First slide

Simple harmonic motion

Question

A particle is acted simultaneously by mutually perpendicular simple harmonic motion. $x = a \cos ω t$ and $y = a \sin ω t$ . The trajectory of motion of the particle will be

Moderate

A

an ellipse
B

a parabola
C

a circle
D

a straight line

Solution

$Given, x = a \cos ω t$ ……(i)
$y = a \sin ω t$ ……..(ii)
Squaring and adding Eqs. (i) and (ii),
$x^{2} + y^{2} = a^{2} (\cos^{2} ω t + \sin^{2} ω t)$
$\Rightarrow x^{2} + y^{2} = a^{2} [∵ \cos^{2} ω t + \sin^{2} ω t = 1]$
This is the equation, of a circle
Clearly, the locus is a circle of constant radius a.

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