A) \[2\pi \sqrt{\frac{x_{1}^{2}+x_{2}^{2}}{v_{1}^{2}+v_{2}^{2}}}\]
B) \[2\pi \sqrt{\frac{x_{2}^{2}-x_{1}^{2}}{v_{1}^{2}-v_{2}^{2}}}\]
C) \[2\pi \sqrt{\frac{v_{1}^{2}+v_{2}^{2}}{x_{1}^{2}+x_{2}^{2}}}\]
D) \[2\pi \sqrt{\frac{v_{1}^{2}-v_{2}^{2}}{x_{1}^{2}-x_{2}^{2}}}\]
Correct Answer: B
Solution :
Let A be the amplitude of oscillation then \[v_{1}^{2}={{\omega }^{2}}({{A}^{2}}-x_{1}^{2})\,\,\] ?(i) \[v_{2}^{2}={{\omega }^{2}}({{A}^{2}}-{{x}^{2}})\,\,\,\] ?(ii) Subtracting Eq. (ii) from Eq. (i), we get \[v_{1}^{2}-v_{2}^{2}={{\omega }^{2}}(x_{2}^{2}-x_{1}^{2})\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\omega =\sqrt{\frac{v_{1}^{2}-v_{2}^{2}}{x_{2}^{2}-x_{1}^{2}}}\] \[\Rightarrow \,\,\,\,\,\,\frac{2\pi }{T}=\sqrt{\frac{v_{1}^{2}-v_{2}^{2}}{x_{2}^{2}-x_{1}^{2}}}\] \[\Rightarrow \,\,\,\,\,T=2\pi \sqrt{\frac{x_{2}^{2}-x_{1}^{2}}{v_{1}^{2}-v_{2}^{2}}}\]
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