The reason the Leaning Tower of Pisa has yet to just topple over is because its Center of Mass (CoM) is still over the base of the tower!! It’s a fascinating concept and one that can lead to several very impressive sculptures, buildings and designs.
The position of the Center of Mass (CoM) of a collection of objects depends on the masses and positions of each individual object. The greater the mass of an object, the greater its impact on the total mass and therefore the greater importance of the object’s position!
xcm = (m1/mt)*x1 + (m2/mt)*x2 …
This formula continues on down until you reach the total (n) number of objects. It basically describes position of the CoM as the sum of the ratios of an object’s mass to the total mass multiplied by the position of the object! You’ll find a couple of examples that we went over here!
If the system, that we are trying to find the CoM for, is in a plane defined by an x and y axis, we need to specify the specific coordinates of the position of the CoM.
There doesn’t have to be any mass at the CoM!! That is, there can be nothing physically at the CoM. The same can be said for any symmetrical object with a hole at its center, like a compact-disc or a donut or ping pong ball. No matter the case, the CoM is located in the center of the empty hole if the mass is distributed evenly.
Now, for motion of the CoM..,
The sum of F(ext) = m(total) x a(CoM)
This tells us that the CoM of a system of objects moves exactly as if the entire mass, m(total), of the system were concentrated at the CoM, and all of the external forces on the system were exerted on that concentrated mass. You can think of the CoM as a tiny blob and all of the external forces are exerted on that blob!!
An exploding cannon shell!!
A civil war reenactment fires a unit of a cannon shell over level ground at a target 200 m away. The cannon is perfectly aimed to hit the target. Air resistance can be neglected. At the highest point of the trajectory, the shell explodes into two identical halves both hitting the ground at the same time. If one falls vertically downward from the explosion. Where does the other half land??
Along these lines, let’s have a look at this scenario!!
(1) The CoM of an empty car of 1050 kg is 2.5 m behind the front of the car. How far from the front of the car will the CoM be when two people sit in the front seat 2.8 m from the front of the car and three people sit in the back seat 3.9 m from the front. Assume the mass of each person is 70 kg.
(2) The distance between a Carbon atom (mc = 12 u) and an Oxygen atom (mo = 16 u) molecule is 1.13 x 10^-10 m. How far from the Carbon atom is the CoM?
(3) A 3.0 kg block (A) is attached to a 1.0 kg block (B) by a spring of negligible mass that is compressed and locked in place. The blocks are sliding with negligible friction along the x-direction at initial constant speed of 2.0 m/s.
(a) At time t = 0, the position of block A and B are x = 1.0 m and x = 1.2 m, respectively. Describe the location of the CoM in the mass-spring system at time t = 0.
(b) At the t = 0 a mechanism released the spring and the blocks begging to oscillate as they slide. Predict the location of the CoM 2.0 s later.
(c) If at t = 2.0 s block B is located at 6.0 m, describe the location of block A.
(4) two lumps of clay moving in opposite directions collide with each other and move off as one with no external forces exerted on the system. The first lump has a mass of 0.40 kg and a velocity of 2.0 m/s and the other has a mass of 1.6 kg and a velocity of -1.0 m/s. What is the velocity of the CoM before the collision?
Also here is the test!! Due Friday, 4/17!
Thanks so much for your hard work!!
Film Clas 