A particle is executing a simple harmonic motion. Its maximum acceleration is $\alpha $ and maximum velocity is $\beta $ . Then, its time period of vibration will be:
A. $\dfrac{{2\pi \beta }}{\alpha }$
B. $\dfrac{{{\beta ^2}}}{{{\alpha ^2}}}$
C. $\dfrac{\alpha }{\beta }$
D. $\dfrac{{{\beta ^2}}}{\alpha }$

Hint: Here we have to use the concepts of simple harmonic motion.
Simple harmonic motion can be defined as an oscillatory motion in which the acceleration of the particle at any point is directly proportional to the displacement of the mean position.

Complete step by step answer:
Maximum acceleration in simple harmonic motion is given by:
${a_{\max }} = {\omega ^2}a = \alpha $
Maximum velocity in simple harmonic motion is given by:
${v_{\max }} = \omega a = \beta $
So,
$
 \dfrac{\alpha }
{\beta } = \dfrac{{{\omega ^2}a}}
{\omega } \\
 = \omega \\
 = \dfrac{{2\pi }}
{T} \\
$
Hence, the time period of vibration is:
$T = \dfrac{{2\pi \beta }}{\alpha }$

So, the correct answer is “Option A”.

Additional Information:
- Not all oscillatory motions are simple harmonic in nature whereas all harmonic motions are periodic and oscillatory. Oscillatory motion is often referred to as the harmonic motion of all oscillatory movements, the most important of which is basic harmonic motion.
- In a simple harmonic motion the return of force or acceleration acting on the particle should always be equal to the displacement of the particle and be oriented towards the equilibrium state.
- In other words, the same equation refers to the position of an object experiencing basic harmonic motion and to one component of the position of an object experiencing uniform circular motion.
- In basic harmonic motion, the speed and displacement of the target is zero at an extreme location.

Note:
Here we have to remember that angular frequency $\omega = \dfrac{{2\pi }}{T}$ . Also if we exchange the values of numerator and denominator in confusion then our answer would be wrong.
In oscillatory motion displacement, velocity and acceleration and force differ in a manner that can be represented by either sine or cosine functions usually referred to as sinusoids.
The amplitude is actually the maximum displacement of the object from the direction of equilibrium.