Question

A particle moves in a circle of radius 5cm with constant speed and time period 0.2πs. The acceleration of the particle is
(A) 15
(B) 25
(C) 36
(D) 5

Answer 1

Velocity is given as a ratio of displacement to time. But when a particle moves in a circular motion then in addition to linear velocity, we do have angular velocity. The direction of linear velocity is always tangential to the direction of motion while in case of angular velocity and angular acceleration, it is always directed towards the center.

Formula used:
The acceleration of the particle is due to the circular motion and therefore is centripetal acceleration $a=\dfrac{{{v}^{2}}}{r}$. By substituting the values of velocity (v) and radius (r) we get the acceleration of a particle (a).

Complete step by step answer:
Given the radius of the circle in which the particle moves,$r=5cm=5\times {{10}^{-2}}m$.The time period of the particle,$T=0.2\pi$s.Therefore the total displacement(d) travelled by the particle is the circumference of the circle in one time period,$d=2\pi r=2\pi \times 5\times {{10}^{-2}}m$.We know that velocity(v) is given as the ratio of displacement(d) to the time taken(t), $v=\dfrac{d}{t}$.
Substituting the given values we get,
$v=\dfrac{2\pi \times 5\times 1{{0}^{-2}}}{0.2\pi } \\\ \Rightarrow v=0.5m/s$
Since a particle moves in a circle the acceleration acting on the particle is due to the centripetal force. The centripetal acceleration for a particle is given as $a=\dfrac{{{v}^{2}}}{r}$.
Substituting the value of velocity(v) and radius(r) we get,
Therefore the acceleration of the particle is,
$a=\dfrac{{{0.5}^{2}}}{5\times 1{{0}^{-2}}}\\\ \therefore a =5m/{{s}^{2}}$

Note: For a particle moving in a circular motion the acceleration is due to the centripetal force. The centripetal acceleration is given as $a=\dfrac{{{v}^{2}}}{r}$. Velocity is defined as the ratio of displacement to the time taken. For a body moving in a circular path in one time period the displacement is equal to the circumference of the circle. Centripetal force is a force acting on a particle moving on a circular path radially inward that enables it to move in a circular path.