Question

A particle moves in a circle of the radius 25cm at two revolutions per second. The acceleration of the particle in \[m/{s^2}\] is:
$
  {\text{A}}{\text{. }}{\pi ^2} \\
  {\text{B}}{\text{. 8}}{\pi ^2} \\
  {\text{C}}{\text{. 4}}{\pi ^2} \\
  {\text{D}}{\text{. 2}}{\pi ^2} \\
 $

Answer 1

Hint: We are given the radius of the circle and also the frequency of revolution of the particle. The acceleration of the particle is equal to the product of radius of the path and the square of the angular velocity of the particle. The angular velocity of the particle can be found by taking the product of frequency of revolution of particle with $2\pi $.

Formula used:
The angular frequency of a body is given as
$\omega = 2\pi f$
Here $\omega $ is known as the angular velocity while f signifies the frequency of revolution of the body.
The acceleration of the body is given as
$a = r{\omega ^2}$
Here r is the radius of the path in which the body rotates.

Complete step-by-step answer:
We are given a particle which is moving in a circle of the radius 25cm. Therefore,
$r = 25cm = 0.25m$
The particle makes two revolutions per second. The frequency of a particle is equal to the revolutions made by the particle per second. Therefore, the frequency of the particle is given as
$f = 2Hz$
From the frequency, we can calculate the angular frequency of the particle by using the formula given in the formula used section. It can be obtained in the following way.
$\omega = 2\pi r = 2\pi \times 2 = 4\pi {\text{ }}rad/s$
Now we have the angular frequency of the particle as well as the radius of the circular path in which the particle moves. Using the formula for acceleration, we can write that
$a = r{\omega ^2} = 0.25 \times {\left( {4\pi } \right)^2} = 4{\pi ^2}{\text{ }}m/{s^2}$
Hence the acceleration of the particle is $4{\pi ^2}{\text{ }}m/{s^2}$. Hence, the correct answer is option C.

So, the correct answer is “Option C”.

Note: It should be noted that we have been asked to find the linear acceleration. This is evident from the units of acceleration given in the question. In case if we had been asked to find angular acceleration then the units of acceleration would have been $rad/{s^2}$.