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 ma_{r}.

F_{net} = ma_{r}

The expression F_{net} = ma_{r} is a vector equation.

We directed the X-axis toward the center of the curvature.

In the y-direction, there are only F_{N,y} and mg. There is no acceleration in the y-direction.

F_{N,y} = mg

F_{N,y} is the component adjacent to the angle and equals the magnitude of the vector times the cosine of the angle. Thus,

F_{N}∙cos(β) = mg (1)

In the x-direction, there is only F_{N,x}. Thus,

F_{N,x} = ma_{r}

or

F_{N}∙sin(β) = ma_{r} (2).

We know that a_{r} can be presented as v^{2}/r.

a_{r} = v^{2}/r (3)

We substitute the equation (3) into the equation (2) and we get

F_{N}∙sin(β) = mv^{2}/r (4).

We divide the equation (4) by equation (1) and obtain

tan(β) = v^{2}/(rg)

Finally, we substitute the known values (please don’t forget that 90 km/h = 25 m/s)

and we get

tan(β) = 25^{2}/(50∙9.81)

β ≈ 51.88^{0}