# Homework Help: A Matter of Wheels and Trains and The Spin of the Earth?

1. Sep 10, 2010

### modulus

A Matter of Wheels and Trains and The Spin of the Earth..!?

This question actually comes from a book of mathematical puzzles. The particular question I'm gonna state, uses the concept of centrifugal force. And though I'm pretty sure a good mathematician should have a clear understanding of physics, I have a feeling, that the concept of centrifugal force has been misused in this question.

1. The problem statement, all variables and given/known data
Question:Two identical trains, at the equator start travelling round the world in opposite directions. They start ogether, run at the same speed and are on different tracks. Which train will wear out its wheel treads first?

The Given Answer:Naturally, the train travelling against the spin of the earth. This train will wear out its wheels more quickly becausethe centrifugal force is less on this train.

2. Relevant equations
I don't think we really need any equations. Just a clear conceptual knowledge of centrifugal forces and friction. This may possibly help:

f = neta*mu

Here, 'neta' is the normal force and should be equal to the train's weight minus the so-called 'centrifugal force'.

3. The attempt at a solution
After reading the question, and some thinking, I figured out it had to do with the spin of the Earth. But the 'centrifugal force' thing really didn't hit me.
My solution was based on the idea that the wheels of any train will try to push the ground in the direction opposite to that in which the train is moving. So, the spin of the Earth is beneficial for the train moving opposite to that spin, because the Earth moves in the direction the wheels try to push it in.
As for the train moving with the Earth's spin, the earth moves in the direction against which the wheels push it in, so the earth provides it with a good amount of resistance (friction), and thus, this train's wheels should wwear out first.

But the answer says, the train moving against should wear out first, beacause it has less centrifugal force on it...?? How does that make sense? They use the word 'centrifugal force' so liberally...as if it's a by-product of circular motion! I thought centrifugal force was a pseudo force?

The questions I want to ask here are:
1. Should the directions of the trains have any effect on the way centrifugal force acts on them (because as far as the direction of this force is concerned, it's just either radially inward or radially outward, right)?
2. If the weight of the trains are the centripetal force in this case, why don't they move in circles the whole time? Because, if the triains exist, they have weight, and that weight acts radially inward. So....that weight can always act as a centripetal force. Basically what I'm saying is, how do we attribute circular motion to only a force which acts along the radius of the circle, why not on the circumference?
3. And finally, which answer is correct? Mine or the book's....or neither?

Any help will be greatly appreciated, thank you!

2. Sep 10, 2010

### Cleonis

Re: A Matter of Wheels and Trains and The Spin of the Earth..!?

Well, the trains do need to overcome air resistance, so they need traction. The thing is, since the velocity with respect to the Earth is the same in both directions, the two trains are dealing with the same amount of air resistence. So any difference in wear cannot be attributed to difference in encountered air resistence.

An object that is co-rotating with the Earth, at the equator, is moving with a velocity of about 1700 km/h. (about 465 meters per second). The amount of centripetal force that is required for that circumnavigation is just a fraction of the Earth's gravity: about 0.3 percent.

The fastest trains today are maglev trains, that have the capacity for travelling at 500 km/h.

As seen from the north pole the Earth is rotating counterclockwise.
- A train travelling with a velocity of 500 km/h westward, with respect to the Earth, is circumnavigating the Earth's axis counterclockwise, with a velocity of about 1200 km/h
- A train travelling with a velocity of 500 km/h eastward, with respect to the Earth, is circumnavigating the Earth's axis counterclockwise, with a velocity of about 2200 km/h

My ball park estimate is that the for the westward moving train the required centripetal force is 0.2 percent of the Earth's gravity, and that for the eastward moving train the required centripetal force is 0.4 percent of the Earth's gravity.