The displacement of a particle is represented by the equation y = 3sinwt-4sin^3wt (where t is the time). The motion of the particle is
[Pls explain how to check whether motion is periodic or not, execute simple harmonic or not and how to calculate time period for this]

Dear Student,
A periodic motion is a motion that repeats after regular time intervals, called the period. If in a period motion, the acceleration $\frac{d^{2}y}{d{t}^2}$ is proportional to the displacement from a fixed point y and the acceleration is directed towards the fixed point, then the motion is simple harmonic.
Given, the displacement is
$$\begin{gathered} y=3\sin \left(\omega t\right)-4{\sin }^3\left(\omega t\right)=\sin \left(3\omega t\right) \\ \frac{dy}{dt}=3\omega \cos \left(3\omega t\right) \\ \frac{d^{2}y}{d{t}^2}=-9{\omega }^2\sin \left(3\omega t\right)=-(3\omega {)}^{2}y \\ Thus, \frac{d^{2}y}{d{t}^2}\propto y \end{gathered}$$
Since,  the acceleration $\frac{d^{2}y}{d{t}^2}$ is proportional to the displacement from a fixed point y and the acceleration is directed towards the fixed point, then the motion is simple harmonic having an angular frequency ${\omega }_o=3\omega$
The time period of this SHM is:
$T = \frac{2\pi }{{\omega }_o}=\frac{2\pi }{3\omega }$
Regards,
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