A) \[\left( \frac{m}{{{m}_{1}}} \right)\sqrt{5g\ell }\]
B) \[\frac{3}{2}\left( \frac{m}{{{m}_{1}}} \right)\sqrt{5g\ell }\]
C) \[\frac{2}{3}\left( \frac{{{m}_{1}}}{m} \right)\sqrt{5g\ell }\]
D) \[\left( \frac{{{m}_{1}}}{m} \right)\sqrt{g\ell }\]
Correct Answer: B
Solution :
Using conservation of momentum, \[{{m}_{1}}{{v}_{1}}+0={{m}_{1}}\frac{{{v}_{1}}}{3}+mv\Rightarrow {{m}_{1}}{{v}_{1}}-\frac{{{m}_{1}}{{v}_{1}}}{3}=mv\] \[(v=\sqrt{5g1}\]to complete motion along vertical circle) \[\frac{2{{m}_{1}}{{v}_{1}}}{3}=mv\Rightarrow v=\frac{2}{3}\frac{{{m}_{1}}{{v}_{1}}}{m}\] or\[{{v}_{1}}=\frac{3}{2}\frac{m}{{{m}_{1}}}\sqrt{5g\ell }\]
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