First slide

Rolling motion

Question

A sphere of outer radius R having some cavity inside is allowed to roll down on an incline without slipping and it reaches a speed $v_{0}$ at the bottom of the incline. The incline is then made smooth by waxing and the sphere is allowed to slide without rolling and now the speed attained is ( $\frac{5}{4}$ ) $v_{0}$ . What is the radius of gyration of the sphere about an axis passing through its centre?

Difficult

A

$\sqrt{\frac{2}{5} R}$
B

$\sqrt{\frac{2}{3} R}$
C

$(\frac{4}{5}) R$
D

$(\frac{3}{4}) R$

Solution

Applying law of CONSERVATION OF ENERGY in the case of rolling
$\frac{1}{2} {mv}_{o}^{2} + \frac{1}{2} I {(\frac{v_{o}}{R})}^{2} = mgh$
Applying law of CONSERVATION OF ENERGY in the case of sliding
$\frac{1}{2} m {(\frac{5}{4} v_{o})}^{2} = mgh$
solve them to get: I = $\frac{9}{16} {mR}^{2}$
now I = m $K^{2}, K = \sqrt{\frac{I}{m}} = \frac{3}{4} R$

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