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=R_E*cosΘ)

I know it has to do with conservation of momentum (L_i=L_f) 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.

Answer 1

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.