Answer to Question #86964 in Mechanics | Relativity for Maria

Question

If a curve with a radius of 93 m is properly banked for a car traveling 69 km/h, what must be the coefficient of static friction for a car not to skid when traveling at 88 km/h?
Express your answer using two significant figures.

Answer 1

A banked curve means a car should do the turn at the design speed without participation of acceleration. This information allows us to find out the angle of incline of the road: 

$\alpha = \arctan\frac{v_0^2}{gR}; \\$

For greater speeds (car sliding upwards): 

$F_n\sin\alpha + F_f\cos\alpha = m\frac{v^2}{R}; \\ F_n\cos\alpha - F_f\sin\alpha - mg = 0; \\$

Remembering that $F_f = {\mu}F_n; \\$ and dividing the equations we get

$\frac{\sin\alpha+{\mu}\cos\alpha}{\cos\alpha - {\mu}\sin\alpha} = \frac{v^2}{gR}; \\ \mu = \frac{v^2-gR\tan\alpha}{gR+v^2\tan\alpha} = \frac{(v^2-v_0^2)gR}{(gR)^2+(vv_0)^2} \approx 0.2. \\$