Two curves on a highway have the same radii. However, one is unbanked and the other is banked at an angle θ. A car can safely travel along the unbanked curve at a maximum speed v0 under conditions when the coefficient of static friction between the tires and the road is 0.563. The banked curve is frictionless, and the car can negotiate it at the same speed v0. Find the angle θ of the banked curve.

θ = 29.38°

Explanation:

The centripetal force is given by the formula;

F_c = F_n(sin θ) = mv²/r

Now, the vertical component of the normal force is; F_n(cos θ)

Now, this vertical component is also expressed as; F_n(cos θ) = mg

Thus, the slope is;

F_n(sin θ)/F_n(cos θ) = (mv²/r)/mg

tan θ = v²/rg

v² = rg(tan θ)

The initial speed will be gotten from the relation;

(v_o)² = μ_s(gr)

Plugging rg(tan θ) for (v_o)², we have;

μ_s(gr) = rg(tan θ)

rg will cancel out to give;

μ_s = (tan θ)

Thus, θ = tan^(-1) μ_s

μ_s is coefficient of static friction given as 0.563

θ = tan^(-1) 0.563

θ = 29.38°

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