What Forces Are at Play to Prevent Slippage Between Two Accelerating Boxes?

[zeion]

Top and bottom box force and motion question

Hello. This is my first time posting, I hope I have done everything correctly.

Homework Statement

A small box is resting on a larger box, which in turn sits on a table. When a horizontal force is applied to the larger box, both boxes accelerate together. The small box does not slip on the larger box.

1) What force causes the small box to accelerate horizontally?
2) If the acceleration of the pair of boxes has a magnitude of 2.5 m/s$$^2$$, determine the smallest coefficient of friction between the boxes that will prevent slippage.

The Attempt at a Solution

1)
Horizontal Fa on the lower box causes the top box to move, but Fa cannot > Fs between the two boxes or top will slip. But if Fs between two boxes is > than Fa then bottom cannot accelerate therefore top will not accelerate. Therefore Fa = Fs.
Fa causes the top box to move, but Fs between top and bottom causes top to not slip.

2)
$$\mu_s = \frac{F_s}{F_n}$$
It does not slip so $$F_A = F_s = ma$$
There is not vertical movement so $$F_n = mg$$
Therefore $$\mu_s = \frac{ma}{mg} = \frac{a}{g} = \frac{2.5m/s^2}{9.8m/s^2} = 0.25$$
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[LowlyPion]

But if Fs between two boxes is > than Fa then bottom cannot accelerate therefore top will not accelerate.

Welcome to PF.

It's really enough to say that the force of friction is a maximum proportional to the normal force that reacts against horizontal forces. So long as the system is not accelerating the top box greater than the maximum frictional force available, then it won't move.

 

[zeion]

Welcome to PF.

It's really enough to say that the force of friction is a maximum proportional to the normal force that reacts against horizontal forces. So long as the system is not accelerating the top box greater than the maximum frictional force available, then it won't move.

Hi. Thank you.
So the force of friction is never greater than the horizontal forces, it can only either be less or equal to them, is that right? I don't understand how horizontal forces affect Fn, I thought Fn was proportional to the perpendicular forces ie. gravity, which is proportional to the object's mass.

 

[LowlyPion]

Hi. Thank you.
So the force of friction is never greater than the horizontal forces, it can only either be less or equal to them, is that right? I don't understand how horizontal forces affect Fn, I thought Fn was proportional to the perpendicular forces ie. gravity, which is proportional to the object's mass.

Fn is just that. Weight down and supporting force up.

Frictional forces are calculated on the basis that their magnitudes can be determined by a proportionality relationship with the normal forces.