Question 57 of 194

A particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by; $x={a}_{1}\cos ⁡\omega t$ and $y={a}_{2}\cos ⁡2\omega t$ traces a curve given by :

ABCD

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Answer: (D)

$x={a}_{1}\cos ⁡\omega t$$y={a}_{2}\cos ⁡\left(2\omega t\right)$$={a}_{2}\left\{2{\cos }^{2}⁡\omega t-1\right\}$$={a}_{2}\left\{2{\left(\dfrac{x}{{a}_{1}}\right)}^{2}-1\right\}$$∴\dfrac{y}{{a}_{2}}=\dfrac{2{x}^{2}}{{a}_{1}^{2}}-1$$∴{a}_{1}^{2}y=2{a}_{2}{x}^{2}-{a}_{1}^{2}{a}_{2}$which is a parabola symmetrical about y-axis opening in upward direction