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Circular Motion & Highway Curves

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Sect. 5-3: Highway Curves: Banked & Unbanked Case 1 - Unbanked Curve: When a car rounds a curve, there MUST be a net force toward the circle center ( a Centripetal Force ) of which the curve is an arc. If there werent such a force, the car couldnt follow the curve, but would (by Newtons 2 nd Law) go in a straight line. On a flat road, this Centripetal Force is the static friction force. Centripetal Force No static friction? No Centripetal Force The Car goes straight! There is NEVER a Centrifugal Force!!! =

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Example 5-6: Skidding on a curve A car, mass m = 1,000 kg car rounds a curve on a flat road of radius r = 50 m at a constant speed v = 14 m/s (50 km/h). Will the car follow the curve, or will it skid? Assume: a. Dry pavement with the coefficient of static friction μ s = 0.6. b. Icy pavement with μ s = Newtons 2 nd Law: F = ma x: F x = ma x F fr = ma R = m(v 2 /r) y: F y = ma y = 0 F N - mg = 0; F N = mg The maximum static friction is F fr = μ s F N Free Body Diagram

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If the friction force isnt sufficient, the car will tend to move more nearly in a straight line ( Newtons 1 st Law ) as the skid marks show. As long as the tires dont slip, the friction is static. If the tires start to slip, the friction is kinetic, which is bad in 2 ways!! 1. The kinetic friction force is smaller than the static friction force. 2. The static friction force points toward the circle center, but the kinetic friction force opposes the direction of motion, making it difficult to regain control of the car & continue around the curve.

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Case 2- Banked Curve Banking curves helps keep cars from skidding. For every banked curve, there is one speed v at which the entire Centripetal Force is supplied by the horizontal component of the normal force F N, so that no friction is required!! Newtons 2 nd Law Tells us what speed v this is: x: F x = ma x F Nx = m(v 2 /r) or Also y: F y = ma y = 0 F N cosθ - mg = 0 F N cosθ = mg So F N = (mg/cosθ) Put this into the x equation: g(sinθ/cosθ) = (v 2 /r) or tanθ = (v 2 /gr)

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Example 5-7: Banking angle a. For a car traveling with speed v around a curve of radius r, find a formula for the angle θ at which a road should be banked so that no friction is required. b. Calculate this angle for an expressway off-ramp curve of radius r = 50 m at a design speed of v = 14 m/s (50 km/h). Newtons 2 nd Law x: F x = ma x F Nx = m(v 2 /r) or F N sinθ = m(v 2 /r) (1) y: F y = ma y = 0 F N cosθ - mg = 0 or F N cosθ = mg (2) Dividing (2) by (1) gives: tanθ = [(v 2 )/(rg)] Putting in the given numbers tanθ = 0.4 or θ = 22º

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