A banked circular road has radius r = 50 m, tanθ = 3/4, coefficient of friction μ = 1/3 and g = 10 m/s². If the maximum safe speed satisfies 9v_max² = 6500N, find N.
Physics · Circular Motion · friction and circular motion
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Question

A banked circular road has radius r = 50 m, tanθ = 3/4, coefficient of friction μ = 1/3 and g = 10 m/s². If the maximum safe speed satisfies 9v_max² = 6500N, find N.

G1 Physics · Accepted Answer

At the maximum safe speed, friction acts down the slope and is limiting: f = μN_R. The standard result is v_max²/(rg) = (sinθ + μcosθ)/(cosθ − μsinθ). Since tanθ = 3/4, sinθ = 3/5 and cosθ = 4/5. Thus v_max²/(rg) = [(3/5)+(1/3)(4/5)] / [(4/5)−(1/3)(3/5)] = (13/15)/(3/5) = 13/9. Therefore v_max² = 50×10×13/9 = 6500/9. Hence 9v_max² = 6500, so N = 1.