Oscillations - Result Question 5
6. When two displacements represented by $y_1=a \sin (\omega t)$ and $y_2=b \cos (\omega t)$ are superimposed the motion is:
[2015]
(a) simple harmonic with amplitude $\frac{a}{b}$
(b) simple harmonic with amplitude $\sqrt{a^{2}+b^{2}}$
(c) simple harmonic with amplitude $\frac{(a+b)}{2}$
(d) not a simple harmonic
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Answer:
Correct Answer: 6. (b)
Solution:
- (b) The two displacements equations are $y_1=a \sin (\omega t)$
$ \begin{aligned} & \text{ and } y_2=b \cos (\omega t)=b \sin (\omega t+\frac{\pi}{2}) \\ & \begin{matrix} y _{eq}=y_1+y_2 \\ \quad=a \sin \omega t+b \cos \omega t \end{matrix} \end{aligned} $
$ =a \sin \omega t+b \sin (\omega t+\frac{\pi}{2}) $
Since the frequencies for both SHMs are same, resultant motion will be SHM.
Now amplitude, $A _{e q}=\sqrt{a^{2}+b^{2}+2 a b \cos \frac{\pi}{2}}$
$ \Rightarrow A _{e q}=\sqrt{a^{2}+b^{2}} $
