Chapter 8, Problem 12Q

Two inclines have the same height but make different angles with the horizontal. The same steel ball rolls without slipping down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

To determine

The incline for which the speed of the ball at the bottom will be greater.

Answer to Problem 12Q

Solution:

The speed of the ball in both cases will be the same.

Explanation of Solution

Given:

The height is the same for both the inclines.

Formula used Total kinetic energy:

  $KE=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$

Where, m is the mass, v is the speed, I is the moment of inertia and $\omega$ .

Moment of inertia of a sphere is:

  $I=\frac{2}{5}m{r}^2$

Angular speed:

  $\omega =\frac{v}{r}$

Calculation:

If h is the height of the two inclines, the total energy of the ball at the top of both is $mgh$ , which is the ball’s potential energy, and where m is the mass of the ball.

At the bottom, all of this energy is converted into kinetic energy, which is given by,

  $KE=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$ , where v is the speed of the ball at the bottom, I the moment of inertia and ω the angular velocity.

I for the spherical ball is $\frac{2}{5}m{r}^2$ and $\omega =\frac{v}{r}$ , where r is the radius of the ball.

Total kinetic energy does not depend on the angle of the incline.

Thus, the speed of the ball in both cases will be the same.

Conclusion

Thus, the speed of the ball in both cases will be the same.