Chapter 5.2, Problem 20PP

To determine

The horizontal force to be applied on an object.

Answer to Problem 20PP

The horizontal force to be applied is $F_A=78\text{ N}$

Explanation of Solution

Given:

The coefficient of kinetic friction between the rug and wooden floor is 0.12 and the weight of the person sitting on the rug is 650 N.

Formula Used:

Here, a force is to be applied horizontally on the rug along with Thomas, to move it along the polished wooden floor with a constant speed. There will be a force opposing the applied force which is due to kinetic friction in this case since the body is in motion. According to Newton’s laws, the frictional force $F_{f,kinetic}$ and the applied force $F_A$ are equal and they oppose each other.

This can be expressed as,

  $F_A=-F_{f,kinetic}$ …… (1)

Now, the kinetic friction force will be proportional to the normal force acting in the body. In this case the normal force is equal to the weight of the body as there are no other forces along the normal.

  $F_{\text{N}}=mg$ …… (2)

Thus, the kinetic friction force can be expressed as,

  $F_{f,kinetic}={\mu }_{\text{k}}{F}_{\text{N}}$ …… (3)

Here ${\mu }_{\text{k}}$ represents the coefficient of kinetic friction.

Based on this, the magnitude of horizontal force to be applied can be deduced as,

  $F_A={\mu }_{\text{k}}{F}_{\text{N}}$ …… (4)

Calculation:

Given the weight of Thomas, sitting on the rug is 650 N. Thus based on (2), the normal force is

  $F_{\text{N}}=650\text{ N}$

Now, the applied force can be calculated by substituting the normal force along with the coefficient of kinetic friction in (4), as

  $\begin{array}{c}{F}_A=0.12\times 650\\ =78\text{ N}\end{array}$

Conclusion:

Thus, the horizontal force to be applied on the rug is 78 N.