Question 332: Work, Energy and Power & Circular Motion

Question

A train has to negotiate a curve of radius $$400 \mathrm{~m}$$. By how much should the outer rail be raised with respect to the inner rail for a speed of $$48 \mathrm{~km} / \mathrm{h}$$ ? The distance between the rails is $$1 \mathrm{~m}$$.

Accepted Answer

To solve this problem, we'll need to apply the concept of banking of roads (or in this case, railway tracks). When a train (or any vehicle) goes around a curve, the outer rail needs to be raised with respect to the inner rail to provide the necessary centripetal force required for circular motion. This prevents the train from slipping or tipping over.
The formula for the elevation (h) of the outer rail required for safe turning at a velocity (v) on a curve of radius (r) and width (d) can be derived from balancing the forces acting on the train:
$$h = \frac{v^2}{g \cdot r} \cdot d$$
where
$h$ is the height by which the outer rail should be raised,
$v$ is the speed of the train,
$r$ is the radius of the curve,
$d$ is the distance between the rails, and
$g$ is the acceleration due to gravity ($9.8 \, 	ext{m/s}^2$).
Firstly, we need to convert the speed from km/h to m/s.
$$ v = 48 \, 	ext{km/h} = \frac{48 	imes 1000}{3600} = 13.33 \, 	ext{m/s} $$
Now, we can plug the values into our equation:
$$ h = \frac{(13.33)^2}{9.8 	imes 400} 	imes 1 $$
Let's calculate this step by step:
Squaring the speed: $(13.33)^2 = 177.7$
Multiplying by the distance between rails and dividing by the product of gravity and the radius:
$$ h = \frac{177.7}{9.8 	imes 400} = \frac{177.7}{3920} = 0.0453 \, 	ext{meters} $$
Converting meters to centimeters (since $1 \, 	ext{meter} = 100 \, 	ext{centimeters}$),
$$ h = 0.0453 \, 	ext{m} 	imes 100 = 4.53 \, 	ext{centimeters} $$
Therefore, the correct option is D) 4.5 cm.

Answer

1.5 cm

Answer

2.5 cm

Answer

3.5 cm

Answer

4.5 cm

Work, Energy and Power & Circular Motion

A train has to negotiate a curve of radius $400 \mathrm{~m}. By how much should the outer rail be raised with respect to the inner rail for a speed of 48 \mathrm{~km} / \mathrm{h} ? The distance between the rails is 1 \mathrm{~m}$.

Choose the correct answer: