question_answer

A small body of mass m slides without friction from the top of a hemisphere of radius r. The point at which the body will be detached from the surface of hemisphere, is :

A) \[\frac{r}{3}\]                                   

B) \[\frac{r}{2}\]

C)  \[\frac{2}{3}r\]                                

D)  2r

Correct Answer: B

Solution :

                Body will be detached if \[\frac{m{{\upsilon }^{2}}}{r}=mg\,\,\cos \theta \]                   ??(i) As           \[{{\upsilon }^{2}}-{{0}^{2}}=2gd\]                 \[\upsilon =\sqrt{2gd}\] and        \[\cos \theta =\frac{h}{r}\] So, from equation (i),                 \[\frac{m\times 2gd}{r}=mg\frac{h}{r}\] or            \[h=2d\] or            \[h=2(r-h)\] i.e.,        \[h=(2/3)r\]