Oscillations - Result Question 22
25. The composition of two simple harmonic motions of equal periods at right angle to each other and with a phase difference of $\pi$ results in the displacement of the particle along [1990]
(a) circle
(b) figures of eight
(c) straight line
(d) ellipse
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Answer:
Correct Answer: 25. (c)
Solution:
- (c) $x=a \sin \omega t$ and $y=b \sin (\omega t+\pi)=-b \sin \omega t$
or, $\frac{x}{a}=-\frac{y}{b}$ or $y=-\frac{b}{a} x$
It is an equation of a straight line.
If two mutually perpendicular S.H.M’s of same frequency be $x=9$, sin $\omega t$ and $y=a_2 \sin (\omega t+\phi)$ then general equation of Lissajous figure is
$ \begin{aligned} & \frac{x^{2}}{a_1^{2}}+\frac{y_2}{a_2^{2}}-\frac{2 x y}{a_1 a_2} \cos \phi=\sin ^{2} \phi \\ & \text{ For } \phi=\pi \frac{x^{2}}{a_1^{2}}+\frac{y^{2}}{a_2^{2}}-\frac{2 x y}{a_1 a_2} \cos (\pi)=\sin 2(\pi) \\ & \Rightarrow \frac{x}{a_1}+\frac{y}{a_2}=0 \quad \frac{x}{a}+\frac{y}{a^{2}}=0 \Rightarrow y=\frac{-a_2}{a_1} x \end{aligned} $
