Two bodies are stacked on one another with friction. Force applied on bottom body less than maximal friction between bodies. Why do the bodies move?

Question

I'm studying Newton's three laws.

Suppose we have two objects: A and B. A is on a smooth surface. B is stacked on top of A.

Between A and B there is friction. The static friction coefficient is some non-0 constant J.

A force F is acting horizontally on A, pulling it to the right.

In response, my understanding is that the static friction with B is going to push A to the opposite direction.

Suppose F is not strong enough to exceed the maximal value of the static friction. This means that |F| = |friction from B applied on A|.

Since both of these forces operate on A at the same size in opposite directions, it seems like A should stay stationary exactly where it is.

However, in actuality I think A will move to the right immediately, and B will stay stacked on top of it.

What am I missing? Why would A (and B on top of it) will move to the right, instead of staying in place?

Answer 1

I think where you go wrong is saying the friction force that A applies to B, call it $f_{AB}$, equals the applied force $F$ if $F$ is not strong enough for the maximum possible static friction force between A and B to be exceeded.

If the maximum static friction force is not exceeded the two blocks will accelerate as one. That acceleration will be, per Newton’s 2nd law,

$$a=\frac{F}{M_{A}+M_{B}}$$

Then since the only external horizontal force acting on B is the static friction force $f_{AB}$, we have from Newton’s 2nd law

$$f_{AB}=M_{B}a=\frac {M_{B}F}{M_{A}+M_{B}}$$

Hope this helps

Answer 2

The assumption $|F| = |f_{B \text{ applied on } A}|$ is incorrect. There is no reason a priori why this should be true. In fact, the only constraint on $B$ is $F_{\text{total}} = ma$, and in many cases (perhaps in many of the past problems that you've solved) we have $a = 0$, and therefore the force would be equal to the friction. However since there's no reason $B$ cannot move, $a = 0$ is not a valid assumption.

We split into two cases.

• $A, B$ does not slide. Then we need the acceleration of $A$ and $B$ to be equal. Thus the acceleration would have to be $\frac{F}{m_A + m_B}$ by analyzing $A+B$ as a single entity. And the only force able to provide $B$ with this acceleration is the friction. This happens when $f = \frac{Fm_B}{m_A + m_B}$ does not exceed the maximum friction, and $A,B$ moves together to the right. Now the friction on $A$ is also $\frac{|F| m_B}{m_A + m_B}$, in the opposite direction of $F$, and therefore the total force on $A$ is equal to $\frac{|F| m_A}{m_A + m_B}$ by subtraction, which indeed satisfies $F_{\text{total}} = m a$.

• $A, B$ slides. In this case the friction is equal to the maximum friction. You can then analyze how each object moves by calculating the forces, and thus the acceleration.