A roller-coaster car and its passengers have a combined mass m. The car is released from rest at point P, a vertical distance H above the lowest point of a circular loop of radius r. The track is rigid and frictionless, and air resistance is negligible.
(a) Calculate the speed of the car at the bottom of the loop in terms of g, H, and any other necessary quantities.
(b) The designers wish the car to maintain continuous contact with the track all the way around the loop. Determine the smallest value of H that satisfies this requirement.
(c) Suppose the loop radius is doubled while all other design features remain unchanged. Qualitatively describe how your answer to part (b) is affected and explain the reason for the change.