question_answer

A body slides without friction from a height h along a track shown in figure, so that it loops the loop in the vertical plane. If the radius of the loop is R, what should be the minimum value of h in terms of R, so that the body is just able to loop the loop. There is no friction between the body and the track :

A)  h=R                                      

B)  h = 2R

C) \[h=\frac{5}{2}R\]                           

D)  h = 4R

Correct Answer: C

Solution :

                Let v be velocity of the body at C, then \[\frac{m{{v}^{2}}}{R}=mg\]                       ?.. (i) At A the potential energy is\[mgh\]and at C it is mg (2r). From law of conservation of energy, this loss in potential energy \[mg(h-2R),\]will be equal to kinetic energy of the ball at C, that is \[mg(h-2R)=\frac{1}{2}m{{v}^{2}}\]                  ...(ii) From Eqs. (i) and (ii), we get                 \[h=\frac{5}{2}R\]