Oscillations

JEE Main level, single correct, medium

49 of 194

Question 49 of 194

Two simple harmonic motions, as shown, are at right angles. They are combined to form Lissajous figures.
x(t)=Asin⁡(at+δ)x(t)=A \sin (a t+\delta)
y(t)=Bsin⁡(bt)y(t)=B \sin (b t)
Identify the correct match below

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Answer: (C) Parameters: A≠B,a=b;δ=π2;\mathrm{A} \neq \mathrm{B}, \mathrm{a}=\mathrm{b} ; \delta=\frac{\pi}{2} ; Curve: Ellipse

From the two mutually perpendicular S.H.M.'s, the general equation of Lissajous figure,
x2A2+y2B2−2xyABcos⁡δ=sin⁡2δ\frac{x^2}{A^2}+\frac{y^2}{B^2}-\frac{2 x y}{A B} \cos \delta=\sin ^2 \delta

x=Asin⁡(at+δ)y=Bsin⁡(bt+r)\begin{aligned} &x=A \sin (a t+\delta) \\ &y=B \sin (b t+r) \end{aligned}
Clearly A≠BA \neq B hence ellipse.

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