Question

A train of mass ${10}^4\text{kg}$ rounds a curve of radius $100\text{m}$ at a speed of $10\text{m/s}$. At what angle the track be banked in order to have no thrust on the rails? ($g=10{\text{m/s}}^2$)

• A. ${\tan }^{-1}\left(\frac{1}{10}\right)$
• B. ${\tan }^{-1}\left(10\right)$
• C. ${\tan }^{-1}\left(\frac{1}{20}\right)$
• D. ${\tan }^{-1}\left(\frac{1}{100}\right)$

Answer 1

The correct option is A ${\tan }^{-1}\left(\frac{1}{10}\right)$

For banking of tracks, the purpose is to settle the requirement of necessary centripetal force at the curve, through the component of normal reaction.
Applying equation of dynamics along centre of circle:
$N\sin \theta =\frac{m{v}^2}{r}...\left(i\right)$
Balancing forces in vertical direction gives:
$N\cos \theta =mg...\left(ii\right)$

Dividing Eq.$\left(i\right)$ and Eq.$\left(ii\right)$
$\tan \theta =\frac{v^2}{rg}...\left(iii\right)$

Putting $v=10\text{m/s}r=100\text{m}g=10{\text{m/s}}^2$ in above equation:-
$\theta ={\tan }^{-1}\frac{v^2}{rg}\therefore \theta ={\tan }^{-1}\frac{100}{100\times 10}={\tan }^{-1}\left(\frac{1}{10}\right)$

Hence option A is the correct answer.