# A circular track of radius 100 m is banked at an angle of 30°. If the coefficient of friction between the wheels of a car and the road is 0.5, then what is the (i) optimum speed of the car to avoid wear and tear on its tires, and (ii) maximum permissible speed to avoid slipping?

### At the optimum speed, the normal reaction's component is enough to provide the needed centripetal force, and the frictional force is not needed. So, the optimum speed is given by

\begin{aligned} v_{0} &=\sqrt{\operatorname{Rg} \tan \theta} \\ &=\sqrt{100 \times 9.8 \times \tan 30} \\ &=23.8 \mathrm{m} / \mathrm{s} \end{aligned}

The maximum permissible speed is given by

\begin{aligned} v_{\max } &=\sqrt{\operatorname{Rg}\left(\frac{\mu_{s}+\tan \theta}{1-\mu_{s} \tan \theta}\right)} \\ &=\sqrt{100 \times 9.8\left(\frac{0.5+0.58}{1-0.5 \times 0.58}\right)} \\ &=38.5 \mathrm{m} / \mathrm{s} \end{aligned}

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