Question

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

• A. $4\pi r^2{\mu }_{k}N$
• 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

Answer: (D)

$-2\pi r{\mu }_{k}N$

The kinetic frictional force at any instantaneous position is

$f_k={\mu }_{k}N$.

$f_k$, acts tangentially opposite to the motion of the pencil.

For elementary displacement $dl$, the work done by $f_k$ is $dW=\overrightarrow{f_k}\cdot \overrightarrow{dl}=-f_{k}dl$

=> W = $-{\int }_0^{2\pi r}{f}_{k}dl=-{\mu }_{k}N{\int }_0^{2\pi r}dl$

= $-\mu N2\pi r=-2\pi r{\mu }_{k}N$