A 2000-kg car is traveling on a banked curved icy road. The road is banked at an

Question

A 2000-kg car is traveling on a banked curved icy road. The road is banked at an angle of 12.0 degrees and has a radius of curvature of 500 m. What is the velocity of the car necessary to travel on the icy road without sliding.

Explanation / Answer

Since the road ic icy, it may be considered frictionless, and since the car is on a curve, there must be a centripital acceleration. You may ask, "What force is causing the centripital acceleration?" The answer is the normal force, specifically the the horizontal component of the normal force. A free-body diagram of the car on the banked curve would show that the normal force acts perpendicular to the road. The horizontal component of the normal force is nsin?, so from Newton's 2nd law, the net centripital force is : ?F(c) = ma(c) = nsin?---->equation (1) We don't know the value of n, but we may find it from the net vertical forces : ?F = 0 = ncos? - mg Solving for n : n = mg / cos?----->equation (2) Plugging (2) into (1) : ma(c) = (mg / cos?)sin? Recall that centripital acceleration, a(c), is equal to v² / r. The equation above becomes : mv² / r = (mg / cos?)sin? But sin? / cos? = tan?, so : v² = rgtan? v = vrgtan? = v(500m x 9.8m/s² x tan12° = 32m/s (rounded)

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