Question

A hollow sphere of radius $R$ lies on a smooth horizontal surface. It is pulled by a horizontal force acting tangentially from the highest point. The distance traveled by the sphere during the time it makes one full rotation is

• A. $2\pi R/3$
• B. $\pi R/3$
• C. $\pi R$
• D. $4\pi R/3$

Answer 1

The correct option is A $2\pi R/3$

torque due to force ,$\tau =F\times R$

we know ,$\tau =I\alpha$

therefore

$F\times R=\left(\frac{2}{3}\right)M{R}^2\alpha$

implies that $\alpha =\frac{3F}{2MR}$

Applying $\theta =\omega t+\frac{1}{2}\alpha t^2$

$2\pi =\frac{1}{2}\alpha t^2$

implies that $t^2=\frac{8\pi MR}{3f}$

suppose $d$ is the distance traveled in the time $t$

Acceleration $a=\frac{F}{M}$

$d=\frac{1}{2}a{t}^2$

$=\frac{1}{2}\times \frac{F}{M}\times \left(\frac{8\pi MR}{3F}\right)$

$=\frac{4\pi R}{3}$