Chapter 12, Problem 1EP

To determine

The sum of the forces acting on the car and the sum of the forces acting on th descending mass.

Answer to Problem 1EP

The sum of the forces for the car is $\left({\mu }_k{m}_{c}g\cos \theta +m_{c}g\sin \theta \right)$ and that for the descending mass is $m_{1}g$.

Explanation of Solution

Draw a diagram to show the components of forces for a car moving in an inclided plane.

        

Write the expression for the sum of the force acting on the car.

    $F=f+F_{\parallel }$        (1)

Here, $F$ is the sum of the forces, $f$ is the frictional force and $F_{\parallel }$ is the parallel force.

Write the expression for parallel force acting on the car.

    $F_{\parallel }=m_{c}g\sin \theta$

Here, $m_c$ is the mass of the car, $g$ is the acceleration due to gravity and $\theta$ is the inclination angle.

Write the expression for the sum of forces acting on the descending mass.

    $T=m_{1}g$

Here, $T$ is the tension of the string and $m_1$ is the mass of the descending body.

Conclusion:

Substitute $m_{c}g\sin \theta$ for $F_{\parallel }$ in equation (1).

    $F=f+m_{c}g\sin \theta$

Thus, the sum of the forces for the car is $\left(f+m_{c}g\sin \theta \right)$ and that for the descending mass is $m_{1}g$.