A) \[{{\sin }^{-1}}\left\{ 1-{{e}^{2}}\left( 1-\cos {{\theta }_{0}} \right) \right\}\]
B) \[{{\cos }^{-1}}\left\{ 1-{{e}^{2}}\left( 1-\cos {{\theta }_{0}} \right) \right\}\]
C) \[{{\tan }^{-1}}\left\{ 1-{{e}^{2}}\left( 1-\cos {{\theta }_{0}} \right) \right\}\]
D) Zero
Correct Answer: B
Solution :
[b] Speed just before collision \[=\sqrt{2g\ell (1-\cos {{\theta }_{0}})}\] Speed just after collision \[=e\sqrt{2g\ell (1-\cos {{\theta }_{0}})}\] \[\frac{1}{2}m{{e}^{2}}[2g\ell (1-\cos {{\theta }_{0}})]=mg\ell (1-\cos \theta )\]From conservation of energy, \[\therefore \theta ={{\cos }^{-1}}\left\{ 1-{{e}^{2}}(1-\cos {{\theta }_{0}}) \right\}\]
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