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
Film Clas 