A solid sphere is in rolling motion. In rolling motion a body possesses translational kinetic energy () as well as rotational kinetic energy () simultaneously. The ratio for the sphere is [2018]

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Solution

Q.
A solid sphere is in rolling motion. In rolling motion a body possesses translational kinetic energy ( $K_{t}$ ) as well as rotational kinetic energy ( $K_{r}$ ) simultaneously. The ratio $K_{t} : (K_{t} + K_{r})$ for the sphere is [2018, 1991]

1 7 : 10

2 5 : 7

3 10 : 7

4 2 : 5

Ans.
(2)

Translational kinetic energy, $K_{t} = \frac{1}{2} m v^{2}$

Rotational kinetic energy, $K_{r} = \frac{1}{2} I ω^{2}$

$∴$ $K_{t} + K_{r} = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2} = \frac{1}{2} m v^{2} + \frac{1}{2} (\frac{2}{5} m r^{2}) {(\frac{v}{r})}^{2}$

$∴$ $K_{t} + K_{r} = \frac{7}{10} m v^{2}$ $[∵ I = \frac{2}{5} m r^{2} (for sphere)]$

So, $\frac{K_{t}}{K_{t} + K_{r}} = \frac{5}{7}$

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