# Relative Acceleration

Ok so I have this problem which I wasn't really sure how to approach; when I looked at the results I understood that it needed newton's 3rd law to be solved and I sort of made it up, but I don't understand why it turns out to be correct(i only have the results, no explanation) and I feel like I've been misusing the law.

## Homework Statement

A platform of mass m2 = 4 kg is at rest over a frictionless horizontal plane; a body of mass m1 = 2 kg is at rest over the platform and there is no friction. This body starts moving, due to an internal motor, to the right with an acceleration a = 6.3 m/s2 with respect to the platform. Calculate the accelerations of m1 and m2 with respect to the plane.

## Homework Equations

Newton's 2nd and 3rd law;
relative acceleration: a = a' + at
where a is the acc in the inertial fixed frame, a' the acc in the moving frame, and at the acc of the moving frame with respect to the fixed one

## The Attempt at a Solution

I'll now call a1 the acceleration of m1 relative to the fixed frame and a2 the acceleration of m2 relative to the fixed frame
In the fixed frame there is a force F1 acting on m1 such that F1 = m1a1
from the relative acc. law: a1 = a + a2 ⇒ F1 = m1(a + a2)
due to newton's 3rd law the resultant of internal forces of the system must be 0, so there should be a force F2 acting on m2 which is opposite to F1: -F1 = F2 = m2a2
Now i get
-m2a2 = m1(a+a2) ⇒ -a2(m2+m1) = m1a ⇒ a2 = -m1a/(m1+m2) = -2.1 m/s^2 (negative so m2 is pushed to the left)
a1 = a + a2 = 4.2 m/s^2

So, what I don't get is, why does newton's 3rd law apply here? I went back to check how my theory book explains it and it talks about points exerting a force on each other, but here how is m1 exerting a force on m2 through his motion?
If I didn't look at the solutions I would have never thought about this way of solving it.
I am pretty sure the key is in the fact, already noted, that the force applied is internal to the system, but I still don't understand D: after all they are two separate bodies and there is no friction between them, so how is this working exactly? Is there some real-world situation that can help me visualize that or some other ways I can think about it?

## Answers and Replies

haruspex
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body of mass m1 = 2 kg is at rest over the platform and there is no friction.
Does it really state that there is no friction between the body and the platform? If so, one wonders how the body achieves movement... a propeller perhaps? And you are right that it would not result in any movement of the platform.
To double check, please post the exact wording of the original, even if not in English.

Some kind of propulsion was the samw thing I thought about.
Btw it does not explicitly say that there is no friction, but it doesn't even say that there is. I added that myself cause the book is quite clear on the fact that when friction is not mentioned one should not take it into account.
Now let's suppose there is friction, how does this change things? In this case there would surely be an interaction between the two bodies, but the fact that I got the correct answers without accounting for it means that the accelerations are independent of the friction coefficient, which sounds quite weird to me. Just a coincidence maybe?

kuruman
Science Advisor
Homework Helper
Gold Member
Some kind of propulsion was the samw thing I thought about.
Btw it does not explicitly say that there is no friction, but it doesn't even say that there is. I added that myself cause the book is quite clear on the fact that when friction is not mentioned one should not take it into account.
Now let's suppose there is friction, how does this change things? In this case there would surely be an interaction between the two bodies, but the fact that I got the correct answers without accounting for it means that the accelerations are independent of the friction coefficient, which sounds quite weird to me. Just a coincidence maybe?
If there is no friction and assuming some kind of propulsion like a propeller or a jet engine, the body will move relative to the ground but the platform will not because the platform is not attached to the platform in the horizontal direction. In this case, the platform's role is to exert a normal force to keep the body in place vertically.

If there is friction, there are equal and opposite horizontal forces that accelerate each mass horizontally. The fact that you got the right answer without needing the coefficient of friction is no coincidence, but as it should be. You can see this quite easily if you consider momentum conservation of the body+platform system. Because there are no horizontal external forces acting on the components of this system, the horizontal acceleration of the center of mass is zero, m1*a1+m2*a2 = 0. This will give you a relation between the accelerations relative to the ground which you can then relate to the given a = 6.3 m/s2 as you have already done.

• conscience and mastrofoffi
You can see this quite easily if you consider momentum conservation of the body+platform system. Because there are no horizontal external forces acting on the components of this system, the horizontal acceleration of the center of mass is zero, m1*a1+m2*a2 = 0. This will give you a relation between the accelerations relative to the ground which you can then relate to the given a = 6.3 m/s2 as you have already done.
Thank you very much, crystal clear.