$A$ train runs along an unbanked circular track of radius $30 \; m$ at a speed of $54 \; km/h$. 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 of the rail?

(D) Radius of the circular track,$r = 30 \; m$.
Speed of the train,$v = 54 \; km/h = 54 \times \frac{5}{18} \; m/s = 15 \; m/s$.
Mass of the train,$m = 10^{6} \; kg$.
The centripetal force is provided by the lateral thrust exerted by the rails on the wheels of the train. According to Newton's third law of motion,the wheels exert an equal and opposite force on the rails,which causes wear and tear.
The angle of banking $\theta$ required to prevent wear and tear is given by the relation:
$\tan \theta = \frac{v^{2}}{rg}$
Substituting the values:
$\tan \theta = \frac{(15)^{2}}{30 \times 9.8} = \frac{225}{294} \approx 0.765$
$\theta = \tan^{-1}(0.765) \approx 37.4^{\circ}$.
(Note: Using $g = 10 \; m/s^{2}$,$\tan \theta = \frac{225}{300} = 0.75$,so $\theta = \tan^{-1}(0.75) \approx 36.87^{\circ}$).