A) Not a simple harmonic
B) Simple harmonic with amplitude \[\frac{a}{b}\]
C) Simple harmonic with amplitude\[\sqrt{{{a}^{2}}+{{b}^{2}}}\]
D) Simple harmonic with amplitude \[\frac{(a+b)}{2}\]
Correct Answer: C
Solution :
The two displacement equations are \[{{y}_{1}}=a\sin (\omega t)\] and \[{{y}_{2}}=b\cos (\omega t)=b\sin \left( \omega t+\frac{\pi }{2} \right)\] \[{{y}_{eq}}={{y}_{1}}+{{y}_{2}}\] \[=a\sin \omega t+b\cos \omega t\] \[=a\sin \omega t+b\sin \left( \omega t+\frac{\pi }{2} \right)\] Since the frequencies for both SHMs are same, resultant motion will be SHM. Now \[{{A}_{eq}}=\sqrt{{{a}^{2}}+{{b}^{2}}+2ab\,\cos \frac{\pi }{2}}\] \[\Rightarrow \]\[{{A}_{eq}}=\sqrt{{{a}^{2}}+{{b}^{2}}}\]
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