A) \[T={{t}_{1}}+{{t}_{2}}\]
B) \[{{T}^{2}}=t_{1}^{2}+t_{2}^{2}\]
C) \[{{T}^{-1}}=t_{1}^{-1}+t_{2}^{-1}\]
D) \[{{T}^{-2}}=t_{1}^{-2}+t_{2}^{-2}\]
Correct Answer: B
Solution :
\[{{t}_{1}}=2\,\pi \sqrt{\frac{m}{{{k}_{1}}}}\] and \[{{t}_{2}}=2\,\pi \sqrt{\frac{m}{{{k}_{2}}}}\] In series, effective spring constant, \[k=\frac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\] \[\therefore \]Time period, \[T=2\,\pi \sqrt{\frac{m\,({{k}_{1}}+{{k}_{2}})}{{{k}_{1}}{{k}_{2}}}}\] ?..(ii) Now, \[t_{1}^{2}+t_{2}^{2}=4\,{{\pi }^{2}}m\,\left( \frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{2}}} \right)=\frac{4\,{{\pi }^{2}}m\,({{k}_{1}}+{{k}_{2}})}{{{k}_{1}}{{k}_{2}}}\] \[t_{1}^{2}+t_{2}^{2}={{T}^{2}}.\] [Using equation (ii)]
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