Derive an expression for maximum speed of a vehicle moving along a horizontal circular track.

Answer: \(v_{max} = \sqrt{\mu_s r g}\)

Consider a vehicle of mass \(m\) moving with speed \(v\) on a flat (horizontal) circular road of radius \(r\). The necessary centripetal force is provided only by the force of static friction \(f_s\) between the tyres and the road, directed towards the centre.

For circular motion: \(f_s = \dfrac{mv^2}{r}\) … (1)

The friction cannot exceed its limiting value: \(f_s \le \mu_s N = \mu_s mg\) … (2) (since the normal reaction \(N = mg\) on a horizontal road)

So \(\dfrac{mv^2}{r} \le \mu_s mg\), i.e. \(v^2 \le \mu_s r g\).

The maximum safe speed is therefore \(v_{max} = \sqrt{\mu_s r g}\). It is independent of the mass of the vehicle. Above this speed friction cannot supply the needed centripetal force and the vehicle skids outwards.