If everyone living on Earth moved to the equator, the length of day would change

Question

If everyone living on Earth moved to the equator, the length of day would change. Calculate the change in the day length. (professor gave the hint to use r=RE*cos)

I know it has to do with conservation of momentum (Li=Lf) and that the day will lengthen by less than a second, but I'm not sure how to estimate the inertia of people living around the world like normal unless I use the moment of inertia for a thin shell (2/3mr2) and I don't know where the equation above is supposed to be used....or what theta he is talking about.

Explanation / Answer

If everyone moved to the Earth's equator, then the moment of inertia of the earth-people system would increase.

Angular momentum of Earth-people system is moment of inertia times angular velocity.

Conservation of angular momentum (given that no external force) means that if moment of inertia increases, then angular velocity decreases - meaning Earth would rotate more slowly - hence the day (time for one rotation of Earth) would INCREASE in duration, at least conceptually.

The moment of inertia of 6 billion people all at the equator is their mass times the square of their distance from Earth's center, or roughly 6E9 * 50 kg/person * 4E13 m = 1.2E25 kg-m^2, where Ex means 10^x (10 to power x). This is about ten trillion trillion.

Earth's moment of inertia, which can be calculated as roughly 2.4E37 kg-m^2, or about one trillion times the moment of inertia increase caused by having all the people move to the equator.

Thus the movement would increase the length of a day by one part in a trillion, or about one-tenth of a microsecond.

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