Question

Q] A railway track goes around a curve having a radius of curvature of 1 km. The
distance between the rails is 1 m. Find the elevation of the outer rail above the inner rail so that there is no side pressure against the rails when a train goes round the curve
at 36 km / hr.​

Answer 1

Given,

• radius of curvature, $\sf{r=1\ km=1000\ m}$

• distance between the rails, $\sf{d=1\ m.}$

• speed of the train, $\sf{v=36\ km\,h^{-1}=10\ m\,s^{-1}}$

Let $\sf{g=10\ m\,s^{-2}.}$

There would be no side pressure against the rails when the train moves along the rail with maximum safe speed, which is given by,

$\sf{\longrightarrow v^2=rg\tan\theta}$

Then the angle of elevation will be,

$\sf{\longrightarrow \theta=\tan^{-1}\left(\dfrac{v^2}{rg}\right)}$

$\sf{\longrightarrow \theta=\tan^{-1}\left(\dfrac{10^2}{1000\times10}\right)}$

$\sf{\longrightarrow \theta=\tan^{-1}\left(\dfrac{1}{100}\right)}$

$\sf{\longrightarrow \theta=0.01\ rad}$

Then the elevation of outer rail will be given by,

$\sf{\longrightarrow h=d\sin\theta}$

$\sf{\longrightarrow h=1\times\sin(0.01)}$

$\sf{\longrightarrow\underline{\underline{h=0.01\ m}}}$

Answer 2

Answer:

• Elevation of outer rail above the inner rail = 0.01 m

Explanation:

Given:

• Radius of curvature (r) = 1 km = 1000 m
• Distance between the rails (d) = 1 m
• Speed of train (v) = 36 km/h

To Find:

• Elevation of outer rail above inner rail (h).

Now, firstly we will convert speed into m/sec,

$\implies{\sf{\dfrac{36\times 5}{18}\;m/sec}}$

$\implies{\bf{10\;m/sec}}$

Now, the angle of elevation would be,

$\implies{\sf{\theta = \tan^{-1}\bigg(\dfrac{v^{2}}{rg}\bigg)}}$

Let, g = 10 m/s

$\implies{\sf{\theta = \tan^{-1}\bigg(\dfrac{10^{2}}{1000\times 10}\bigg)}}$

$\implies{\sf{\theta = \tan^{-1}\bigg(\dfrac{100}{10000}\bigg)}}$

$\implies{\sf{\theta = \tan^{-1}\bigg(\dfrac{1}{100}\bigg)}}$

$\implies{\boxed{\sf{\theta =0.01\;rad}}}$

Now, the elevation of outer rail will be,

$\implies{\sf{h=d\sin \theta}}$

$\implies{\sf{h=1\times \sin\times 0.01}}$

$\implies{\boxed{\sf{h=0.01\;m}}}$