question_answer

A small solid sphere of radius \[r\] rolls down an incline without slipping which ends into a vertical loop of radius \[R\]. Find the height above the base so that it just loops the loop

A) \[\frac{5}{2}R\]           

B) \[\frac{5}{2}(R-r)\]

C) \[\frac{25}{10}(R-r)\]    

D) \[\frac{27}{10}R-\frac{17r}{10}\]

Correct Answer: D

Solution :

[d]  The minimum velocity at P, top of the loop, should be\[v=\sqrt{g(R-r)}\] , if the sphere keeps on rolling at top \[v=\omega R\] \[mgh=\frac{1}{2}m{{v}^{2}}+\frac{1}{2}I{{\omega }^{2}}+mg(2R-r)\] \[=\frac{1}{2}mg(R-r)+\frac{1}{2}\left( \frac{2}{5} \right)m{{R}^{2}}2{{\omega }^{2}}+mg(2R-r)\] \[=\frac{7}{10}mg(R-r)+mg(2R-r)\left[ \omega R=v=\sqrt{g(R-r)} \right]\] \[=\frac{mg}{10}(27R-17r)\,\,or\text{ }h=\frac{1}{10}(27R-17r)\]