Question

For a particle in SHM, if the amplitude of the displacement is $\alpha$ and the amplitude of velocity is $v$ , the amplitude of acceleration is
A. $v\alpha$
B. $\dfrac{{{v^2}}}{\alpha }$
C. $\dfrac{{{v^2}}}{{2\alpha }}$
D. $\dfrac{v}{\alpha }$

Answer 1

We need to apply the equation of displacement of the particle from its equilibrium location in simple harmonic motion to get the expressions of velocity and acceleration of a particle in simple harmonic motion. The expression of velocity is given by the first derivative of the displacement with respect to time, and the expression of acceleration is given by the second derivative of the displacement with respect to time.

Complete step by step answer:
The SHM would have the form, because the amplitude of displacement of the SHM is $\alpha$ :
$x = \alpha \sin \left( {\omega t + \phi } \right)$
Thus, $v = \dfrac{{dx}}{{dt}} = \omega \alpha \cos \left( {\omega t + \phi } \right)$
$\Rightarrow$ Amplitude of velocity $= \omega \alpha = v$
$\Rightarrow \omega = \dfrac{v}{a}$
Similarly, in SHM, the particle's acceleration is
$\dfrac{{dv}}{{dt}} = - {\omega ^2}\alpha \sin \left( {\omega t + \phi } \right)$
The negative sign meant that acceleration and displacement were moving in opposite directions. The motion of a particle executing S.H.M. is accelerated in practise because its velocity changes by a constant or variable number.
$\Rightarrow$ Amplitude of acceleration $= {\omega ^2}\alpha = {\left( {\dfrac{v}{\alpha }} \right)^2}\alpha = \dfrac{{{v^2}}}{\alpha }$
Hence, the correct option is: (B) $\dfrac{{{v^2}}}{\alpha }$ .

Additional Information:
1. At the equilibrium position, SHM, the velocity is at its highest.
2. At the extreme position, velocity is zero.
3. Furthermore, when the velocity is maximal, the acceleration is zero, which means the acceleration is zero anytime the item is in its initial position or the particle's displacement is zero.
4. The acceleration is greatest at the extreme position because the displacement is greatest.

Note:
It's important not to mix up acceleration with maximum acceleration, as well as speed and maximum speed. The formulas for the two terms are not the same. When calculating frequency from angular velocity, be cautious.