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A nonuniform ball of radius $R$ and radius of gyration about geometric center $= R / 2,$ is kept on a frictionless surface. The geometric center coincides with the centre of mass. The ball is struck horizontally with a sharp impulse $= J .$ The point of application of the impulse is at a height ‘h’ above the surface. Then

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The ball will roll purely if $h = 5 R / 4 .$

The ball will roll purely if $h = 3 R / 2 .$

The ball will slip on surface for all cases.

There will be no rotation if $h = R .$

answer is B, D.

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Detailed Solution

$J = m v . . . . . . . . . . . (1)$

$J \times r = (\frac{m R^{2}}{4}) ω . . . . . . . . . . . (2)$

Solving , $ω = \frac{v}{R}, t h i s c o n d i t i o n s a t i s f i e s p u r e r o l l i n g .$

Since $r = \frac{R}{4} w e g e t h = \frac{5 R}{4}$

if $h = R, r = 0$

$ω = 0$ (from eq.-2)

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