Question

A nonuniform ball of radius $R$ and radius of gyration about geometric centre $=R/2$, is kept on a frictionless surface. The geometric centre 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

• A. the ball with slip on surface tor all cases
• B. the ball will roll purely if $h=5R/4$
• C. the ball will roll purely if $h=3R/2$
• D. there will be no rotation if $h=R$

Answer 1

Answer: (B), (D)

the ball will roll purely if $h=5R/4$
there will be no rotation if $h=R$

$I=m{K}^2$ $(where.k=\frac{R}{2}=radiusofgyration)$
$=m{\left(\frac{R}{2}\right)}^2$
$=\frac{1}{4}m{R}^2$
Ball will roll purely if $v=R\omega$
$\therefore \left(\frac{J}{m}\right)R=\left[\frac{J(h-R)}{\left(1/4\right)m{R}^2}\right]$
Solving this equation, we get $h=\frac{5R}{4}$
Further, ifbaU is struck at centre afmass, there will be no rotation only translation.