A train at latitude λ in the northern hemisphere is moving due north with a speed v along
a straight and level track. Which rail experiences the larger vertical force? Show that the ratio
R of the vertical forces on the rails is given approximately by:
R= 1+ 8Ωvh sin λ / ga
where h is the height of the centre of mass of the train above the rails which are a distance a
apart, g is the acceleration of free fall, and Ω is the angular velocity of the Earth’s rotation.
1. Coriolis force: 2mΩv sin λ
2. Weight of train, mg.
The Attempt at a Solution
I have figured out that the eastward track should experience a larger force, as the Earth is rotating from west to east and the train will tend to veer towards the east. I understand that the Coriolis force vector points to the right on the Northern Hemisphere.
For the second part of the question, I considered the forces acting on the tracks. The tracks should experience a vertical force due to the weight of the train, and a horizontal force due to the Coriolis force. However if the Coriolis force acts horizontally, how does it contribute to the vertical force?
Or should I consider the 3D rotation vector with a coordinate system that corresponds to (x,y,z)=(north, east, upwards) as shown by this diagram? (from http://en.wikipedia.org/wiki/Coriolis_effect)
I thought that if I considered only the component along the z-axis to obtain the vertical force. But how would it relate to the spacing, a, between the tracks?