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?
- 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 L0 = L2. 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).