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Q 9CQ

Expert-verifiedFound in: Page 353

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.

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.

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.

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