# Acceleration of center of mass

• Kevodaboss
In summary: Alright, that's what I expected. Thanks for the help!In summary, the three blocks are at rest on a level surface and are subjected to forces of equal magnitude acting in the same direction. The location of the force does not affect the magnitude of the center-of-mass acceleration. The blocks are ranked according to magnitude of center-of-mass acceleration, from largest to smallest.
Kevodaboss

## 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!

Kevodaboss said:
the magnitudes of the accelerations would be the same
Well done.
Kevodaboss said:
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?

haruspex said:
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...

Kevodaboss said:
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?

Kevodaboss said:
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?

Kevodaboss said:
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
haruspex said:
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!

## What is acceleration of center of mass?

Acceleration of center of mass is the rate of change of the velocity of the center of mass of a system. It is a measure of how quickly the center of mass is moving and in what direction.

## How is acceleration of center of mass calculated?

The acceleration of center of mass can be calculated using the formula a = F/m, where a is the acceleration, F is the total force acting on the system, and m is the total mass of the system.

## What factors affect the acceleration of center of mass?

The acceleration of center of mass is affected by the total mass of the system and the magnitude and direction of the net force acting on the system.

## Why is the acceleration of center of mass important?

The acceleration of center of mass is important because it allows us to understand the overall motion of a system. It can also help us analyze the forces acting on a system and predict its future motion.

## What is the relationship between acceleration of center of mass and Newton's Second Law?

According to Newton's Second Law, the acceleration of center of mass is directly proportional to the net force acting on a system and inversely proportional to the mass of the system. This means that a system with a greater mass will require a greater force to achieve the same acceleration as a system with a smaller mass.

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