Newton's third law conceptual worked example

Block A with mass m sits on top of block B with mass 2m in an elevator. The elevator is moving downward and slowing down. All right, when we have this diagram over here, it's moving downward and slowing down, so that means it's accelerating upwards. The magnitude of the Earth’s force of gravity on block A is F_sub_g_sub_a. The force magnitude exerted on block A by block B is F_sub_a, and the force magnitude exerted on block B by block A is F_sub_b.

How do the force magnitudes compare? This is fascinating! So pause this video and see if you can figure it out on your own.

All right, well, let's just draw a free body diagram for block A. So I have block A right over here. Now, there are a couple of things we know. We know that this whole system is moving downward and slowing down. We know that because it says the elevator is moving downward and slowing down, and block A sitting on block B is sitting in the elevator. If you're moving downward and slowing down, that means that your net acceleration is upwards, which tells you that your net force on block A needs to be upwards.

This tells you if we have some forces that are acting upwards, the magnitude of those are going to be larger than the forces acting downwards. So for block A, what are the forces that are acting downwards on it? Well, the force of gravity is acting on block A, and that's that right over there. Its magnitude is capital F_sub_g_sub_a.

And what's the upward force acting on block A? Well, they say the force magnitude exerted on block A by block B is F_sub_a, so the magnitude here is F_sub_a. We immediately know because we have a net acceleration upwards—remember, we're moving down but we're slowing down—so it's a net acceleration upwards. That immediately tells us that the magnitude of F_sub_a is going to be greater than the magnitude of the force of gravity on A.

So we can immediately get rid of some of these choices, but they also are comparing F_sub_b here. How can we involve F_sub_b in this inequality? Well, the key here is Newton's third law. Whatever force B is exerting on A, and we know we call that F_sub_a or its magnitude as F_sub_a, A is going to exert, from Newton's third law, an equal and opposite force on B.

So this is going to be magnitude; the arrows point downward, but it's going to have the same magnitude F_sub_b. Thus, the magnitude of these two forces are the same. Another way to think about it is F_sub_a is equal to F_sub_b by Newton's third law. Since we know that the whole system is accelerating upwards—it's moving downwards but it's slowing down—we know that this thing is going to be greater than the force of gravity on A.

We see that that is choice B and we’re done.