Question

A sphere of mass $M$ rolls without slipping on a rough surface with centre of mass has constant speed $v_o$. If mass of the sphere is $m$ and its radius be $R$, then the angular momentum of the sphere about the point of contact is

• A. $\frac{9}{5}M{v}_{o}R(-\hat{k})$
• B. $\frac{4}{5}M{v}_{o}R(-\hat{k})$
• C. $\frac{7}{5}M{v}_{o}R(-\hat{k})$
• D. $\frac{8}{5}M{v}_{o}R(-\hat{k})$

Answer 1

The correct option is C $\frac{7}{5}M{v}_{o}R(-\hat{k})$

Draw a diagram of given problem.

Calculate angular momentum of sphere about point of contact.

Given,
Mass of sphere$=m$
Radius of sphere$=R$
Velocity of sphere$=v_o$
Angular momentum of the sphere about point of contact,
${\overrightarrow{L}}_p=I_{cm}{\omega }_0(-\hat{k})+M{v}_{0}R(-\hat{k})\left(i\right)$
As the sphere is in pure rolling motion.
$\omega =v_o/R$
Put these all above value in equation $\left(i\right)$
${\overrightarrow{L}}_P=[\frac{2}{5}M{R}^2\left(\frac{v_0}{R}\right)+M{v}_{0}R](-\hat{k})$
$=\frac{7}{5}M{v}_{0}R(-\hat{k})$