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Question: A rail track is banked for speed v by making the outer rail h higher than the inner rail. The distan...

A rail track is banked for speed v by making the outer rail h higher than the inner rail. The distance between the rails is d. The radius of curvature of the track is r, then
A. hd=v2rg\dfrac{h}{d} = \dfrac{{{v^2}}}{{rg}}
B. tan⁡(sin⁡−1hd)=v2rg\tan \left( {{{\sin }^{ - 1}}\dfrac{h}{d}} \right) = \dfrac{{{v^2}}}{{rg}}
C. tan⁡−1hd=v2rg{\tan ^{ - 1}}\dfrac{h}{d} = \dfrac{{{v^2}}}{{rg}}
D. hd=v2dg\dfrac{h}{d} = \dfrac{{{v^2}}}{{dg}}

Explanation

Solution

Draw a free body diagram for the scenario and calculate the net forces in order to balance the centrifugal force we should prevent toppling.

Complete step by step answer:
When vehicles traveling with some speed take a turn, then while taking the turn it experiences centrifugal force in the opposite direction of the turn. If the speed of the vehicle is very high, then the centrifugal force will also be high and can topple the vehicle leading to an accident. Therefore, to prevent these accidents banking of tracks is done at the turns.
Banking of the terms are done by increasing the height of that side of the track which is opposite to the turn. Due to this, the net contact force of the vehicle with the ground increases which causes an increasing frictional force with the ground, and thus preventing the vehicle from toppling.
The height of the banks depend on the radius of curvature of the track, the velocity of the vehicle, the width of the vehicle (distance between the wheels for a car are the distance between the rails for a train). To find the relation between them we must look at their force body diagrams calculating the net force to prevent toppling.
tan⁡θ=hd\tan \theta = \dfrac{h}{d}
mg\sin \theta = \dfrac{{m{v^2}}}{r}\cos \theta \\\ g\sin \theta = \dfrac{{{v^2}}}{r}\cos \theta \\\ \tan \theta = \dfrac{{{v^2}}}{{rg}} \\\
∴hd=v2rg\therefore \dfrac{h}{d} = \dfrac{{{v^2}}}{{rg}}

So, the correct answer is “Option A”.

Note:
The banking is made only once and cannot be altered every time. Therefore, the maximum speed limit that a vehicle should have to prevent toppling is fixed and cannot be altered, and therefore to prevent any accident the maximum speed limit should be always followed.