A) \[2{{\pi }^{2}}m{{a}^{2}}{{v}^{2}}\]
B) \[{{\pi }^{2}}m{{a}^{2}}{{v}^{2}}\]
C) \[\frac{1}{4}m{{a}^{2}}{{v}^{2}}\]
D) \[4{{\pi }^{2}}m{{a}^{2}}{{v}^{2}}\]
Correct Answer: B
Solution :
[b] The kinetic energy of a particle executing S.H.M. is given by \[K=\frac{1}{2}m{{a}^{2}}{{\omega }^{2}}\,{{\sin }^{2}}\omega t\] \[=\,\,<\frac{1}{2}m{{\omega }^{2}}{{a}^{2}}{{\sin }^{2}}\omega t>\,=\frac{1}{2}m{{\omega }^{2}}{{a}^{2}}<{{\sin }^{2}}\omega t>\] \[=\frac{1}{2}m{{\omega }^{2}}{{a}^{2}}\left( \frac{1}{2} \right)\] \[=\frac{1}{4}m{{\omega }^{2}}{{a}^{2}}\,\,\,\,\,\,\,\,\,\left( \therefore <{{\sin }^{2}}\theta >=\frac{1}{2} \right)\] \[=\frac{1}{4}m{{a}^{2}}\,{{(2\pi v)}^{2}}\,\,\] \[(\therefore \,\,\omega =2\pi v)\] or \[<K>\,={{\pi }^{2}}m{{a}^{2}}{{v}^{2}}\]
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