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A train runs along an unbanked circular track of radius $30 \mathrm{~m}$ at a speed of $54 \mathrm{~km} / \mathrm{h}$. The mass of the train is $10^6$ 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 the rails?
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Here, lateral thrust exerted by the rails to wheel is providing the necessary centripetal force and the train would exert an equal and opposite thrust (by Newton's $3^{\text {rd }}$ law) on the rails
$\therefore \tan \alpha=v^2 / r \mathrm{~g}$
Given, $r=30 \mathrm{~m}, v=54 \mathrm{~km} / \mathrm{h}=15 \mathrm{~m} / \mathrm{s}, \mathrm{g}=10 \mathrm{~ms}^{-2}$.
$\alpha=\tan ^{-1} v^2 / r g\left[\tan ^{-1}(15)^2 /(30 \times 10) \approx 37^{\circ}\right]$
$\therefore \tan \alpha=v^2 / r \mathrm{~g}$
Given, $r=30 \mathrm{~m}, v=54 \mathrm{~km} / \mathrm{h}=15 \mathrm{~m} / \mathrm{s}, \mathrm{g}=10 \mathrm{~ms}^{-2}$.
$\alpha=\tan ^{-1} v^2 / r g\left[\tan ^{-1}(15)^2 /(30 \times 10) \approx 37^{\circ}\right]$
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