Question

A train is moving with a speed of 12 m/s on rails which are 1.5 m apart. To negotiate a curve radius of 400 m, the height by which the outer rail should be raised with respect to the inner rail is (Given, \( g = 10 \, \text{m/s}^2 \)):

• A. 6.0 cm
• B. 5.4 cm
• C. 4.8 cm
• D. 4.2 cm

Answer 1

The Correct Option is B

To determine the height by which the outer rail should be raised, we use the concept of railway banking, which involves balancing the centripetal force needed to negotiate a curve with the gravitational force on the inclined plane. The formula for the height \(h\) of the outer rail is given by:

\(h = \frac{v^2b}{gR}\)

where:

• \(v\) is the velocity of the train (\(12 \, \text{m/s}\)), 
• \(b\) is the distance between the rails (\(1.5 \, \text{m}\)),
• \(g\) is the acceleration due to gravity (\(10 \, \text{m/s}^2\)),
• \(R\) is the radius of the curve (\(400 \, \text{m}\)).

Plugging these values into the formula, we have:

\(h = \frac{(12)^2 \cdot 1.5}{10 \cdot 400}\)

Simplifying:

\(h = \frac{144 \cdot 1.5}{4000}\)

\(h = \frac{216}{4000}\)

\(h = 0.054 \, \text{m}\)

Converting this into centimeters, we find:

\(h = 5.4 \, \text{cm}\)

Thus, the height by which the outer rail should be raised is 5.4 cm.

Therefore, the correct answer is 5.4 cm.

Answer 2

For a train moving around a curve, the required banking angle θ is given by:

\[\tan \theta = \frac{v^2}{Rg}\]

where \(v = 12 \, \text{m/s}\), \(R = 400 \, \text{m}\), and \(g = 10 \, \text{m/s}^2\).

Substitute the values:

\[\tan \theta = \frac{12^2}{10 \times 400} = \frac{144}{4000} = \frac{h}{1.5}\]

where \(h\) is the height by which the outer rail should be raised over the inner rail, and the distance between the rails is 1.5 m.

Solving for \(h\):

\[h = \frac{144 \times 1.5}{4000} = 5.4 \, \text{cm}\]

Thus, the required height is 5.4 cm.