Time for another Fermi problem. There was a recent story in Science News that talked about the effects of the Chilean quakes on the Earth's rotation. The basic idea is that some ginmourmous amounts of rock moved closer to the Earth's center. Since the angular momentum of the Earth is conserved, the angular rotation rate would increase. The estimated change of the day was by about 1.26 microseconds.

## Could all the cars in the USA be used to change the rotation of the Earth?

Well, I shouldn't have phrased the question that way. Of course 1 car technically is all you need to change the rotation of the Earth - but by how much. Suppose everyone in the USA got in their car and drove East at the same time. How much would this change the rotation rate of the Earth?

Starting points:

- What is the rotation rate of the Earth now? I will call it omega-0, but the value is about 1 rotation in 23.9345 hours (don't confuse this with 1 rotation 24 hours - that is how long it takes the Sun to be back in the same position in the sky).
- Should I assume the Earth is a sphere? Well, I am going to even though it isn't. Also, I don't know the mass distribution of the Earth. Maybe I will just assume uniform density - again this is wrong (but it is wrong in a direction favorable to no change in motion).
- There are about 300 million people in the USA. I am going to totally estimate 100 million cars.
- I will use an average car the same weight as a Crown Victoria. Why? Because this is my estimation. How about 1,800 kg?
- These cars will be moving West at a speed of 60 mph (27 m/s).

The basic physics idea is that the Earth is rotating and has angular momentum. If the cars are moving with respect to the rotating Earth, the angular momentum of the two must be constant. For a rigid object (the Earth) the angular momentum is (I will assume rotation about a fixed axis because that makes things so much easier):

If I assume the Earth is on a fixed rotation axis (which it totally is not), then the angular momentum vector and the angular velocity vector are in the same direction with the moment of inertia as a scalar value. (here is an example of a case where L and omega are not in the same direction). But, back to the Earth. If the Earth is a uniform sphere, then the moment of inertia about a fixed axis is:

What about a car? If I consider a car to be a point particle, then as it move around the Earth, it would have an angular momentum of:

But what is the angular velocity for the car? Let me draw a picture.

So first, the car is stationary with respect to the rotating Earth. (here I am just showing one typical car) Then, the car drives with a velocity v with respect to the surface of the Earth. Before the car moves, the angular momentum of the Earth plus cars is: (magnitude only since the direction does not change)

Since the cars have the same angular velocity as the Earth, I can add the moments of inertia together. Also, I have included an *n* for the number of cars. Now that the cars are moving, the angular momentum would be:

Angular momentum is conserved. This means that L_{0} = L_{2}. I want to solve for the change in angular velocity. To do this, I will set the two angular momentum expressions equal to each other.

I know, that seems like some messy algebra - but it really isn't bad. How can I check to see if I made a mistake. First, what if the cars are going at zero m/s? If I put in v = 0 m/s, then the change in angular speed is zero rad/s. That is good. Also, if the mass or the number of cars goes to zero, the same thing happens. Also, the units check out. Now to just put in the numbers. Doing this I get:

If I know omega, I can find the time for one rotation (a day) as:

Using this for omega-0 and omega-1, I can get the two times:

If I calculate these two times and subtract them, I get a change in time of 3.7 x 10^{-10} seconds. So there is your answer. No where near the effect of the Chilean quake (even if you change my assumptions by a little).

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Dear Rhett,

I'd like to see this put to the test to verify your prediction.

How realistic is it for you to arrange for all the cars to traverse the States, say, next weekend which is Easter when most folk have some spare time? I am available to help out on Saturday, but I am busy on a Ferrari racing day on Good Friday.

And do you have any research grants to pay for the gas?

Good luck, see you next week.

Chris, London

@Chris,

I have a car - This weekend is good for me. What time? You will be responsible for your own gas.

This is very similar to an item I read several years ago before the construction of the Three Gorges Dam commenced. The article mentioned that, due to man's penchant for hydroengineering, dams have stored more and more water away from the center of rotation of the earth, slowing down its rotation by a miniscule, but still measurable, amount. Of course, holding back all that water has other effects as well, so who can say?

@Peter,

Yes - I remember seeing something like this as well. I was considering recalculating it.

Your question was about all the cars in the United States. This solution assumes the cars have a distance from the axis of rotation equal to the "radius" of the Earth. To the actual distance is cosine(latitude) * Radius.

But that would only lesson the effect.

can somebody clear one of my doubts, Is enviornment is also rotating with earth or only earth is rotating?

Dear Prof. Allain,

If I hold a theoretical ruler that extends from the earth to the surface of the moon. If from earth I push it one inch in the direction of the moon. To me, theoretically speaking, it should penetrate into the moon's surface by one inch simultaneously. It appears to me that if that's the case, isn't that the momentum or motion or force travels faster than the speed of light. Your explanation is much appreciated.

M. Wong

In the intro paragraph you have the cars driving East but in the starting points the cars driving West. In the drawing, it looks like the cars drive East, in the direction of the rotation of the Earth. Please clarify.

This is very similar to an item I read several years ago before the construction of the Three Gorges Dam commenced. The article mentioned that, due to man's penchant for hydroengineering, dams have stored more and more water away from the center of rotation of the earth, slowing down its rotation by a miniscule, but still measurable, amount. Of course, holding back all that water has other effects as well, so who can say?