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Q.
A particle of mass $m$ oscillates along $x$-axis according to equation $x = a \sin \, \omega$ . The nature of the graph between momentum and displacement of the particle is
$x=a \sin \omega t$ or $\frac{x}{a}=\sin \omega t \ldots$ (i)
Velocity, $v=\frac{d x}{d t}=a \omega \cos \omega t \ldots$ (ii)
$\frac{v}{a \omega}=\cos \omega t$
Squaring and adding (i) and (ii), we get
$\frac{x^{2}}{a^{2}}+\frac{v^{2}}{a^{2} \omega^{2}}=\sin ^{2} \omega t+\cos ^{2} \omega t$
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2} \omega^{2}}=1$
It is an equation of ellipse.
Hence, the graph between velocity and displacement is an ellipse.
Momentum of the particle $= mv$
$\therefore $ The nature of graph of the momentum and displacement is same as that of velocity and displacement.