First slide

Rolling motion

Question

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

Moderate

A

2 : 5
B

7 : 10
C

10 : 7
D

5 : 7

Solution

Translational kinetic energy of sphere
$K_{t} = \frac{1}{2} {mv}_{c}^{2}$ -----------------(i)
Summation of translational and rotational kinetic energy
$K_{t} + K_{r} = \frac{1}{2} {mv}_{c}^{2} + \frac{1}{2} I_{0} ω^{2}$
For solid sphere, $I_{0} = \frac{2}{5} {mr}^{2} and in case of rolling$
$v_{c} = ωr$
It means $\frac{K_{t}}{K_{t} + K_{r}} = \frac{{mω}^{2} r^{2}}{{mω}^{2} r^{2} + \frac{2}{5} {mr}^{2} ω^{2}} = \frac{1}{1 + \frac{2}{5}} = \frac{5}{7}$

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