Example: Banked Curve Physics

Question:

A circular racetrack of radius \(300\,\mathrm{m}\) is banked at an angle of \(15^\circ\). If the coefficient of friction between the wheels of a race-car and the road is 0.2, what is the (a) optimum speed of the race-car to avoid wear and tear on its tyres, and (b) maximum permissible speed to avoid slipping?

Solution:

On a banked road, the horizontal component of the normal force and the frictional force contribute to provide centripetal force to keep the car moving on a circular turn without slipping.

(a) The optimum speed \(v_o\) (where no friction is needed) is given by: \[ v_o = \sqrt{R g \tan \theta} \] With \(R = 300\,\mathrm{m}\), \(\theta = 15^\circ\), \(g = 9.8\,\mathrm{m/s^2}\): \[ v_o = \sqrt{300 \times 9.8 \times \tan 15^\circ} = 28.1\,\mathrm{m/s} = 101.2\,\mathrm{km/h} \]

(b) The maximum permissible speed \(v_{\text{max}}\) is given by: \[ v_{\text{max}} = \sqrt{R g \frac{\mu_s + \tan \theta}{1 - \mu_s \tan \theta}} \] With \(\mu_s = 0.2\): \[ v_{\text{max}} = \sqrt{300 \times 9.8 \times \frac{0.2 + \tan 15^\circ}{1 - 0.2 \times \tan 15^\circ}} = 38.1\,\mathrm{m/s} = 137.2\,\mathrm{km/h} \]