First slide

Rolling motion

Question

A ball of radius r rolls inside a hemispherical shell of radius R. It is released from rest from point A as shown in figure and the ball starts rolling without slipping. The angular velocity of centre of the ball in position B about the centre of the shell is:

Moderate

A
$2 \sqrt{\frac{g}{5 (R - r)}}$
B

$\sqrt{\frac{10 g}{7 (R - r)}}$
C
$\sqrt{\frac{2 g}{5 (R - r)}}$
D
$\sqrt{\frac{5 g}{2 (R - r)}}$

Solution

Loss in P.E. = Gain in K.E.
$m g (R - r) = \frac{1}{2} m v^{2} (1 + \frac{K^{2}}{r^{2}}) a n d v = (R - r) ω F o r a s o l i d s p h e r e K^{2} = \frac{2 r^{2}}{5} \Rightarrow v = \sqrt{\frac{10 g (R - r)}{7}} ω = \frac{v}{R - r} = \sqrt{\frac{10 g}{7 (R - r)}}$

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