Question

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?
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Answer 1

For a car on a banked road we have the relation gives by the following expression:

$v = \sqrt{rg (\frac{p + \tan \alpha }{1 - p \tan \alpha } )}$

Where p is coefficient of friction and all other symbols have relevant usual meaning here.

(i)
Optimum speed( to avoid wear and tear) is given by,

$\sqrt{rg \tan( \alpha ) } \\ \\ = \sqrt{100(10) \tan(30) } \\ \\ = \sqrt{1000( \frac{1}{ \sqrt{3}) } } \\ = 577.35m {s}^{ - 1}$

(ii)

Maximum presmissable speed to avoid slipping,

$v = \sqrt{(100)(10) (\frac{0.5 + \tan(30) }{1 - (0.5) \tan(30) } )} \\ = \sqrt{1000( \frac{0.5 + \frac{1}{ \sqrt{3} } }{1 - 0.5 ( \frac{1}{ \sqrt{3} } ) }) } \\ \\ = 1514.6 \frac{m}{ {s}^{} }$

Answer 2

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