We model the car as a point mass and express the acceleration vector in terms of the Serret-Frenet frame. Then the equation of motion in the direction of the normal vector is (see the free -body diagram)
man=N SIN α+H cos α
Since the car does not move in the vertical direction we can apply the equilibrium condition
We now solve these two equations for the normal force N and the force of static friction H:
H = man cosα −mg sin α ,
N = man sin α +mg cosα .
The car does not slide if the condition of static friction
is satisfied. With an = v2/r this leads to the allowable region of the velocity:
This is displayed for µ = 0.3 as a function of the angle α in the figure.