Question

A particle moves with constant speed v along a circular path of radius r and completes the circle in time T. Then find the acceleration of the particle.
A. \[\dfrac{{2\pi v}}{T}\]
B. \[\dfrac{{2\pi r}}{T}\]
C. \[\dfrac{{2\pi {r^2}}}{T}\]
D. \[\dfrac{{2\pi {v^2}}}{T}\]

Answer 1

Hint: Before we start addressing the problem, we need to know about acceleration. Acceleration is defined as the rate at which velocity changes with time, in terms of both speed and direction of a particle. The unit of acceleration is \[m{s^{ - 2}}\].

Formula Used:
To find the acceleration the formula is,
\[a = {\omega ^2}r\]
Where, r is radius and \[\omega \]is the angular velocity.

Complete step by step solution:
Consider a particle that is moving with constant speed v along a circular path of radius r and completes the circle in time T. We need to find the acceleration of the particle. The formula to find the acceleration of the particle is,
\[a = {\omega ^2}r\]……. (1)
The relation between the linear and angular velocity is given as,
\[v = r\omega \]
If we rewrite this equation for\[\omega \] then it will become,
\[\omega = \dfrac{v}{r}\]
Substitute the value of \[\omega \] in equation (1) we obtain,
\[a = {\left( {\dfrac{v}{r}} \right)^2}r\]
\[\Rightarrow a = \dfrac{{{v^2}}}{r}\]
\[\Rightarrow a = \omega v\]
\[\therefore a= \dfrac{2\pi}{T}v\]
Therefore, the acceleration of the particle is \[\dfrac{{2\pi }}{T}v\].

Hence, option A is the correct answer.

Note: Here in this problem it is important to remember that if a particle is moving in a circular motion, then the acceleration is taken as angular acceleration. Angular Acceleration is defined as the rate of change of angular velocity with respect to the time taken. The angular velocity is usually expressed in radians per second square. It can be represented in the equation as, \[\alpha = \dfrac{{\Delta \omega }}{{\Delta t}}\]. This angular acceleration is also known as rotational acceleration.