A) \[\sin \,\,\omega t - cos \omega t\]
B) \[si{{n}^{2}}\,\omega t\]
C) \[\sin \,\,\omega \,t +sin 2\,\,\omega t\]
D) \[\sin \,\,\omega \,t -sin 2\,\,\omega t\]
Correct Answer: A
Solution :
One of the conditions for SHM is that restoring force (F) and hence acceleration should be proportional to displacement (y). Let, \[\operatorname{y}\, =\,\,sin\,\,\omega t -\,\,\cos \omega t\] \[\frac{dy}{dt}=\omega \cos \,\,\omega t+\omega \,\,\sin \,\,\omega t\] \[\frac{{{d}^{2}}y}{d{{t}^{2}}}=-{{\omega }^{2}}\sin \,\,\omega t\,\,+\,\,{{\omega }^{2}}\,\cos \,\,\omega t\] or \[\operatorname{a} =- {{\omega }^{2}}\,(sin \omega t\,\,-\,\,cos\,\,\omega t)\] \[\operatorname{a} = -{{\omega }^{2}}y\] \[\Rightarrow \,\,\,\,\,a\propto -y\] This satisfies the condition of SHM. Other equations do not satisfy this condition.
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