Question

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 μs = 0.423. The banked curve is frictionless, and the car can negotiate it at the same maximum speed v0. Find the angle θ of the banked curve.

Answer 1

You'll be using these two equations:
Equation 1: Fc=μsFN where μs is the static friction given, FN is your normal force, and Fc is your centripetal force.

Equation 2: tan θ=V^2/rg where V is your speed, r your radius and g your constant of 9.8.

So you are basically replacing variables
1) Fc=μsFN, Fc can also be equal to mv^2/r so place that in the equation 1. Also replace your Fn which is also equal to mg you'll get

mv^2/r=0.423*mg.

Isolate the 0.423 to be 0.423=mv^2/r/mg. The mass (m)cancels out and you'll be left with V^2/rg=0.423 or V^2=.423rg now plug this into the second equation

2)tan θ=.423rg/rg both "rg's" cancel out so you'll be left with tan θ=.423 which then you'll isolate θ by doing the inverse with .423 to get θ=tan-1(.423) to get your answer which should be 22.9 or 23 degrees.

Good Luck!!