At the top of a vertical loop, when v equals the minimum speed to maintain contact, what is the normal force on the car?

At the top of a vertical loop, when v equals the minimum speed to maintain contact, what is the normal force on the car?

• A. N = 0
• B. N = m g
• C. N = 2 m g
• D. N = m g - m v^2 / R

Answer: (A)

At the top of the loop, both gravity and the normal force point toward the center of the circle (downward). The car’s inward (centripetal) acceleration is v^2/R, so the sum of inward forces must equal m v^2/R: N + mg = m v^2/R. The normal force is a contact force and cannot pull the car toward the center, so it cannot be negative. The minimum speed to stay in contact occurs when the track just stops pushing on the car, i.e., N = 0. Setting N to zero gives m v^2/R = mg, so v^2 = gR. Under this threshold condition, gravity alone supplies the needed centripetal force, and the normal force is zero. Hence the normal force at the top when the speed is at the minimum to maintain contact is zero. Other expressions would describe other speeds (when N would be positive for faster speeds), but at the exact minimum speed that keeps contact, the normal force vanishes.

At the top of the loop, both gravity and the normal force point toward the center of the circle (downward). The car’s inward (centripetal) acceleration is v^2/R, so the sum of inward forces must equal m v^2/R: N + mg = m v^2/R. The normal force is a contact force and cannot pull the car toward the center, so it cannot be negative. The minimum speed to stay in contact occurs when the track just stops pushing on the car, i.e., N = 0. Setting N to zero gives m v^2/R = mg, so v^2 = gR. Under this threshold condition, gravity alone supplies the needed centripetal force, and the normal force is zero. Hence the normal force at the top when the speed is at the minimum to maintain contact is zero.

Other expressions would describe other speeds (when N would be positive for faster speeds), but at the exact minimum speed that keeps contact, the normal force vanishes.