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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.

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

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2 Answers 2

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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

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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.

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