Question

Maximum acceleration of an object in simple harmonic motion is $24m/{s^2}$ and maximum velocity is $16m/s.$ Then amplitude of SHM is
$(1)\,\dfrac{{32}}{3}m$
$(2)\,\dfrac{3}{{32}}m$
$(3)\,\dfrac{2}{3}m$
$(4)\,\dfrac{3}{2}m$

Answer 1

Hint: The simple Harmonic Motion is an Oscillatory motion under which the retarding force is proportional to the amount of the displacement from an equilibrium position. The Amplitude is the maximum displacement of an object from the equilibrium position. Here we will find the correlation between the maximum velocity and the maximum acceleration to find the amplitude.

Complete step by step answer:
Any simple Harmonic Motion can be expressed by –
$x = A\sin (\omega t)$
Where A is the amplitude and $\omega $ is the angular frequency.
We know that the Velocity is the change in displacement with respect to time.
$v = \dfrac{{dx}}{{dt}} = \dfrac{d}{{dt}}\left( {A\sin (\omega t)} \right)$
The maximum velocity, ${v_{\max }} = A\omega $
Given that –
$
  {v_{\max }} = A\omega = 16m/s \\
  A\omega = 16m/s{\text{ }}....{\text{ (a)}} \\
 $
Also, the acceleration is the change in the velocity with respect to time.
Acceleration $ = \dfrac{{dv}}{{dt}} = - A{\omega ^2}\sin (\omega t)$
So, the maximum acceleration
 $
  {a_{\max }} = A{\omega ^2} = 24m/{s^2} \\
  A{\omega ^2} = 24m/{s^2}{\text{ }}.....{\text{ (b)}} \\
 $
Take ratio of the equation (a) and the equation (b)
$\dfrac{{A\omega }}{{A{\omega ^2}}} = \dfrac{{16}}{{24}}$
Same terms from the denominator and the numerator cancel each other, so remove “A” and from the numerator and the denominator.
$ \Rightarrow \dfrac{1}{\omega } = \dfrac{2}{3}$
Do-cross multiplication and make the term the subject –
$ \Rightarrow \omega = \dfrac{3}{2}{s^{ - 1}}$
Place the above value in the equation (a)-
$A \times \dfrac{3}{2} = 16$
Make the subject “A” –
$ \Rightarrow A = 16 \times \dfrac{2}{3}$
Simplify the above equation –
$ \Rightarrow A = \dfrac{{32}}{3}m$

So, the correct answer is “Option 1”.

Additional Information:
Both the acceleration and the velocity are the vector quantities and therefore it can have positive, negative values. It may be zero also. If the acceleration is negative, it is called decelerating of the object.

Note:
Always remember the correct formula for the maximum acceleration and the velocity. When there is maximum displacement of the angle, the sine function is always equal to one. Always frame the correct equation based on the given data, check it twice. Do not forget to place the correct unit in the resultant answer.