At what angle θ should it be banked for maximum control of the racing car?

Question

A civil engineer is to design a road curve at an 88.3 m radius to carry racing cars traveling at 21 m/s.

µ = 0.13 (of  the car thin)

At what angle θ should it be banked for maximum control of the racing car? The coefficient of friction of the surface is 0.13 and the acceleration due to gravity is 9.8 m/s².

Answer 1

\(V_{max} = \sqrt{\frac{rg(\mu_s + tan \theta)}{(1-\mu_stan \theta)}}\)

\(\frac{V^2}{rg} = \frac{\mu_s+ tan\theta}{1 - \mu_s tan\theta}\)

\(\frac{(21)^2}{88.3 \times 10} = \frac{0.13 + tan\theta}{1 - 0.3 tan \theta}\)

\(\frac{441}{883} = \frac{0.13 + tan\theta}{1-0.13 tan\theta}\)

\(0.49 = \frac{0.13 + tan\theta}{1 - 0.13tan \theta}\)

\(0.49 - 0.063 tan\theta = 0.13 + tan \theta\)

\(0.36 = 1.063 tan\theta\)

\(tan\theta = \frac{0.36}{1.063}\)

\(tan\theta = 0.3386\)

\(\theta = tan^{-1} (0.3386)\)