10.8 Friction between horizontal surfaces

The acceleration of block, “ $a_A$ ”, is :

$$a_A=\frac{F}{m}$$

The acceleration of plank, “ $a_B$ ”, is :

$$a_B=0$$

There is friction between block and plank

The friction force in this case will influence the motion as it acts as external force on each individual body. The resulting motion of the bodies, however, would depend on the nature of friction (static, limiting or kinetic). In the figure below, the forces on the block and plank are shown separately for each of them.

In order to determine the direction of frictions at the interface, we go by two simple steps. We first consider the body on which external force (F) is applied. The direction of friction on the body (A) is opposite to the external force. We determine friction on the other body (B) by applying Newton’s third law of motion. Friction on plank (B) is equal in magnitude, but opposite in direction.

Once direction of friction is known, we need to know the nature of friction. For this, we first calculate limiting friction between the bodies so as to compare it with external applied force "F". The limiting friction is given by :

$$F_S=\mu N=\mu mg$$

Important thing to realize here is that limiting friction depends only on the mass of the block, “m”, and is independent of the mass, “M”, of the plank. Now, we should compare external force with friction to determine its magnitude. Finally, we analyze motion in following manner :

1 : $F\le F_S\left(F\le {\mu }_{s}mg\right)$

In this case, friction self adjusts to external force, “F”. Hence, static friction is given by :

$$\Rightarrow f_s=F$$

As there is no relative motion, the block and plank move together as a single unit. The friction forces at the interface are internal forces for the combined body. The free body diagram of the combined body is shown here.

The common acceleration of the block and plank, “a”, is :

$$a=a_A=a_B=\frac{F}{m+M}$$

2 : $F>F_S\left(F>{\mu }_{s}mg\right)$

The block and plank move with different accelerations and have relative motion. The friction between the interfaces is, therefore, kinetic friction.

The free body diagrams of block and plank are shown here.

The acceleration of block, “ $a_A$ ”, is :

$$a_A=\frac{F-{\mu }_{k}mg}{m}$$

The acceleration of block, “ $a_B$ ”, is :

$$aB=\frac{{\mu }_{k}mg}{M}$$

An external force is applied on the plank

The external force is applied on the plank as shown in the figure below. There are two possibilities : (i) there is no friction between "A" and "B" or (ii) there is friction between "A" and "B".

There is no friction between block and plank

In this situation, the external force on block, “B”, accelerates only the plank. As there is no friction between the interface of block and plank, there is no external force on the block in horizontal direction. As such, block, “A”, is not accelerated. The free body diagrams of “A” and “B” are as shown in the figure.