Planetary Angular Momentum and Latitude
Date: Fall 2013
Does angular momentum of Earth vary based on latitude? Earth rotates faster at the equator than the poles because the circumference is greater at the equator. Therefore, should the angular momentum of Earth be greatest there due to the larger radius and velocity? Following this logic the poles would have a smaller angular momentum because they spin more slowly and the radius of the Earth is smaller. How can this be reconciled with conservation of angular momentum?
The angular momentum of the earth is that of the entire earth. When you calculate angular momentum, you generally assume that the body carrying the momentum is one object. In the same fashion, if a rock is moving through space, you calculate the momentum of it as momentum=(total mass)*velocity. If you broke a small piece off the rock, this piece would have a smaller mass and less momentum. The momentum of each of the two pieces would still add up to the same momentum as before, however, so there would be no violation of the conservation of momentum. Each section of the rock contributes to the total momentum of the rock. In a similar fashion, a rotating object can be broken down into pieces, and each piece contributes to the total angular momentum of the earth.
Kyle J. Bunch, PhD, PE
Thank you for your question. I think it is important to distinguish between the angular momentum of Earth as a solid ball, versus the angular momentum of some object which is located at some latitude on Earth. Neither view is completely true, but both have their uses.
Earth as a rotating mass has one value of angular momentum at any moment in time. The same is true for a brick which located at some latitude and is spinning with the earth. However if you carry that brick from a latitude which is close to the equator to a latitude which is further away (and closer to a pole) the brick will lose angular momentum as you suggest. To a very tiny degree, the earth will spin faster as a result or your transporting the brick, thus the total angular momentum of the earth will remain the same. When you transport the brick, you will exert forces upon the brick which allow the momentum to be transferred between the brick and Earth. This is the same as an ice skater who spins faster as she pulls in her arms. Her arm muscles exert forces between her body and her hands. Her hands will lose some angular momentum as she applies force to pull them in, but her whole body spins faster so the total angular momentum is conserved (except for loss due to friction.)
Similar bricks located at different latitudes will have different angular momentum.
Conservation of momentum means that an object neither gains nor loses momentum with time unless some external force acts upon the object. In the example of the brick and the ice skater's hands, those external forces are easy to see.
There is no requirement that all of the different parts of an object have the same angular momentum. It would be difficult for that to be true for most objects.
In the case of the earth, it actually very gradually loses angular momentum with time due to the gravitational forces of the sun and the moon. If we neglect that, Earth would maintain constant angular momentum with time so would neither gain nor lose it.
I hope I have helped to clarify this.
The entire planet has a total angular momentum. This total angular momentum does not have to be evenly distributed over the planet. An even simpler device is a solid disk spinning around its axis. The portion of the disk that is near the edge contributes more to the total angular momentum than does the portion of the disk near the center.
Conservation of angular momentum means that the total angular momentum of the object does not change over TIME unless the object experiences torque from a source outside the object. Although different parts of Earth contribute different amounts of angular momentum to the total quantity, that total remains constant from day to day.
Dr. Ken Mellendorf
Illinois Central College
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