First slide

Rolling motion

Question

A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of
mass is K. If radius of the ball be R, then the fraction of total energy associated with its rotational energy
will be

Moderate

A

$\frac{K^{2}}{K^{2} + R^{2}}$
B

$\frac{R^{2}}{K^{2} + R^{2}}$
C

$\frac{K^{2} + R^{2}}{R^{2}}$
D

$\frac{K^{2}}{R^{2}}$

Solution

Moment of inertia, $I = m K^{2}$
$∵$ Ball is rolling without slipping, $v = R ω$
Translational kinetic energy, $K_{R} = \frac{1}{2} I ω^{2} = \frac{m K^{2} v^{2}}{2 R^{2}}$
$∴$ Total kinetic energy $= \frac{1}{2} m v^{2} + \frac{m K^{2} v^{2}}{2 R^{2}}$
Required fraction $= \frac{\frac{m K^{2} v^{2}}{2 R^{2}}}{\frac{1}{2} m v^{2} + \frac{m K^{2} v^{2}}{2 R^{2}}} = \frac{K^{2}}{R^{2} + K^{2}}$

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