How does wind direction affect Earth's rotation time?

[levaphysics]

Earth system can be described as consisting of two parts. First, a homogeneous Earth with a certain radius, a certain mass and a rotation time of 24 hours and also an atmosphere in the form of a spherical shell. Atmospheric shell assumed to have thickness of 30 km and with a homogeneous mass distribution. Angular momentum of the planets plus the atmospheric scale is constant.
At one time the winds are such that the atmospheric shell rotates with angular velocity 0.40 ∙ 10-6 rad / s from east to west relative to the underlying earth.
On another occasion, the winds blowing in a different direction so that the atmospheric shell has an angular velocity of 0.40 ∙ 10-6 rad / s from west to east relative to the underlying earth.
How much change Earth's rotation time (time for one lap) between the two dates?

 

[D H]

You need to show some work before any of us can help you.

What are the relevant equations here, and what have you done to attempt to solve this problem?

 

[levaphysics]

I need help, i f somebody show me how to start thinking on this problem. i guess that if an object is moving in free space, with no external forces or torques acting on it, then it will rotate about its centre-of-mass. So we can represent the total instantaneous motion of a rigid body by a combination of the linear velocity of its centre of mass and its rotation about its centre of mass.
The angular velocity vector W(t) can be derived from the angular position, as a function of time
W(t) = d theta /dt

 

[Redbelly98]

It looks like conservation of angular momentum is to be applied here. Linear velocity & momentum are not relevant.

You'll need to express the angular momentum of the solid Earth and of the atmosphere in an equation. But to do that, you would have to find the masses of the Earth and of the atmosphere.