Will angular velocity of earth be Affected By this? [duplicate]

Question

If all the humans, animals, and all other beings came together to North Pole of earth, Will it affect the angular velocity Of earth in any means?

Answer 1

Yes.

Now you need to do your homework and estimate by how much. You need to estimate the angular momentum of the Earth. As a rough approximation you could do that by assuming a uniform density sphere. That's not correct, but it will do for an initial estimate that will probably get the magnitude correct. That is, it will get the $X$ in the angular momentum being $A\times 10^X$. Probably won't get X wrong by more than 1.

The sources of error in that will include several factors.

• The Earth is not perfectly spherical but bulges at the equator due to rotating.
• The Earth is not perfectly smooth.
• The Earth is not uniform density.

Then estimate the mass of all living things. That's going to introduce a huge source of uncertainty since it would require knowing the mass of lots of things that are difficult to be confident of. The mass of plants is difficult for example, since it changes quite significantly due to seasonal exchanges with the atmosphere.

Then as a very rough estimate of the change in angular velocity you could treat the movement to the north pole as removal of the mass but with the same angular momentum. The result would be the Earth would be a very slightly smaller sphere of smaller mass. In order to keep the same angular momentum it will need to spin slightly faster.

This introduces further approximations because moving large masses around on the Earth will change the Earth's shape. That's really tough to be accurate about. But for now just ignore it. Probably over the time scales of interest this is smaller error than the previously listed uncertainties.

Along the way you will need to work out the formula for angular momentum as a function of angular velocity for a perfect sphere of uniform density. Then you need to work out the change in angular velocity to keep the same angular momentum with smaller mass.

If I've done my (quite hasty) calculations correctly it looks a little like so. If you move $\frac{\delta}{M}$ of the mass of the Earth to the north pole, the angular velocity will increase by $\frac{4}{3} \frac{ \delta}{ M}$. Since the Earth is just under $6 \times 10^{24}$ kg, if we moved 4.5 billion tonnes ($ 4.5 \times 10^{12}$ kg or about 4.5 cubic km of water) to the north pole the length of the day would decrease by very roughly 1 part in $10^{12}$.