The ratio of maximum acceleration to maximum velocity in a simple harmonic motion is $10{s^{ - 1}}$. At, $t = 0$ the displacement is $5m$. What is the maximum acceleration? The initial phase is $\dfrac{\pi }{4}$​.
$\left( a \right)$ $500m/{s^2}$
$\left( b \right)$ $750\sqrt 2 m/{s^2}$
$\left( c \right)$ $750m/{s^2}$
$\left( d \right)$ $500\sqrt 2 m/{s^2}$

Hint So this problem can be solved by using the equation of harmonic which is $y = A\sin \left( {\omega t + \dfrac{\pi }{4}} \right)$and also by using the velocity formula which is equal to$v = A\omega \cos \left( {\omega t + \dfrac{\pi }{4}} \right)$. So by using both of these and equating we will get the maximum acceleration.
Formula used
The harmonic equation can be represented by
$y = A\sin \left( {\omega t + \dfrac{\pi }{4}} \right)$
Velocity,
$v = A\omega \cos \left( {\omega t + \dfrac{\pi }{4}} \right)$
And acceleration will be equal to
$a = - A{\omega ^2}\sin \left( {\omega t + \dfrac{\pi }{4}} \right)$
Here,
$y$, will be the harmonic motion
$v$, will be the velocity
$a$, will be the acceleration
And $\omega $, will be the angular velocity.

Complete Step By Step Solution We have been given the ratio between the maximum acceleration and the maximum velocity. And there is the displacement also given. So we have to just find the maximum acceleration.
 Let the equation for the harmonic motion will be represented as
$y = A\sin \left( {\omega t + \dfrac{\pi }{4}} \right)$
And also we know,
Velocity is equal to,
$v = A\omega \cos \left( {\omega t + \dfrac{\pi }{4}} \right)$
And the acceleration of the body will be equal to
$a = - A{\omega ^2}\sin \left( {\omega t + \dfrac{\pi }{4}} \right)$
Since we have been given that the ratio of maximum acceleration to maximum velocity in a simple harmonic motion is$10{s^{ - 1}}$ which will be represented by $\omega $
Therefore, substituting the value of time
That is $t = 0$,
Inharmonic motion equation, we get
$5 = A\sin \left( {\dfrac{\pi }{4}} \right)$
So from here,
$ \Rightarrow A = 5\sqrt 2 $
Now, we will find the maximum acceleration and for this we have
Maximum acceleration $ = A{w^2}$
Therefore, substituting the values, we get
$ \Rightarrow 5\sqrt 2 \times 10 \times 10$
On solving the above, the final value will be equal to
$ \Rightarrow 500\sqrt 2 m/{s^2}$

Therefore, $500\sqrt 2 $is the maximum acceleration, and hence, the option $D$is the right choice.

Note When harmonic motion occurs harmonic functions are the waveforms of the motions that result. In other cases, for example, various musical instruments, when the player stimulates the instrument in a certain way, multiple wavelengths of response are being stimulated at the same time. In that case, the waveform produced will be a combination of different frequencies.