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 ${\mu }_k$).

• A. $2\pi r{\mu }_{k}N$
  
• B. $\pi r{\mu }_{k}N$
  
• C. $-\pi r{\mu }_{k}N$
  
• 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 $F_k={\mu }_{k}N$

${\overrightarrow{F}}_k$ acts tangentially opposite to the motion of the pencil.

For an infinitesimal displacement $\overrightarrow{dl}$, the work done by pencil is,

$dW=\overrightarrow{F_k}.\overrightarrow{dl}$

$\Rightarrow dW=-F_k\left(dl\right)$

Now, the work done by the frictional force in drawing a circle of radius $r$,

$\Rightarrow W=-{\int }_0^{2\pi r}{F}_{k}dl$

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

$\Rightarrow W=-{\mu }_{k}N[2\pi r-0]$

$\therefore W=-2\pi {\mu }_{k}Nr$

Hence, option (d) is the correct answer.