Question

q1 Example 5.11 A circular racetrack of
radius 300 m is banked at an angle of 15°
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?​

Answer 1

$\begin{gathered}\sf : \implies \: a \: circular \: race \: track \: of \: 300m \: \\ \\ \sf : \implies \: banked \: at \: an \: angle \: of \: 15 \degree \\ \\ \sf : \implies \: coefficient \: of \: friction \: between \: the \: wheels \: of \: race \: car \: and \: road = 0.2 \\ \\ \sf : \implies \: \mu \: = 0.2\end{gathered}$

$\begin{gathered}\sf : \implies \: \orange{{ \underline{1) = optimum \: speed \: of \: race \: car \: to \: avoid \: wear \: and \: tear \: on \: its \: tires \: }}} \\ \\ \sf : \implies \: \tan( \theta) = \frac{ {v}^{2} }{rg} \\ \\ \sf : \implies \: {v}^{2} = rg \tan( \theta) \\ \\ \sf : \implies \: {v}^{2} = 300 \times 10 \times \tan(15 \degree) \\ \\ \sf : \implies \: {v}^{2} = 300 \times 10 \times 0.26 \\ \\ \sf : \implies \: {v}^{2} = 780 \\ \\ \sf : \implies \: v \: = \sqrt{780} \approx \: 28\end{gathered}$

$\begin{gathered}\sf : \implies \: \orange{{ \underline{2) = maximum \: permissible \: speed \: to \: avoid \: slipping}}} \\ \\ \sf : \implies \: v_{m} \: = \sqrt{Rg \: ( \frac{ \mu \: + \tan( \theta) }{1 - \mu \: \tan( \theta) } }) \\ \\ \sf : \implies \: v_{m} \: = \sqrt{300 \times 10( \frac{0.2 \times \tan(15 \degree) }{1 - 0.2 \tan( 15 \degree) } } ) \\ \\ \sf : \implies \: v_{m} \: = \sqrt{3000( \frac{0.2 \times 0.26}{1 - 0.2 \times 0.26} } ) \\ \\ \sf : \implies \: v_{m} \: = \sqrt{1458.7} = 38.19 \: ms {}^{ - 1}\end{gathered}$

Answer 2

Answer:

A circular racetrack of radius 300 m is banked at an angle of 15o. If the coefficient of friction between the wheels of a race car and the road is 0.2.

Explanation: