# A car moves around a curved banked road

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.

###### Solution.

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

or

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