Question

Find the work done by the frictional force in drawing a circle of radius $r$ by a pencil of negligible mass with a normal pressing force $N$ (coefficient of friction $\mu$)

Answer 1

The kinetic frictional force at any instantaneous position is $f_k={\mu }_{k}N$
${\overrightarrow{f}}_k$ acts tangentially opposite to the motion of the pencil. For elementary displacement $d\overrightarrow{l}$ the work done by $dW={\overrightarrow{f}}_{k}d\overrightarrow{l}=-f_{k}dl$
$\Rightarrow W=-{\int }_0^{2\pi r}{f}_{k}dl=-{\mu }_{k}N{\int }_0^{2\pi r}dl=-{\mu }_{k}N2\pi r=-2\pi r{\mu }_{k}N$