Chapter 12, Problem 20A

Beginning from a rest position, a solid disk A, a solid ball B, and a hoop C, race down an incline. Rank them in order of finishing: winner, second place, and third place.

To determine

To rank:The result of race, down an incline inorder of finishing.

Answer to Problem 20A

The rank of the race, down an incline in an order of finishing is solid ball B, solid disk A, hoop C.

Explanation of Solution

Given:

Solid disk A, a solid ball B and a hoop C are initially at rest. They race down along an inclined plane.

Formula used:

Moment of inertia of rigid body about an axis of rotation is defined as the sum of products of the mass and the square of the distance from the axis of rotation.

The expression for moment of inertia is,

  $I={\displaystyle \sum _{i=1}^m{m}_{\text{i}}{r}_{\text{i}}^2}$

Here, $m_{\text{i}}$ is the mass of the particle and $r_{\text{i}}$ is the distance of the particle from the axis of rotation.

Calculation:

The expression for moment of inertia of solid disk A is,

  $I_{\text{disk}}=\frac{1}{2}m{r}^2$

Here, $m$ is the mass of the disk and $r$ is the radius of the disk.

The expression for moment of inertia of solid ball B is,

  $I_{\text{ball}}=\frac{2}{5}m{r}^2$

Here, $m$ is the mass of the ball and $r$ is the radius of the ball.

The expression for moment of inertia of hoop C is,

  $I_{\text{hoop}}=m{r}^2$

Here, $m$ is the mass of the hoop and $r$ is the radius of the hoop.

The objects have least moment of inertia produce more speed.

By comparing the values of moment of inertia $I_{\text{hoop}}>I_{\text{disk}}>I_{\text{ball}}$ .

Conclusion:

Thus, the rank of the race down an incline in an order of finishing is solid ball B, solid disk A, hoop C.