A particle of mass m oscillates along x -axis according to equation $x=a\sin \omega t$. The nature of the graph between momentum and displacement of the particle is

$$\begin{gathered} x=a\sin \omega t\text{ or }\frac{x}{a}=\sin \omega t ......\left(i\right) \\ \text{ Velocity, }v=\frac{dx}{dt}=a\omega \cos \omega t \\ \frac{v}{a\omega }=\cos \omega t ......\left(ii\right) \\ \text{ Squaring and adding (i) and (ii), we get } \\ \frac{x^2}{a^2}+\frac{v^2}{a^2{\omega }^2}={\sin }^2\omega t+{\cos }^2\omega t \\ \frac{x^2}{a^2}+\frac{v^2}{a^2{\omega }^2}=1 \\ \text{ It is an equation of ellipse. } \\ \text{ Hence, the graph between velocity and displacement is an ellipse.} \\ \text{ Momentum of the particle }=mv \\ \therefore \text{ The nature of graph of the momentum and displacement is same as that of velocity and displacement. } \end{gathered}$$