A curved road is banked at an angle β, and there is no friction between the road and the car’s tires. The road has a radius of curvature of 50 m.
A car has a speed of 90 km/h.
Find angle β.
According to Newton’s Second Law for uniform circular motion, the net force acting on the car equals mar.
Fnet = mar
The expression Fnet = mar is a vector equation.
We directed the X-axis toward the center of the curvature.
In the y-direction, there are only FN,y and mg. There is no acceleration in the y-direction.
FN,y = mg
FN,y is the component adjacent to the angle and equals the magnitude of the vector times the cosine of the angle. Thus,
FN∙cos(β) = mg (1)
In the x-direction, there is only FN,x. Thus,
FN,x = mar
FN∙sin(β) = mar (2).
We know that ar can be presented as v2/r.
ar = v2/r (3)
We substitute the equation (3) into the equation (2) and we get
FN∙sin(β) = mv2/r (4).
We divide the equation (4) by equation (1) and obtain
tan(β) = v2/(rg)
Finally, we substitute the known values (please don’t forget that 90 km/h = 25 m/s)
and we get
tan(β) = 252/(50∙9.81)
β ≈ 51.880