How to calculate gyroscopic inertia this setup?

[SpaceThoughts]

This text refers to the photo below.

The wheel in the middle can rotate vertically without friction. The construction around it can rotate the wheel horizontally as well, also without friction.

When the blue weight is falling while the wheel is not rotating, I can calculate the kinetic energi for the wheel using the formula for rotating kinetic energy (E rotation = 0,5 x I x W^2.) inserting the moment of inertia for a short rod, rather than a cylinder (1/12 x m x L^2)

However, when the wheel is rotating, the blue weight is falling slower because the gyroscope slows down the process. So this formula has to be modified to include the speed of the wheel in order to be able to calculate the actual kinetic energy of the mass rotating horizontally. What does this modified formula look like?

I am not a professor in physics, and would appreciate a simple answer, if possible;-)

Thanks in advance.

 

[LightHero]

The modified equation for calculating the kinetic energy of the mass rotating horizontally is: E rotation = 0.5 x I x (W + ω)^2, where I is the moment of inertia for a short rod (1/12 x m x L^2), W is the angular velocity of the wheel in radians per second, and ω is the angular velocity of the mass in radians per second.