How do you find the coefficient of static friction of a car on a banked curve without skidding?

Question

If a curve with a radius of 88m is perfectly banked for a car traveling 75Km/h, what must be the coefficient of static friction for a car not to skid when traveling at 95Km/h?

Answer 1

Let:

a be the angle of inclination of the slope to the horizontal,

u be the coefficient of static friction,

m be the mass of the car,

v be the speed of the car,

r be the radius of  the curve,

R be the normal reaction of the slope on the car,

F be the force of friction down the slope.

Resolving horizontally and vertically:

R sin(a) + F cos(a) = mv^2 / r ...(1)

mg + F sin(a) = R cos(a) ...(2)

In addition:

F / R <= u ...(3)

From (2):

mg = R cos(a) - F sin(a) ...(4)

From (1):

mv^2 / r = R sin(a) + F cos(a) ...(5)

Eliminating m from (4) and (5):

rg[ R sin(a) + F cos(a) ]= v^2 [ R cos(a) - F sin(a) ]

F [rg cos(a) + v^2 sin(a) ] = R[ v^2 cos(a) - rg sin(a) ]

Substituting for F / R in (3):

u >= [ v^2 cos(a) - rg sin(a) ] / [ rg cos(a) + v^2 sin(a) ] ...(6)

Putting u = 0 for ideal banking:

v^2 cos(a) - rg sin(a) = 0

tan(a) = v^2 / (rg) ...(7)

(7) gives:

tan(a) = (750 / 36)^2 / (88 * 9.81)

a = 26.69 deg.

(6) gives:

u >= [ (950 / 36)^2 cos(a) - (88 * 9.81) sin(a) ] / [ (88 * 9.81) cos(a) + (950 / 36)^2 sin(a) ]

u >= 0.216.