Question

The work done by the frictional force on a surface in drawing a circle of radius $r$ on the surface having coefficient of friction ${\mu }_k$ by a pencil of negligible mass with a normal pressing force $N$ is :

• A. 0.0
• B. $-2\pi r^2{\mu }_{k}N$
• C. $-3\pi r^2{\mu }_{k}N$
• D. $-2\pi r{\mu }_{k}N$

Answer 1

The correct option is D $-2\pi r{\mu }_{k}N$

The kinetic frictional force at any instantaneous position is $f={\mu }_{k}N$ (Tangentially)

For elementary displacement $dl$, the work done by $f$​ is $dW=\overrightarrow{f}.\overrightarrow{dl}$
$dW=-fdl$

$-ve$ sign indicates that $f,dl$ are in opposite direction.
$\int dW=-\int fdl$
${\int }_0^{W}dW=-{\mu }_{k}N{\int }_0^{2\pi r}dl$
$\Rightarrow W=-2\pi r{\mu }_{k}N$