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Q 9CQ

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Found in: Page 353

### College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

# Question: A ball slides up a frictionless ramp. It is then rolled without slipping and with the same initial velocity up another frictionless ramp (with the same slope angle). In which case does it reach a greater height, and why?

The height reached by the ball during rolling is greater than during sliding.

See the step by step solution

## Step 1: Height reached by the ball during sliding.

During sliding of the ball, the translational kinetic energy of the ball is converted to the potential energy.

$$\begin{array}{}K.E{._{trans.}} &= P.E.\\\frac{1}{2}m{v^2} &= mgh\\h &= \frac{{{v^2}}}{{2g}}\end{array}$$

Here, h is the height reached, v is the velocity of the ball, m is the mass of the ball, and g is the acceleration due to gravity.

## Step 2: Height reached by the ball during rolling without friction.

During rolling of the ball without friction, the rotational and translational kinetic energy of the ball is converted to potential energy.

$$\begin{array}{c}K.E{._{trans.}} + K.E{._{rot.}} &= P.E.\\\frac{1}{2}m{v^2} + \frac{1}{2}I{\omega ^2} &= mgh\\\frac{1}{2}m{v^2} + \frac{1}{2} \times \left( {m{r^2}} \right) \times {\frac{v}{{{r^2}}}^2} &= mgh\\m{v^2} &= mgh\\h &= \frac{{{v^2}}}{g}\end{array}$$

Clearly, the height reached by the ball during rolling is greater than during sliding.