question_answer

Fig. 4(HT).9 shows a loop-the-loop track of radius r. A car (without engine) starts from a platform at a distance h above the top of the loop and goes around the loop without falling off the track. Find the minimum value of h for a successful looping. Neglect friction.

Answer:

                Suppose \[\upsilon \] is speed of the car at the topmost point of the loop. \[\therefore \]   \[mgh=\frac{1}{2}m{{\upsilon }^{2}}\]    or    \[m{{\upsilon }^{2}}=2mgh\]                                  ...(i)   At the top of the loop, \[mg+R=\frac{m{{\upsilon }^{2}}}{r}=\frac{2mgh}{r}\] where R is the normal reaction.                    For h to be minimum, R = minimum = 0 ; \[\therefore \]   \[\frac{2mg{{h}_{\min .}}}{r}=mg\]            ;                       \[{{h}_{\min }}=r/2\]