A) \[2\pi \sqrt{\frac{{{\operatorname{v}}_{1}}^{2}-{{\operatorname{v}}_{2}}^{2}}{{{\operatorname{x}}_{1}}^{2}-{{\operatorname{x}}_{1}}^{2}}}\]
B) \[2\pi \sqrt{\frac{{{\operatorname{x}}_{1}}^{2}+{{\operatorname{x}}_{1}}^{2}}{{{\operatorname{v}}_{1}}^{2}+{{\operatorname{v}}_{2}}^{2}}}\]
C) \[2\pi \sqrt{\frac{{{\operatorname{x}}_{1}}^{2}-{{\operatorname{x}}_{1}}^{2}}{{{\operatorname{v}}_{1}}^{2}-{{\operatorname{v}}_{2}}^{2}}}\]
D) \[2\pi \sqrt{\frac{{{\operatorname{v}}_{1}}^{2}+{{\operatorname{v}}_{2}}^{2}}{{{\operatorname{x}}_{1}}^{2}+{{\operatorname{x}}_{1}}^{2}}}\]
Correct Answer: D
Solution :
particle velocity executing SHM is given by \[\operatorname{v}=\sqrt{{{A}^{2}}-{{x}^{2}}}\] \[{{\operatorname{v}}_{1}}=\omega \sqrt{{{A}^{2}}-{{x}^{2}}_{1}}{{\operatorname{v}}_{2}}=\omega \sqrt{{{A}^{2}}-{{x}^{2}}_{2}}\] \[{{\operatorname{v}}_{1}}^{2}={{\omega }^{2}}{{A}^{2}}-{{\omega }^{2}}-{{x}^{2}}_{1}{{\operatorname{v}}_{2}}^{2}={{\omega }^{2}}\left( {{A}^{2}}-{{x}^{2}}_{2} \right)\] \[{{\operatorname{v}}_{1}}^{2}={{\omega }^{2}}{{A}^{2}}-{{\omega }^{2}}{{x}^{2}}_{1}{{\operatorname{v}}_{2}}^{2}={{\omega }^{2}}{{A}^{2}}-{{\omega }^{2}}{{x}^{2}}_{2}\]\[{{\operatorname{v}}_{1}}^{2}{{\operatorname{v}}_{2}}^{2}={{\omega }^{2}}{{x}^{2}}_{2}-{{\omega }^{2}}{{\operatorname{x}}^{2}}\] \[{{\operatorname{v}}_{1}}^{2}-{{\operatorname{v}}_{2}}^{2}={{\omega }^{2}}\left( {{x}^{2}}_{2}-{{\operatorname{x}}^{2}}_{1} \right)\] \[\omega ={{\frac{{{\operatorname{v}}_{1}}^{2}-{{\operatorname{v}}_{2}}}{{{x}^{2}}_{2}-{{\operatorname{x}}^{2}}}}^{2}}\] \[\operatorname{T}=2\omega \sqrt{\frac{{{x}^{2}}_{2}-{{\operatorname{x}}^{2}}_{1}}{{{\operatorname{v}}_{1}}^{2}-{{\operatorname{v}}_{2}}^{2}}}\]
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