First slide

Rolling motion

Question

When a sphere rolls without slipping the ratio of its kinetic energy of translation to its total kinetic energy is

Moderate

A

1 :7
B

1 : 2
C

1 : 1
D

5 : 7

Solution

$\frac{E_{k}}{E} = \frac{\frac{1}{2} {mv}^{2}}{\frac{1}{2} {mv}^{2} + \frac{1}{2} {Iω}^{2}}$
$= \frac{\frac{1}{2} {mv}^{2}}{\frac{1}{2} {mv}^{2} + \frac{1}{2} \times \frac{2}{5} {mr}^{2} \times (v^{2} / r^{2})}$
$(∵ I = \frac{2}{5} {mr}^{2} and V = rω)$
$= \frac{1}{1 + (2 / 5)} = \frac{5}{7} or 5 : 7$

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