Question

Derive an expression for the maximum speed acquired by vehicle on a banked circular track of radius r with coefficient of friction μ between the wheels and road.

Answer 1

Consider a vehicle of weight mg moving on circular level road of radius r with constant velocity v. While taking the round, the tyres of the car tend to leave the road and move away from the centre of curve. In such case, forces of friction f₁ and f₂ will act inward towards the two tyres. If R₁ and R₂ are the normal reactions of ground on the tyres, then

f₁ = μ_s R₁ and f₂ = μ_s R₂

where μ_s = coefficient of static friction

Now total frictional force,

f = μ_s R₁ + μ_s R₂

f = μ_s (R₁ + R₂)

f = μ_s R

where R = reaction of ground on the vehicle.

As total frictional force f cannot exceed μ_s R, hence f ≤ μ_s R

This frictional force will provide the necessary centripetal force f = mv²/r

Hence the maximum speed, vmax = √μ_srg

 If the vehicle is driven at a speed greater than vmax, then it will skid and goes off the road in a circle of radius greater than r as the maximum available friction is inadequate to give required centripetal force.