Question

A civil engineer has to design a circular banked racing track on which cars can move up to speed of 360 km/hr with coefficient of fraction miu = 1/8 and radius 4 km . Angle of banking for safe racing should be

Answer 1

If the maximum speed of the cars can be 360 km/hr with a coefficient of friction µ = 1/8 and radius 4 km then the angle of banking for safe racing should be 82.85°.

Explanation:

The maximum permissible speed with which the cars can move on the banked racing track, Vmax = 360 km/hr

The radius of the circular track, r = 4 km

The coefficient of friction, µ = $\frac{1}{8}$

Let the angle of banking for safe racing be “θ”.

• We know that the required centripetal force to keep the cars moving on the circular banked racing track without slipping is provided by the horizontal component of the normal reaction and the frictional force.  

• The weight of the cars on the track is balanced by the vertical component of the normal reaction.

Therefore, the formula for the maximum permissible speed, which prevents the car from slipping is given by,

Vmax = √[{rg(tanθ+µ)} / {1 - µtanθ}]

Substituting the given values in the above formula, we get

360 = √[{4*9.8(tanθ + $\frac{1}{8}$)} / {1 – ($\frac{1}{8}$)*tanθ}]

Squaring both sides

⇒ 129600 = {4 * 9.8 * (tanθ + $\frac{1}{8}$)} / {1 – ($\frac{1}{8}$)*tanθ}

⇒ 3306.12 = [tanθ + $\frac{1}{8}$] / [1 – ($\frac{1}{8}$)*tanθ]

⇒ 3306.12 – 413.26 tanθ = tanθ + $\frac{1}{8}$

⇒ tanθ + 413.26 tanθ = 3306.12 – $\frac{1}{8}$

⇒ 414.26 tanθ = 3305.99

⇒ tanθ = 7.98

⇒ θ = tan⁻¹ (7.98)

⇒ θ = 82.85°

Thus, the angle of banking for safe racing should be 82.85°.

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