Question

A highway curve with radius R = 274 m is to be banked so that a car traveling v = 25.0 m/s will not skid sideways even in the absence of friction. At what angle should the curve be banked?

Answer 1

The curve should be banked at 13.1°

Explanation:

The free body diagram for the car on the road is shown in the image attached to this solution.

ΣFₓ = 0, ΣFᵧ = 0,

The forces in the x-direction include the Nₓ (x-component of the Normal Reaction of the road on the car) and the (mv²/r) force keeping it in circular motion.

Nₓ = N sin θ and mv²/r

The forces in the y-direction include the Nᵧ (y-component of the normal reaction of the road on the car) and the weight of the car, mg

Nᵧ = N cos θ and mg

Putting the equations together,

N sin θ = mv²/r (eqn 1)

N cos θ = mg (eqn 2)

Divide (eqn 1) by (eqn 2)

(N sin θ)/(N cos θ) = (mv²/r)/(mg)

tan θ = v²/gr

v = 25 m/s

r = 274 m

Tan θ = 25²/(9.8 × 274) = 0.2328

θ = Tan⁻¹ (0.2328) = 13.1°