Question

A particle is moving in a circle such that it completes 40 revolutions in 40s

Answer 1

The particle is moving in a circular motion, so we can use the formula for angular velocity to calculate the distance covered in given time. Displacement is the shortest path between two points, so the ratio of $\dfrac{\left| \text{displacement} \right|}{\text{distance}}\le 1$. Convert between the units as required.

Formula Used:
$\omega =\dfrac{\theta }{t}$

Complete step-by-step solution:
Given, the particle completes 40 revolutions in $40s$. Radians covered in 40 revolutions is $2\pi \times 40$.
Its angular velocity will be-
$\omega =\dfrac{\theta }{t}$ - (1)
Here,
$\omega$ is the angular velocity
$\theta$ is the angular displacement
$t$ is time taken

Substituting values for the particle in the above equation, we get,
$\begin{aligned} & \omega =\dfrac{2\pi \times 40}{40} \\\ & \Rightarrow \omega =2\pi \,rad\,{{s}^{-1}} \\\ \end{aligned}$

The angular velocity of the particle is $2\pi \,rad\,{{s}^{-1}}$
$\begin{aligned} & 2\min 20\text{ }s=(2\times 60+20)s \\\ & \Rightarrow 2\min 20\text{ }s=140s \\\ \end{aligned}$

In $120s$, from eq(1) it covers angular displacement of-
$\begin{aligned} & 2\pi \text{ }rad\text{ }{{s}^{-1}}=\dfrac{\theta }{140} \\\ & \Rightarrow \theta =2\pi \times 140 \\\ \end{aligned}$

The particle covers $2\pi \times 140\text{ rad}$. This means it completes 140 revolutions at $140s$. In a circular motion, the starting point is the ending point of a path. So we can say that the particle ended from where it started. The displacement is the shortest path between two point and since the points coincide $displacement=0$
$\therefore \dfrac{\left| \text{displacement} \right|}{\text{distance}}=0$

The ratio $\dfrac{\left| \text{displacement} \right|}{\text{distance}}$ is 0. Therefore, the correct option is (A).

Additional information:
For a body to move in a circular motion, a centripetal force acts on it which is given by- ${{F}_{c}}=\dfrac{m{{v}^{2}}}{r}$. It acts towards the centre of rotation, towards the centre of the circle traced by the body.

Note:
In circular motion, the particle covers a full angle, $2\pi$in one revolution. The velocity is tangential to the circular path. Displacement is always shorter or equal to the distance. Since displacement is a vector quantity, it can either be positive or negative depending on the direction of motion of the object.