Question

Maximum speed of a particle in simple harmonic motion is v${}_{max}$.The average speed of particle in SHM is equal to
A. ${\displaystyle \frac{v_{max}}{2}}$
B. ${\displaystyle \frac{v_{max}}{\pi }}$
C. ${\displaystyle \frac{\pi v_{max}}{2}}$
D. ${\displaystyle \frac{2v_{max}}{\pi }}$

Answer 1

Hint:We know that the general equation for a particle performing SHM is $x=A\sin \left(\omega t+\varphi \right)$. On differentiating we get the equation for velocity $v\left(t\right)=|A\omega \cos \left(\omega t+\varphi \right)|$. To get the average speed of the particle we integrate this equation for one time period, avg speed $={\displaystyle \frac{1}{T}}\underset{0}{\overset{T}{\int }}v(t)dt$.

Complete step by step answer:
We know that the general equation for a particle performing simple harmonic motion is given as $x=A\sin \left(\omega t+\varphi \right)$.
Where x is the displacement, A is the amplitude, $\omega$ is the angular frequency, $\varphi$ is the initial phase and t is time.

Differentiating this equation with respect to time to get the equation for velocity $v\left(t\right)=|A\omega \cos \left(\omega t+\varphi \right)|$.
To get the average speed of the particle we use the following equation,
avg speed$={\displaystyle \frac{1}{T}}\underset{0}{\overset{T}{\int }}v(t)dt$.
Substituting the values we get,
avg speed$={\displaystyle \frac{1}{T}}\underset{0}{\overset{T}{\int }}A\omega \cos \left(\omega t+\varphi \right)$
We know that time period T can be written as ${\displaystyle \frac{2\pi }{\omega }}$.

Therefore the equation becomes,
Avg speed $={\displaystyle \frac{1}{2\pi /\omega }}\underset{0}{\overset{{\displaystyle \frac{2\pi }{\omega }}}{\int }}A\omega \cos \left(\omega t+\varphi \right)$
Avg speed $={\displaystyle \frac{2A\omega }{\pi }}$
Max value of velocity is obtained when $\cos \left(\omega t+\varphi \right)=1$ and is equal to $v_{max}=A\omega$

Therefore the average speed of particles in SHM is ${\displaystyle \frac{2v_{max}}{\pi }}$.

Note:For a particle performing SHM the acceleration of the body is directly proportional to the displacement of the body from the mean position. The general equation for displacement of a particle performing simple harmonic motion is $x=A\sin \left(\omega t+\varphi \right)$. Differentiating this equation we get the equation for velocity with time and acceleration of the particle. Average speed is defined as the rate of change of distance with respect to time. It can be written as a ratio of total distance travelled to the total time taken.