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## Homework Statement

A small box of mass [itex]m_{1}[/itex] resting on a plank, of mass [itex]m_{2}[/itex] and length [itex]L[/itex], which itself rests on a frictionless, horizontal surface. Both box and board are stationary when a constant force [itex]F[/itex] is applied to the board.

Take [itex]g[/itex] to be the acceleration due to gravity and [itex]F_{\text{f}}[/itex] to be the magnitude of the frictional force between the board and the box.

(A picture can be found on page 26, question 13 here, though the problem is not the same)

Assuming the coefficient of static friction between the box and the board is unknown, what is the magnitude, and the direction, of the acceleration of the box in terms of [itex]F_{\text{f}}[/itex] (relative to the surface)?

## Homework Equations

The equations needed to solve the problem are newtons second law (where mass = a constant), [itex] F=ma [/itex]. The inequality dealing with friction cannot be used as we are not allowed to use the coefficient of friction between the box and board.

## The Attempt at a Solution

The frictional force [itex]F_{\text{f}}[/itex] must be equal to [itex]m_{1}a_{1}[/itex], where [itex]a_{1}[/itex] is the acceleration on the box. However I am looking for the acceleration relative to the table (surface).

Because the force [itex]F[/itex] accelerates the plank, and the box rests on top of it, the box's mass must be taken into account, so [itex]F=a_{2}(m_{1}+m_{2})[/itex] where [itex]a_{2}[/itex] is the acceleration on the plank. [itex]F_{\text{f}}[/itex] must act in the same direction as [itex]F[/itex] or the box would accelerate opposite to the direction of the plank and go flying off the end, so the total force on the box must be [itex]F+F_{\text{f}}=m_{1}a_{1}[/itex].

Is this reasoning correct? The question implies that [itex]F_{\text{f}}[/itex] can be expressed independently of [itex]F[/itex] or [itex]a_{2}[/itex]. Any thoughts?

Edit: I just asked the lecturer if we were supposed to assume that the box did not slip and he said "Understanding that point is part of the question!" Does that mean we are supposed to assume that, or not?

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