Question

What is the ratio of maximum acceleration to the maximum velocity of a simple harmonic motion?

Answer 1

In order to this question, to find the ratio of maximum acceleration to the maximum velocity of S.H.M, we will first write the displacement equation, and then we will differentiate the equation to get the velocity and similarly again differentiate the velocity equation to get the acceleration. Now, we will easily find the ratio between them.

Complete answer:
Simple harmonic motion, or SHM, is a type of periodic motion in which the restoring force is proportional to the displacement and acts in the opposite direction as the displacement.
According to the simple harmonic motion-
The displacement will be, $x = A\sin (\omega t)$
Here, $\omega t$ is the phase of the motion.
Now, we have displacement already, so we will differentiate the above equation to get the velocity:-
Maximum velocity is, $v = \dfrac{{dx}}{{dt}} = A\omega \cos (\omega t)$
Now, we have the maximum velocity, so we will again differentiate to get the maximum acceleration:-
Maximum acceleration is, $a = \dfrac{{{d^2}x}}{{d{t^2}}} = - A{\omega ^2}\sin (\omega t)$
Now, we have both maximum velocity and maximum acceleration as well, so we can find the of maximum acceleration to the maximum velocity-
$\therefore |\dfrac{a}{v}| = \dfrac{{A{\omega ^2}}}{\omega } = \omega$
Hence, the ratio of maximum acceleration to the maximum velocity of a simple harmonic motion is $\omega$ .

Note: The ratio of maximum acceleration to the maximum velocity of a simple harmonic motion is also known as the angular frequency. The number of revolutions an object makes in a given unit of time is known as angular frequency. In that respect, it is similar to frequency, but in terms of how many times a whole period of motion is turned in radian units.