1. Point Q is at a height R above the ground. Thus, the difference in height between points P & Q is 4 R. Hence the difference in gravitational potential energy of the block between these points is 4 mg R. Since the block starts from rest at P its kinetic energy at Q is equal to its change in potential energy. By the conservation of energy.
\(\frac{1}{2}\)mv2 = 4 mg R
v2 = 8g R
At Q, the only forces acting on the block are its weight ring acting
downward and the force M of the track on block acting in radial direction. Since the block is moving in a circular path, the normal reaction provides the centripetal force for circular motion.
N = \(\frac {mv^2}{R}\)= \(\frac {m \times 8gR}{R}\) = 8 mg
The loop must exert a force on block equal to eight times the blocks weight.
2. For the block to exert a force equal to its weight against the track at the top of the loop,
\(\frac {mv^2}{R}\)= 2 mg
or v2 = 2gR
The block must be released at a height of 3 R above the bottom of the loop.
