A car rounds a banked curve as shown in the figure below. The radius of curvatur

Question

A car rounds a banked curve as shown 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. Determine the range of speeds the car can have without slipping up or down the road. (Use any variable or symbol stated above along with the following as necessary: g.) v_min = v_max = Find the minimum value for mu_s such that the minimum speed is zero. (Use any variable or symbol stated above along with the following as necessary: g.) mu_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)]

Vmin = [gR(tan - µs)/(µs tan + 1)]

in same method with changed direction of friction force we get;

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

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