First slide

Simple hormonic motion

Question

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

Moderate

A

Circle
B

Hyperbola
C

Ellipse
D

Straight line passing through origin

Solution

$x = a \sin ω t or \frac{x}{a} = \sin ω t . . . . . . (i) Velocity, v = \frac{d x}{d t} = a ω \cos ω t \frac{v}{a ω} = \cos ω t . . . . . . (i i) Squaring and adding (i) and (ii), we get \frac{x^{2}}{a^{2}} + \frac{v^{2}}{a^{2} ω^{2}} = \sin^{2} ω t + \cos^{2} ω t \frac{x^{2}}{a^{2}} + \frac{v^{2}}{a^{2} ω^{2}} = 1 It is an equation of ellipse. Hence, the graph between velocity and displacement is an ellipse. Momentum of the particle = m v ∴ The nature of graph of the momentum and displacement is same as that of velocity and displacement.$

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