# Acceleration of center of mass

## Homework Statement

Three identical rectangular blocks are at rest on a level, frictionless surface. Forces of equal magnitude that act in the same direction are exerted on each of the three blocks. Each force is exerted at a different point on the block (indicated by the symbol "x"), as shown in the top-view diagram attached (http://imgur.com/a/zhLKj). The location of each block's center of mass is indicated by a small circle.

a. For each of the blocks, draw an arrow on the diagram above to indicate the direction of the acceleration of the block's center of mass at the instant shown. If the magnitude of the acceleration of the center of mass of any block is zero, state that explicitly. Explain.

b. Rank the blocks according to magnitude of center-of-mass acceleration, from largest to smallest. If any two blocks have the same magnitude center-of-mass acceleration, state so explicitly. Support your ranking by drawing a point free-body diagram for each block.

## Homework Equations

Conceptual problem

## The Attempt at a Solution

I noticed that this question was posted before, but the answers for it were regarding the angular acceleration, not the center-of-mass acceleration.

So I was wondering, for part a, does the location of the force even matter? And if so, how? My thinking is that the magnitudes of the accelerations would be the same, except that for block 1 it would point slightly to the right, block 2 straight forward, and block 3 to the left. Is my reasoning right or wrong?

Thanks for taking the time to read my question and help me out!

haruspex
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the magnitudes of the accelerations would be the same
Well done.
except that for block 1 it would point slightly to the right, block 2 straight forward, and block 3 to the left
Why? What force is acting in the oblique direction?

Why? What force is acting in the oblique direction?

Well, now that I think about it, none... however by intuition it seems like exerting force on the side of the block causes it to turn in a circular fashion...

haruspex
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Well, now that I think about it, none... however by intuition it seems like exerting force on the side of the block causes it to turn in a circular fashion...
It will, but as long as the force applied does not change direction the acceleration won't.
The difficulty in this question is that intuition is misleading. In the real world, if you were to push on the flat surface of such an object in the A and C cases:
- as soon as it starts to rotate you will find that the force you are exerting is now at angle; so you need to think in terms of, say, a round stud protruding from the object, so that your push on the stud can maintain direction
- the pushing is easier than in case B because it yields; but the question specifies F as constant, so you have to be prepared to move your finger much faster in the A and C cases in order to experience the same resistance.

• Mikoto and Kevodaboss
Ah, I see, thanks! So both the magnitude and direction of the center-of-mass accelerations are all equal?

haruspex
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Ah, I see, thanks! So both the magnitude and direction of the center-of-mass accelerations are all equal?
Yes.

Thanks! So why doesn't torque play a role here? Or is it just that the torque doesn't affect the linear acceleration?

haruspex
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is it just that the torque doesn't affect the linear acceleration?
Torque about the mass centre (torque is generally only meaningful in respect of a specified axis) does not affect linear acceleration of the mass centre.

• Mikoto
Torque about the mass centre (torque is generally only meaningful in respect of a specified axis) does not affect linear acceleration of the mass centre.

Alright, that's what I expected. Thanks for the help!