Question: Would it be possible to slow the Earth's rotation down so that it rotates at a rate of once every 25 hours by applying a force at the surface of the Earth in the opposite direction to the earth's spin? If so how much force would be required?
It is certainly theoretically possible to slow
down the rotation of the Earth to a rotation rate of once every 25 hours be applying a couple
at the surface.
A couple is a pair of equal forces applied to an object and the
moment of the couple or torque is the product of one of these forces and the distance
between them.
The actual amount of force within the couple will depend on how
quickly you want to slow it down. A rapid decrease of rotational speed will obviously need a
very large force.
The torque producing angular acceleration (or deceleration) is
given by the equation:
Torque = Moment of inertia of object x Angular
acceleration.
Now the Earth is approximately spherical and so its moment of inertia
about the polar axis is:
Moment of inertia = 2/5 Mr2
where r is the
mean radius of the Earth (6.4x106 m) and M its mass (6x1024 kg).
Angular
deceleration = [5 x Torque]/2Mr2
If you want the rotation rate to decrease
from one rotation in twenty four hours (2p/86400 radians per second) to one rotation every
25 hours (2p/90000) in one year the angular deceleration would
be:
[7.27x10-5 – 6.98x10-5]/3.15x107 = 2.91x10-
6/3.15x107 = 9.24x10-14 radians s-2.
The torque
on the Earth would then be [2/5]6x1024x(6.4x106)2x9.24x10-14 =
9.1x1024 Nm
How you would actually achieve this torque is another
matter.
If it was applied by two large forces at opposite ends of the diameter of the
planet then the value of these forces would be Torque/diameter of Earth =
9.1x1024/12.8x106 = 7x1017 N!
However it is thought
that the Earth is actually slowing down very very slowly due to the increase in mass from
meteorites falling onto its surface. The actual rate could be determined by considering the
conservation of angular momentum of the system.