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A solid sphere of mass ‘M’ and radius ‘R’ is initially at rest. Solid sphere is gradually lowered onto a truck moving with constant velocity $V_{0}$ .
The final speed of sphere’s centre of mass in ground frame when eventually pure rolling sets is (in multiples of $\frac{V_{0}}{7})$ ?

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answer is 2.

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Detailed Solution

$v = v e l o c i t y o f c e n t r e o f s p h e r e w r t t r u c k$

Applying conservation of angular momentum about point P on the truck:

$M V_{0} R = M v R + I ω = M v R + \frac{2}{5} M R^{2} ω$ .....(1)

for pure rolling, $v = ω R$ .... (2)

solving (1) and (2),

$v = \frac{5}{7} V_{0}$

velocity wrt ground:

$v' = V_{0} - v = \frac{2}{7} V_{0}$

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