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7.46. Riding a Loop-the-Loop. A car in an amusement park ride rolls without friction around the track shown in the figure. It starts from rest at point A at a height h above the bottom of the loop. Treat the car as a particle. (a) What is the minimum value of h (in terms of R) such that the care moves around the loop without falling off at the top (point B)? (b) If h = 3.50R and R = 20.0 m, compute the speed, radial acceleration, and tangential acceleration of the passengers when the car is at point C, which is at the end of a horizontal diameter. Show these acceleration components in a diagram, approximately to scale. F ma Identify: Apply K1 + U1 + WNC = K2 + U2 to relate h and vB . Apply the minimum speed required at B for the car not to fall off the track. at point B to find 2 2 At B, a vB / R , downward. The minimum speed is when n 0 and mg mvB / R . The Set up: minimum speed required is vB gR . K1 0 and Wother 0 . Execute: 2 1 (a) KA + UA = KB + UB applied to points A and B gives U A U B 2 mvB . The speed at the top must be at least gR . 1 5 mg (h 2R) mgR, or h R. 2 2 Thus, (b) Apply KA + UA = KC + UC to points A and C. U A UC (2.50) Rmg K C , so vC (5.00) gR (5.00)(9.80 m/s2 )(20.0 m) 31.3 m/s. arad vC2 49.0 m/s 2 . R The tangential direction is down, the normal The radial acceleration is force at point C is horizontal, there is no friction, so the only downward force is 2 gravity, and atan g 9.80 m/s . Evaluate: If h 2 R , then the downward acceleration at B due to the circular motion is greater than g and the track must exert a downward normal force n. n increases as h increases and 5 hence vB increases.