# Finding the Center of mass

1. May 2, 2014

### antigen123

1. The problem statement, all variables and given/known data
Image 1 Shows both system 1 and system 2 at rest
Images 2 and 3 show answer choices:
Given M2 > M1, which of the following pendulum systems has the GREATEST change in potential energy from a state of rest?

Passage says: The greatest Potential energy is always associated with the system whose center of mass is is displaced from its resting state.

2. Relevant equations

3. The attempt at a solution
I got the correct answer (D)by process of elimination. however I am still having trouble locating the center of mass just by observing an object(where would the center of mass be on this object?). I feel this is because of a lack of a solid conceptual understanding of what it is. Is the center of mass a point where net torque is zero and net force is zero??

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2. May 2, 2014

### FaroukYasser

Hi,
first of all, to clear you confusion :) . The center of mass is a single point in any object where the WHOLE weight force of an object is considered to act on. what you said about net torque and net force being zero is called the state of equilibrium which is a different thing.
Now for the question,
Let us assume that the mass of the lighter object is (m) and the heavier (2m) alright? We know that the change in PE = mgh where m is the mass of the object (will be 2m for the heavier object) and g is the gravitational acceleration and h is the increase in height. Now that we have that set up let us proceed, For Part (a), The heavy ball is at the top, it is then moved so it makes a 45 degrees with the horizontal, if you imagine this, you will know that the object's height has increased a little. Attached is what happens, basically:
Sin(45) = New height / L >>> New Height = LSin(45) >>> New height = 0.7L
This means tat the object was before at L height but now its height is 0.7L therefore it has risen 0.3L upwards. But not only the heavy ball rose a 0.3L, since the small ball is connected to it, it rose 0.3L too so the total change in potential energy is:
[(2m) x (g) x (0.3L)] + [(m) x (g) x (0.3L)] = 0.6mgL + 0.3mhL = 0.9mgl
Now try the rest yourself
Hint, When two of them have changed, The lower one will both increase in height because the one above it increased in height AND because it now makes an angle of 45 degrees so the lower one would have an increase of 0.3L + 0.3L = 0.6L

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3. May 2, 2014

### haruspex

It's the 'average position' of the mass. If you have two objects of masses M and m, and the distance between their centres of mass is x, where is the common centre of mass? It must be on aline joining the two mass centres. Suppose it is distance y from M. To get the average mass position we have to add up the mass*distance values and divide by the total mass:
y = (0*M + x*m)/(M+m) = mx/(M+m). Similarly, the distance from the mass centre of m is x - y = Mx/(M+m).