10.8 Friction between horizontal surfaces  (Page 3/4)

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

$$a_A=0$$

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

$$a_B=\frac{F}{M}$$

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 is applied. The direction of friction on the body (B) is opposite to the external force. We determine friction on the other body (A) by applying Newton’s third law of motion. Friction on block (A) is equal in magnitude, but opposite in direction.

Once direction of friction is known, we need to know the nature of friction. Unlike in the previous case when external force is applied on the block, the situation here is different. The plank carries another mass of block over itself. The external force (F) is not completely used to overcome friction at the interface, but to move the combined mass together. As such, external force can not be directly linked to static friction as in the case when force is applied on the block.

For this reason, we shall adopt a different strategy. We shall assume that friction is static friction. If the analysis of force does not support this assumption, then we correct the assumption accordingly. Now, the limiting friction is given by :

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

Let us assume that " $f_S$ " be the static friction between the surfaces. 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 :

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

We observe that friction is the only force on the block. Then, applying Newton's second law for the motion of block, we have :

$$f_S=ma$$

Clearly, if " $f_S$ " so calculated is less than or equal to limiting friction, then block and plank indeed move together. However, if " $f_S$ " is greater than limiting friction, then block and plank move with different accelerations. In such case, friction between "A" and "B" is kinetic friction.

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

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

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

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

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

One external force each is applied on block and plank

In this case, external forces act on the block and plank separately. It is, therefore, not possible in this case to compare limiting force with external force as there are two of them, which are acting on two different bodies.