Question

A spherical body of radius ${}^{\prime }{R}^{\prime }$ rolls on a horizontal surface with linear velocity ${}^{\prime }{v}^{\prime }$. Let $L_1$ and $L_2$ be the magnitudes of angular momenta of the body about centre of mass and point of contact $P$. Then,

• A. $L_2=2L_1$; if radius of gyration $K=R$
• B. $L_2=2L_1$; for all cases
• C. $L_2>2L_1$; if radius of gyration $K<R$
• D. None of these

Answer 1

The correct option is A $L_2=2L_1$; if radius of gyration $K=R$

Angular momentum about center of mass:

$L_1=I\omega =M{K}^2\omega$

angular momentum at the point of contact:

$L_2=I\omega +MRv=M{K}^2\omega +MR\left(R\omega \right)$

$\because (v=R\omega )$

$L_2=M{K}^2\omega +M{R}^2\omega$

$L_2=M\omega (K^2+R^2)$

$\therefore L_2=2L_{1}forK=R$

$L_2>L_{1}forK>R$