Question

A force ${\displaystyle \frac{m{v}^2}{r}}$ is acting on a body of mass m moving with a speed v in a circle of radius r. What is the work done by the force in moving the body over half the circumference of the circle?
$\text{A}\text{. }{\displaystyle \frac{m{v}^2}{r}}\times \pi r$
$\text{B}\text{.}$ Zero
$\text{C}\text{. }{\displaystyle \frac{m{v}^2}{r^2}}$
$$\text{D}\text{. }{\displaystyle \frac{\pi r^2}{m{v}^2}}$$

Answer 1

Hint – As the body is in circular motion, so the centripetal force, i.e., ${\displaystyle \frac{m{v}^2}{r}}$ is directed towards the centre and at every point it is perpendicular to the displacement. Keep this point in mind to solve the question.
Formula used - $W=FS\cos \theta$

Complete step-by-step answer:
We have been given that a body of mass m is moving with a speed v in a circle of radius r.
As, we know when a body moves in a circular path then centripetal force acts on it which is given by ${\displaystyle \frac{m{v}^2}{r}}$
This centripetal force is perpendicular to the displacement of the body at every point.
So, as we know the work done is given by $W=FS\cos \theta$ , where F is the force acting on the body and S is the displacement and theta is the angle between them and W is the work done.
Now, here as mentioned above the angle between the centripetal force and the displacement is 90 degrees.
Keeping the value of theta, we get-
$$\begin{gathered} W=FS\cos {90}^{\circ } \\ \Rightarrow W=0\{\because \cos {90}^{\circ }=0\} \end{gathered}$$
Therefore, the work done is Zero.
Hence, the correct option is B.

Note – Whenever such types of questions appear, then first write all the things given in the question and then by using the formula, $W=FS\cos \theta$ and also by knowing the angle between the force and displacement find the work done. In this case the work done is zero, as the force is perpendicular to displacement.