Ncert Exercises & Solutions

5.31
A train runs along an unbanked circular track of radius 30 m at a speed of 54 km/h. The mass of the train is 10⁶ kg. What provides the centripetal force required for this purpose - The engine or the rails? What is the angle of banking required to prevent wearing out of the rails?

Given,

Radius of circular bend, r = 30 m,

$S p e e d o f t r a i n = 54 k m h^{- 1} = 54 \times \frac{5}{18} = 15 m s^{- 1}$ mass of train, m = 10⁶ kg
Let angle of banking be $θ$

The centripetal force is provided by the lateral thrust by the rails on the wheels.

According to Newton’s third law of motion, the train exerts an equal and opposite thrust on the rails causing its wear and tear.
The angle of banking is given by,

$\tan θ = \frac{v^{2}}{r g}$
$= \frac{{(15)}^{2}}{30 x 10} = \frac{225}{300}$
$θ = \tan^{- 1} (0.75) = 36.87 °$

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