Question

A small block of mass m is taken slowly up a fixed hemisphere from P to Q and then slowly down along its surface from Q to S by force F, which at each point of the surface, is tangential to the surface. If the coefficient of friction is $\mu$, then the work performed by the force F in taking the block from P to S along the hemispherical surface of radius R is?

• A. $2\mu mgR$
• B. $\mu mgR$
• C. $\pi \mu mgR$
• D. $3\pi \mu mgR$

Answer 1

The correct option is A $2\mu mgR$

Work done by $F$ is equal to frictional force ($f$).

Work done, $dW$ is

$dW=f\left(Rd\theta \right)=\mu mg\sin \theta Rd\theta =\mu mgR\sin \theta d\theta$

$dW={\int }_0^{\pi }\mu mgR\sin \theta d\theta =\mu mgR[\cos \theta {]}_0^{\pi }=\mu mgR[1-(-1)]=2\mu mgR$