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I'm trying to understand the two box problem, where a small box is stacked on top of a larger box. The larger box then experiences a push, causing both boxes to accelerate in the direction opposite to the applied force.

Now my understanding of the FBD of the smaller box is that there's Fg, Fn, which cancel out, and a force of static friction that supposedly causes the acceleration.

I can't seem to wrap my head around why friction would cause acceleration. Can anyone help me out? Thanks in advance!

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  • $\begingroup$ Mathematically, I understand that if Fa is right for example, Fk (with floor) and Fs (with small box) of the large box go left. By N3LM, Fs experienced by the small box should go right. But this isn't very intuitive to me $\endgroup$
    – Sam Liu
    Dec 5, 2020 at 21:06

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Remember that friction opposes the relative motion between two bodies. When a force is applied on the bottom box, it wants to slide to the right. However, the top box (box B) is stationary, and will 'want to' oppose sliding to the right. Therefore, box B will exert a force on box A to oppose the sliding. It will thus exert a static friction force $\vec f_{\text{B on A}}$ leftwards on box A. Due to Newton's third law, box A will exert an equal and opposite force on box B, denoted by $\vec f_{\text{A on B}}$.

It is the static friction $\vec f_{\text{A on B}}$ that will cause box B to accelerate rightwards.

enter image description here

(I neglected the vertical forces so the diagram is clearer)

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  • $\begingroup$ @user256873 Thanks for the response! That does make sense. I have learned though that friction can never cause acceleration so I was a bit confused about that part $\endgroup$
    – Sam Liu
    Dec 5, 2020 at 21:23
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    $\begingroup$ @Sam Liu Things you're told in High School that "never happen" just might :) Students are sometimes mislead to simplify the topics. Also, if this fully answers your question, please consider 'accepting the answer.' $\endgroup$
    – user256872
    Dec 5, 2020 at 21:24
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a force of static friction that supposedly causes the acceleration. I can't seem to wrap my head around why friction would cause acceleration.

It is interesting that you have been mistakenly taught that friction does not cause acceleration. Friction is a force and forces cause acceleration, so friction causes acceleration. That acceleration may be in any direction relative to the velocity.

Contrary to what you have in mind friction is essential for motion in daily life. When you walk or run you accelerate due to friction. When you drive a car or ride a train you accelerate due to friction. When you stand on one of those conveyor belt walkways in an airport you accelerate due to friction. Imagine all of those activities on a slippery surface.

Friction opposes slipping, not all motion is slipping.

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  • $\begingroup$ How exactly does friction allow for acceleration when walking or running if you don't mind explaining? From what I understand, when we push backwards on the ground, doesn't the ground push back on the walker, causing them to move forwards? How does friction come into play here? Is that reaction force actually friction? Thanks a lot $\endgroup$
    – Sam Liu
    Dec 6, 2020 at 14:53
  • $\begingroup$ @Sam Liu yes. The reaction force is actually friction. Think about how difficult it is to walk on slippery surfaces. $\endgroup$
    – Dale
    Dec 6, 2020 at 14:56
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    $\begingroup$ Hmm ok thanks. It really isn't very intuitive that the force that pushes you forward is friction. Thanks for your help. $\endgroup$
    – Sam Liu
    Dec 6, 2020 at 16:12
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Unbalanced forces always cause accelerations (although we may not think about them that way when the speed is decreasing). Here's a couple of scenarios to think about.

Take a conveyor belt which is super slick. There's basically zero friction for the objects on it. You allow a book to drop onto the belt from rest. The belt moves underneath, but the object remains stationary.

Now the belt is changed for a normal belt with lots of friction. As the belt is moving, the book from before is dropped onto it. It hits the belt with zero horizontal speed. After a second, the belt has whisked it away. Where did the acceleration come from?

As the underneath block in your problem is moving, this is basically the same problem as the conveyor belt. The (moving) object is supplying friction to bring another object toward its speed. The only difference is we think of the belt as moving at one particular speed, while the box in the problem is accelerating as well.

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  • $\begingroup$ That's a nice analogy, thanks $\endgroup$
    – Sam Liu
    Dec 5, 2020 at 21:23
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This assumes the force applied to the lower box is sufficiently small so that the upper box does not slide on the lower box. (You are not doing the equivalent of jerking the tablecloth leaving the dishes on top in place.)

As the lower box moves say to the right, the upper box with respect to the lower box is moving left; the force of friction on the upper box opposes this relative motion and hence is to the right.

Consider a frame of reference attached to the accelerating upper box. In this frame the upper box is at rest and experiences an inertial force to the left which is countered by a frictional force to the right.

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