Question

The work done by the frictional force on a pencil in drawing a complete circle of

 radius $r = \dfrac{1}{\pi }$ metre on the surface by a pencil of negligible mass with a normal

 pressing force $N = 5N(\mu = 0.5)$

(A) $ + 4J$

 (B) $ - 3J$

 (C) $ - 2J$

 (D) $ - 5J$

Answer 1

Hint: In order to solve the problem we need to know about what is work done by frictional force and knowledge about normal force , direction of normal force , direction of frictional force.

Work done by frictional force is

$dw$

\[\]$dw = \overrightarrow {{F_k}} .dl$

Here $\overrightarrow {{F_k}} $ is the frictional force opposite to the applied force

Due to opposite in direction we can say that

$dw = - \overrightarrow {{F_k}} .dl$

Given:

a pencil in drawing a complete circle of radius $r = \dfrac{1}{\pi }$ metre on the surface by a pencil of negligible mass with a normal pressing force $N = 5N$

and ${\mu _k} = 0.5$(coefficient of kinetic friction)

To find the work done by the frictional force on a pencil

Complete step by step solution:

As per the given question first, we will draw the diagram as per the given situation,

 

 Let the frictional force applied be $\overrightarrow {{F_k}} $​​

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

For elementary displacement $dl$

The work done by $\overrightarrow {{F_k}} $is $dw$

 $dw = \overrightarrow {{F_k}} .dl = - \overrightarrow {{F_K}} .dl$

Integrating on both side, we get

$\int_0^W {dw} = - \int_0^{2\pi r} {\overrightarrow {{F_K}} .dl} $

$\int_0^W {dw} = - {\mu _k}N\int_0^{2\pi r} {dl} $ as we know $(\overrightarrow {{F_K}} = {\mu _k}N)$

 $W = - {\mu _k}N2\pi r$

Put value of ${\mu _k},r,N$

$W = -0.5 \times 5 \times 2 \times \pi \times \dfrac{1}{\pi }$

$W = - 5J$ is the work done by the frictional force on a pencil in drawing a complete circle.

So the correct answer is (d).

Note: We see friction in our day to day activities. Following are the ten examples of friction in daily life:

1. Walking

2. Writing

3. Scating

4. Lighting a matchstick

5. Driving of the vehicle on a surface

6. Applications of breaks in the vehicle to stop it

7. Flight of aeroplanes

8. Drilling a nail into the wall

9. The dusting of the carpet by beating it with a stick

10. Sliding on a garden slide