A car rounds a banked curve as in the figure below. The radius of curvature of t

Question

A car rounds a banked curve as in the figure below. The radius of curvature of the road is R, the banking angle is theta, and the coefficient of static friction is mu_s. (a) Determine the range of speeds the car can have without slipping up or down the banked surface. (Use the following as necessary: theta, mu_s, R, and g.) V_min = V_max = (b) Find the minimum value for mu_s such that the minimum speed is zero. (Use the following as necessary: theta, R, and g.) mu_s = (c) What is the range of speeds possible if R = 100 m, theta = 11.5 degree, and mu_s = 0.120 (slippery conditions)theta V_min = m/s V_max = m/s

Explanation / Answer

(A).

F_vertical = ma

N sin - µs N cos = m (Vmin)²/R

N(sin - µs cos ) = m (Vmin)²/R

[mg/(µs sin + cos )](sin - µ cos ) = m (Vmin)²/R

divided both side by m/cos ,

g(tan - µs)/(µs tan + 1) = (Vmin)²/R

Vmin = [gR(tan - µs)/(µs tan + 1)]
Vmax = [gR(tan + µs)/(-µs tan + 1)]

(B).

if Vmin = 0

Fx = 0

µs m g cos - mg sin = 0

divided both side by m g cos ,

µs = tan

(C).

Vmin = [gR(tan - µs)/(µs tan + 1)]

Vmax = [gR(tan + µs)/(-µs tan + 1)]

Vmin = [9.8*100(tan 11.5 - 0.120)/(0.120* tan 11.5 + 1)] = 8.9349 m/s

Vmax = [9.8*100(tan 11.5 + 0.120)/(-0.120* tan 11.5 + 1)] = 18.0254m/s

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