect to the ground, Harry is moving at his speed with respect to the train plus the train’s speed relative to the ground. That is, to a Hermione, on the ground, Harry is moving, 80 kph + 5 kph = 85 kph.

It is always important to specify the frame of reference when stating a speed. Now, normally, the speed we talk about is relative to the Earth without thinking about it. In the above photo, if Bowser, going 35 kph, passes Mario going 30 kph, Bowser is actually going 5 kph relative to Mario.

When specifying the motion of an object, it is also important to specify the direction of this motion. Direction can be described as North, South, East, West, upwards or downwards to represent a frame of reference. The position of an object in linear motion at any moment is given by the x coordinate of the object. For freely falling objects, we use the y-axis.

It is here where we recognize the difference between displacement and distance. Displacement is defined by the change in position of an object. That is, the displacement is exactly how far an object is from its starting point. To see the distinction between total displacement and total distance, imagine Holly walls 70 m East to a coffee shop and then 30 m West to a park bench to enjoy her cup. The total distance is equal to 100 m. However, since she is only 40 m from her star ing position, her displacement is 70 m – 30 m = 40 m.

Displacement is a quantity with both magnitude and direction. This is called a vector.

The most obvious observation about the motion of a race car, a race horse or a 100 m track racer is how fast they are moving relative to the ground. This is speed and velocity.

Speed refers to how far an object travels in a given time interval, regardless of direction. This is referred to as a rate of change of position and is given in appropriate units for distance and time.

Any speed is defined as the total distance travelled along its path divided by the time it takes to travel that distance.

Speed = distance/time

If Ben travels 240 km in 3 hours, what is his speed?

The terms velocity and speed are sometimes used interchangeably in every day life. Yet, in Physics, we distinguish the two by simply saying speed is a positive number, or the magnitude of velocity. Velocity, on the other hand, signifies magnitude and direction, making velocity another vector! Velocity is, therefore defined in terms of displacement rather than total distance.

Velocity = displacement/time

Is average speed necessarily equal to average velocity??

Nope!!

If Parker Posey walks 40 m East across City Market then walks 30 m West to the Milk section in 70!seconds.

Speed = (40 + 30)/70 = 1.0 m/s

Velocity = (40 – 30)/70 = 0.14 m/s

Questions!!

The position of Wanita as a function of time is plotted moving along the x-axis of a coordinate system. During a 3 second time interval, her position changes from x(1) = 50 m to x(2) = 35 m. What is Wanita’s average velocity.

(x(2) – x(1))/t = (30.5 – 50)/3 = -19.5/3 = -6.5 m/s

The displacement and average velocity are negative indicating Wanita is moving to the left on the axis. Wanita’s average velocity is 6.5 m/s to the left.

How far can Carla travel in 2.5 hours on her bicycle if her average velocity is 18 km/h? We are going to use x = vt.

(a) 18 km/h x 2.5 h = 45 km = 45000 m

According to an Old Wives Tale, every 5 seconds between a lightning flash and the following thunder gives the distance to the flash in miles. Assuming the light arrives in essentially no time at all, estimate the speed of sound in m/s from this rule!

1.609 km/mile x 1 mile/ 5 s x 1000 m/ 1km

= 321.8 m/s, close to 343 m/s!!

Mario and Bowser are moving toward each other at different velocities!!

We have now begun to delve into the finer points of linear motion with constant velocity!!

To understand constant velocity, we must compare it to something more familiar, speed. Speed is a measure of how Janet or Felicia is moving. Velocity, however, is not another name for speed even though they have the same units. Velocity is the change in position, including direction, divided by the time it took to make that change.

With speed, the length is the distance traveled to the water fountain and back. In velocity, the length is how far and in what direction that point ended up from where it started, i.e. displacement. Therefore, your trip to the water fountain and back leads to a displacement of zero (0)!!

Check out these notes and diagrams that show Position as a function of Time for swimmers, Janet and Felicia!

Average velocity!! Phil runs 50 m in a straight line at 5.0 m/s. Phil then continues to move in the same direction, jogging at 2.0 m/s for an additional 50 m. He then turns around and walks at .80 m/s back to where he started.

(a) Calculate the time spent (1) running (2) jogging (3) walking.

(b) Calculate the average velocity for the (4) run and jog together (5) jog and walk together (6) run, jog and walk together.

How can you tell Mario is accelerating??

🚀 Acceleration!!! 🛬

When an object’s speed or direction of motion changes, that is, whenever it’s velocity changes, we say that an object accelerated or undergoes an acceleration!! Therefore we can say that acceleration is the rate of change in velocity. If Mario’s speed increases 2 m/s every second that means that his acceleration is 2 m/s^2.

So, in many important situations the speed, direction of motion, or both, can change as the object moves. If we look at Mario, game designer added a bit of fire leaving his exhaust which is a solid indication that he has accelerated. Also, we see that he is turning his steering wheel which means he is also accelerating by changing the direction of his motion!!

Acceleration doesn’t have to mean increasing speed!! In everyday language, “acceleration” is used to mean “speeding up” and “deceleration” is used to mean “slowing down”. In our space, however, acceleration refers to any change in velocity and so includes both speeding up and slowing down. Actually, speeding up is accelerating in the direction of motion and slowing down is accelerating in the opposite direction of motion. We will always use the term acceleration to describe any change in velocity!

Now, as you will see in these examples we have a rule and several equations that describe constant acceleration.

🚦Rule 🚦

When an object moving in a straight line speeds up, its velocity and acceleration have the same sign (both positive or both negative). When the same object slows down, it’s velocity and acceleration have opposite signs.

Free Fall

Mario in Free Fall!!

Perhaps the most important case of constant acceleration is the motion of falling objects near the surface of the Earth. Object’s in this category have a constant downward acceleration, if we ignore air resistance.

Free fall is an idealization, but many real life situations embody the concept quite well. Some examples are basketball players leaping for a jump ball, a high diver descending toward the water and a leaping frog in midair. In each case, the falling object experiences minimal air resistance because its speed is relatively slow with a small cross section and falls with a constant downward acceleration.

However a leaf falling toward the Earth or a hawk descending at high speeds with its wings folded, do reach a terminal velocity where the effects of air resistance can no longer be ignored. In this case, the upward air resistance balances the downward force of gravity and the hawk can no longer accelerate!!

Some simple rules!!

1.) An object in free fall has a constant downward acceleration, g, that is equal to 9.8 m/s^2.

2.) the magnitude of this acceleration is the same no matter the size of an object!

3.) This acceleration is the same no matter if the object is moving up, moving down or monetarily at rest!!

4.) For free fall problems, we choose to use the coordinate y instead of x. In this example, we also chose positive y to be upward and and the starting position to be y(o) = 0.

5.) when the object is dropped from rest and falls freely, it travels a greater distance in successive equal time intervals. It accelerates downward.

A pictorial and chart representation of distance and speed as an object falls!! Here we find that v = at and y = 1/2gt^2!!

Acceleration due to Gravity (g)

As we have discussed, the downward force of gravity causes objects that are moving upward or even monetarily at rest to accelerate downward.

If you toss a ball straight up in the air, as it ascends, it’s velocity is positive (upward) and it’s acceleration is negative (downward). Since the velocity and acceleration have opposite signs, the ball slows down. At maximum height, the ball is momentarily at rest so velocity is zero but it is still accelerating because it’s velocity is changing and it’s direction is changing.

If the acceleration at maximum height were equal to zero, the ball would reach this point and then stop in midair!! This does not happen and therefore acceleration is not zero!!

The acceleration due to gravity is always a positive number because it is the magnitude of the acceleration due to gravity. Acceleration is a vector and therefore it is the direction that determines the sign in you chosen coordinate plane. The value of the quantity of g is always positive.

Finally, we can apply the KINEMATICS EQUATIONS to free fall by changing our coordinate plane to the vertical direction, using -g as the acceleration and rewriting the kinematics equations. But first, always remember to Celebrate your Givens!!

Here are some problems that will are great examples of what will be on the test!! These problems build off the lab!!

AP!! Constant Acceleration and Free Fall: Motion Buddies!

How do you know the car is accelerating??

Acceleration is another one of those palabras that spends time in the realm of the everyday lexicon.

Go faster!! Speed up!! Slow down!! Turn left!!

Each of the above phrases involves an acceleration which essentially is a change in velocity. So we are moving from constant velocity to a change in speed or a change in direction.

This is very important!!

Acceleration is a vector and, just like velocity, it has a magnitude and a direction. So a change in direction is an acceleration the same way an increase or decrease in speed is an acceleration. Therefore, in our vehicles, there are three accelerators, (a) the gas pedal (b) the brake (c) the steering wheel.

Maggie is accelerating!

So the position 🆚 time graph for an accelerating object is not a straight line. Why?!?! It’s because the slope is not constant as velocity is changing! So let’s delve!! Here is a motion analysis for Jaylene on her scooter going down Pine St.

I love Rules!!

🚨So we have come up to a rule that will help you throughout the course.🚨

When an object, moving in a straight line, speeds up, its velocity and acceleration have the same sign (+,+ or -,-). When an object moving in a straight line slows down, its velocity and acceleration have opposite signs (+,- or -,+).

So let’s take it up a notch and analyze a world record frisbee catch by one amazing puppy!!

In taking some liberties, here is the golden analysis I came up with and it’s really incredible!!

THE KINEMATIC EQUATIONS

🚨🚨Super Important Notes🚨🚨

The link above ⬆️ will take you to everything you need to know about a special set of equations that apply to motion with constant acceleration, including some very important examples. Like we said, most acceleration in nature is constant.

FREE FALL!!

🚨 Super Important NOTES Part Deux 🚨

And perhaps the most important case of constant acceleration is the motion of objects near the surface of the earth. That is, free fall occurs when only the pull of gravity affects an object’s fall with a constant downward acceleration. Again, it is vital to understand all of the examples and physics speak! Ask questions!

This acceleration due to gravity, g, is the same whether the object is moving up, moving down or even momentarily at rest when it switches directions at its maximum height. Do you think mass affects this acceleration??

Many real life situations come close to this ideal like a high diver descending into the pool or a bullfrog in midair. With relatively low speed, there is minimal air resistance and definitive free fall.

If a falling object, at high speed, has a large cross section, we cannot ignore air resistance. In this case, the motion is no longer free fall because acceleration is not constant. In nature, a hawk plummeting at high speeds reaches its terminal velocity at which the effects of air resistance balances the downward pull of gravity, and therefore can no longer accelerate.

With objects in free fall, we can use the Kinematic Equations in the vertical direction with 9.8 m/s^2 as the constant acceleration due to gravity, or g. On many occasions, it is perfectly acceptable to round to 10 m/s^2 especially in multiple choice situations. So, in the vertical direction, if something is dropped, initial velocity and initial position are both zero, then vertical displacement equation becomes y = 1/2(g)t^2.

The Acceleration due to Gravity or “little g”

The value of g at sea level on earth differs from the acceleration due to gravity on other planets, spacecrafts and even on Mt. Everest!

This is the acceleration due to gravity that affects objects moving upwards, downwards and objects momentarily at rest!!

If we consider upward to be positive and Terry tosses a tennis ball straight up, it has a positive velocity and a negative acceleration. According to our rule, if acceleration and velocity have opposite signs, the ball slows down until the highest point when v = 0 m/s as it stops momentarily to change direction. Acceleration cannot be zero at this point or else the ball would just stop in midair!!! As the ball descends, velocity and acceleration are both negative and the ball speeds up as it falls!!

Projectile Motion

Projectile Motion is the amalgamation of constant horizontal velocity and free fall! In essence, the combination of the ideas that comprise the two sets of notes presented above!! The only force acting on a projectile is the downward pull of gravity!! Here is the Golf Lab and DERIVED EQUATIONS for maximum height of the ball flight and total horizontal displacement and these are universal. This is a great introduction to how we manipulate equations to develop relationships between variables that we are given!!

The water fountain and the leaping ballet dancer each follow a curved path so that their direction is constantly changing. Therefore, we can say that they are accelerating at all points in their motion! But this acceleration is only in the vertical direction so horizontal motion is constant.

For projectile motion, we will always place an axis along the direction of the acceleration due to gravity. Therefore, we can make the direction of the horizontal component of velocity the other axis. So projectile motion is two dimensional motion; motion in a plane. When the ballet dancer leaps, her velocity vector has a horizontal component to the right that remains constant. She also has a velocity vector that points upward which changes due to the acceleration of gravity pulling downwards. An important feature is that the dancer will slow down to zero at her maximum height as quickly as she will speed up on the way down from maximum height. Therefore, corresponding heights in her parabolic arc will have will have the exact same velocity, only a different direction!!!

Let’s play Get the Concept for the changing velocity of a projectile!!

Q: An object launched at some initial speed and angle follows a parabolic arc. At what point during the trajectory is the magnitude of its velocity the smallest? At what point, if any, will the velocity of the object be zero?

A: The magnitude of the velocity is smallest at the peak of its motion. The horizontal component of velocity is constant throughout the trajectory. At maximum height, the vertical (y) component of the velocity is zero (0), so regardless of the magnitude of the horizontal (x) component of velocity, the magnitude of the velocity vector must be the smallest. If the (x) component of velocity is zero, then the tota velocity, both magnitude and direction, is zero (0) at the peak. Because the (x) component of the velocity does not change, horizontal velocity can only be zero if the object was launched straight up!!

So, what do we know about projectile motion?

(1) At maximum height, vertical velocity v(y) is equal to zero.

(2) At maximum height, time (t) is equal to half the total flight time.

(3) Horizontal velocity, v(x), is constant because there is zero (0) horizontal acceleration.

(4) Vertical velocity decreases on the way up, increases on the way down at the same constant rate of 9.8 m/s^2. Sometimes we will round this value to 10 m/s^2 and refer to it as “g”. “g” is always a positive number but generally we call downward the “negative direction”. Therefore, acceleration in the vertical direction a(y) = -g.

(5) We resolve the initial velocity into its vector components and treat both the horizontal and vertical directions ((x) and (y)) as superstar individual problems that we know how to solve using the kinematics equations!!

(6) Time (t) is the same for both directions and cannot be negative and is the bridge between directions.

(7) Since “g” slows things down at the same rate as it speeds things up, vertical velocity (v(y)) is the same at mirroring vertical positions and times during projectile flight.

(8) The magnitude of total velocity is smallest at the peak of motion because the (y) component is zero. If the (x) component is zero, then total velocity is zero. This is a scenario that occurs when a ball is thrown straight up!

(9) We derived the universal derived equations that can be applied to projectile motion!!

FACTOR OF CHANGE!!

Two golf balls are hit from the same point on a flat field. Both are hit it an angle of 30 degrees above the horizontal. Ball 2 has twice the initial velocity of ball 1. Balls 1 and 2 land a displacement d(1) and d(2), respectively, from the initial point. Predict the relationship between the displacements d(1) and d(2), neglecting air resistance.

We will venturing trough these scenarios from the AP workbook in the near future!

Hello (Hola, Bonjour, Selam, Namaste) and welcome to Physics!!

The Simpsons!!

What can the Simpsons or any other cartoon teach us about Physics?? I remember my physics teacher, Mitch Johnson, famously saying, “Cartoons are funny because they defy the laws of Physics!” This statement had such a profound influence on me that I wrote an essay to the University of Oxford with it as the theme!

However, we can also learn a lot from proper physics in cartoons as well. The Simpsons are extremely thoughtfully created, living in the balance between physically incorrect scenes and the proper physics we see in the photo! But dare I say that the video has a more of a humorous quality?

Let’s delve into The Simpsons being launched over the sharks! Will the entire family clear the sharks? Why? How can Bart make it across because it looks like he should fall straight down? Well, Matt Groening is a creative genius so he has a few tricks up his sleeve! With the ramp and the water ski rope and handle in the drawing, we are meant to believe that the Marge, Bart, Lisa, Maggie and Homer all took off together with the same speed and were, indeed, all on Homer’s back, maybe?

So, after careful analysis, we can actually consider The Simpsons as projectiles. Projectile motion is defined as motion in two dimensions (2D or Planar) where the only external force on the system is gravity. In these problems we are never really concerned with how the projectile is launched (i.e. kicked, thrown, catapulted, sneezed or hit) as we are with some information as soon as the projectile is in the air, such as the magnitude and direction of the velocity vector.

Since gravity is the only force acting on the Homer and the gang, and gravity is always directed straight down, we know that we can consider motion in this direction to be straight up and straight down. Furthermore, since there are no forces acting in the horizontal direction, we know that the velocity must then remain constant, a concept that will become apparent to you as we get along in the class.

Therefore, by knowing the vector or the components of the vector of velocity, we can predict if and by how much they will avoid the sharks and how far past the sharks they will land!!

Let’s look at a real water ski jump!!!

So, what is Physics??

Some people will try to tell you that it is the study of matter and energy and the interaction between the two. Wait, I missed that because I dozed off!!

I prefer to think of Physics as the Konomi Code to the universe.

Way back in 1985, Kazuhisa Hashimoto was working on the arcade game Gradius. During testing, he developed a shortcut that allowed him to get to where he needed to without dying. The game he developed was too hard! If you entered the code, referred to as the Konomi Code, you would receive the full amount of power-ups usually attained throughout the game!

So if you entered the code, you would be afforded all the necessary power-ups. Most notably, you would receive 30 lives in the classic NES game, Contra!

So, the code has permeated pop culture throughout the years. It appears in various forms in Netflix, Google Hangouts, Wreck-it Ralph, The Incredibles, Archer, Fortnite and even the M&S Christmas website would drop down funny characters. For now, just say, “up up down down left right left right B A Start” to Siri and see what she says!!

So, I prefer to think of Physics as the Konomi Code to the universe. We have the cheat codes and hopefully, by the end of the class, you will see the world just a little bit differently!!

The same thing can be said for Michael Jordan’s foul line dunk! How long do you think he was in the air?

Barry Bee Benson

You may ask, “Why the heck is there a bee in the blog?” And the answer may surprise you but hopefully it dazzles you as well!

Remember we mentioned the term vector earlier when describing velocity? This means velocity has a magnitude AND direction. Speed can be considered the magnitude of velocity with any directional descriptor (up, down, left, right, east or west) serving to describe the direction. Well, it turns out that position and displacement are also vectors. So if I walk to the movie theater, I can that I traveled 33 meters north to the cinema.

It turns out that the same type of lingo can be used to describe the displacement of a bee during pollination of a bunch of flowers. If a bee zig zags across a small field sipping on nectar, he uses this concept of vectors to find his way home to the hive.

Does the bee have to revisit each of the flowers it pollinated to find his way home?

Why is the sky blue??

There is nothing more soothing than a bluebird day, especially in the fall where that high pressure lowers the humidity and gives us incredibly beautiful blue sky! But why? Why is the sky blue? Why are clouds white? Why are sunsets red? We have some beautiful sunsets over Lake Champlain that fill up our Instagram feeds with various hashtags. Light coming from the sun is made up of all the colors of the spectrum each of a different wavelength and appears to the human eye as white light. Ever stare at the sun, well don’t, but if you did, you would see pretty much white. Visible light appears white because it is a combination of all colors. In the visible spectrum, discovered by Isaac Newton, violet and blue are of short wavelengths and red and orange lace larger wavelengths. Remember ROY-G-BIV?

Now, the atmosphere is made up of all sorts of molecules, dust, smoke and all other things we pollute it with. But mostly nitrogen and oxygen. As sunlight passes through the atmosphere, it strikes these very small molecules in which shorter wavelengths are absorbed and then kicked out in a pattern called Rayleigh scattering. Largest wavelengths continue to pass through with out being scattered and, voila, blue sky!! You can think of them as optical tuning forks!

Manhattanhenge!!!

During sunset, the light from the sun has to travel much farther through the atmosphere to reach the earth due to the geometry of its position. This allows for plenty of time for the blue light to be scattered all over and the red light to pass on through to your eye because of your current angle to the sun!!

Pink sky at night Shepherd’s delight? Well, kinda? Weather travels from West to East. Really good sunsets occur with high pressure squash all the molecules down to the surface so more and more blue light is scattered so red really pops!! So a really good sunset in the western sky is going to travel eastward for the morning in the East. The high pressure will be here in Vermont and high pressure means beautiful sunny day!

Does it rain more often on the weekends or is that an Old Wives Tale? It’s true!! It’s 22% more likely to rain on Saturday than Monday on the East coast of the US. As humans, we produce more pollution during the work week that act as microscopic starters for rain droplets!! Therefore, Monday through Wednesday are bright and sunny and then it’s buckets on the weekend!! Check out this video!!

Here are a couple video resources that will help you in understanding!!!

Crash Course

Bozeman Science

Isaac Newton

On Christmas Day 1642, the year Galileo died, Isaac Newton was born in his mother’s farmhouse prematurely and barely survived. He was not particularly exceptional and dropped out of school to work on the family farm, which he hated as he preferred books given to him by his pharmacist neighbor. Isaac’s uncle sensed a spark and prompted him to study at Cambridge where he graduated with no particular distinction.

In 1665, a great plague swept through England forcing a 23 year old Isaac to continue his studies at home. It is here where he laid the foundations for physics extending from Earth to the Heavens. He developed Universal Gravitation, Calculus and his iconic Laws of Motion as well as explored the nature of light.

He is somewhat of a superhero and on his grave in Westminster Abbey, it reads, “Mortals rejoice that there has existed such and so great an ornament of the human race.”

In honor of the great thinker, I have developed a Newton of Fig Newtons lab!!!

Vuvuzelas!!

We will be doing the amazing Speed of Sound lab soon!

Welcome (Bienvenidos, Tena Yistilin, Bem Vindu, Failte) to AP Physics!!

The Simpsons!!

I remember so vividly when my high school physics teacher, Mitch Johnson, said, “Cartoons are funny because they defy the laws of Physics.” He said it so matter-of-factly that I only wrote an entire essay about it on my application to Oxford! The Simpsons, in all its genius, echo the notion with the couch gag depicted above. Why is it funny?

Now check out this amazing world record setting trampoline video!! The amount of our year contained in this video is astonishing!! While you’re here, this basketball dribbling video is just about everything!

52 wing flaps/sec!!!

I happen to love hummingbirds!! They have beautiful iridescent throats that dazzle the eye. They are acrobats with a mind-bending figure 8 wing flap allowing for lift on both the up and down strokes lending to their characteristic float. They are designed for quick elusive movements due their relatively small moment of inertia around the roll axis. They are very similar to the snitch in Harry Potter in this regard!! They are essentially Physics machines! So, in order to truly understand the hummingbird, or any other loving organism, or the world around us or even the physical universe, we need to understand Physics and it’s principles!

What is Physics??

If you look up the definition of Physics, you’ll probably find something to the likes of “the science of matter and energy and the interactions of the two”. I’m sorry, I missed that as I just fell asleep!

I prefer to think of it as the Konomi Code to the Universe!!

Back in 1985 Kazuhisa Hashimoto was working on the arcade game Gradius. During testing, he found the game incredibly challenging so he developed a shortcut that allowed him to easily get to where he needed to be without dying. The code, referred to as the Konomi Code, gives the player a full set of power-ups usually afforded throughout the game!

If you enter “up up down down left right left right B A Start” you will attain the necessary power-ups; most notably you will receive 30 lives in the classic NES game Contra.

Now, this code has permeated throughout pop culture with appearances in Wreck-it Ralph, Bank of Canada, Archer, Gravity Falls, Fortnite, The Incredibles, the 2016 Marks and Spencer Christmas site in England, Google Hangouts and various Conde Naste UK sites. However, you can just grab your iPhone and say the code to Siri!!!

In Physics, we manipulate the Konomi Code to the universe. We have the cheat codes. So, hopefully, by the end of the class you will you will look at the world just a bit differently. In particular, this video of Flanders and how it relates to the padding underneath the backboards in the gym!! Why is it so funny!?!

Let’s see what I mean in the next couple of examples!

Lewis Hamilton on three wheels!!

The 2020 British Grand Prix looked like it was going to be smooth sailing for the great Lewis Hamilton. Throughout a brilliant race, Hamilton had built a lead of over 30 seconds. Well, you no what they say, you can never count your chickens before they’re hatched! What happened next was truly one of the most exciting finishes in all of sport!! Check out the race recap narrated by the best there is, David Croft and Martin Brundle!

Incredible!!!

We want to know if/when and where Max Verstappen catches up to Lewis!! Let’s delve! And you can’t find out the results here!!

Well, that was super fun!! Now let’s analyze the trend that’s sweeping the nation…Maria Kart in a laundry basket on a treadmill into a pool!!

Linearization!!

Here is our first topic!! It involves a process of linearization which is an important and powerful tool for data analysis and the development of relationships!!

Here is the Jumping Jumper Elves lab to practice linearization!

I would absolutely recommend watching this video concerning graphs of position/velocity/acceleration as a function of time

Below you will find links to YouTube channels that are very useful to the course!!

Flipping Physics

Matt Anderson

AP Daily