Question

A metre gauge train is moving at a speed of $36\text{km/hr}$ along a curved track of radius $10\sqrt{3}\text{m}$. Find the height of the outer rail with respect to the inner rail, so that there is no side pressure on the rails. Take $g=10{\text{m/s}}^2$ and the width of the track to be $1\text{m}$.

• A. $0.8\text{m}$
• B. $1\text{m}$
• C. $0.6\text{m}$
• D. $0.5\text{m}$

Answer 1

The correct option is D $0.5\text{m}$

FBD of the train on the curved track is shown below. Let the angle of banking be $\theta$.


To avoid side pressure on rails, dependency on frictional force for providing centripetal force should be eliminated, by opting for banking of track.

For a metre gauge train, the distance between outer and inner rail is $AB=1\text{m}$.

The train is moving with speed $v=36\text{km/hr}\times \frac{5}{18}=10\text{m/s}$,
and $r=10\sqrt{3}\text{m}$

From the FBD,
$N\sin \theta =\frac{m{v}^2}{r}$....$\left(i\right)$
$N\cos \theta =mg$.....$\left(ii\right)$

Dividing (i) by (ii),
$\tan \theta =\frac{v^2}{rg}=\frac{{10}^2}{10\sqrt{3}\times 10}=\frac{1}{\sqrt{3}}$
$\Rightarrow \theta ={\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)={30}^{\circ }$

From triangle ABC in the figure,
$h=AB\times \sin {30}^{\circ }=1\text{m}\times \frac{1}{2}=0.5\text{m}$