Angular momentum of bicycle wheel

[taxidriverhk]

Homework Statement

Assume there is a person sitting on a stool that is free to rotate, the person holds a wheel in which its center of mass is directly above the center of mass of the person plus the stool, then the person rotates the wheel counterclockwise at 6π rad/s. Suddenly, the person flips the wheel over and the angular speed of the wheel changes to 6π rad/s, but in clockwise direction. Given that the moment of inertia of the wheel is 0.6 kgm², and that of the person plus the stool is 12 kgm²

a. Find the angular speed of the person plus the stool when the person flips the wheel over.

b. How much work is done by the person on flipping the wheel?

Homework Equations

L_i = L_f
I_iω_i = I_fω_f

The Attempt at a Solution

a.
Moment of inertia of the person plus the stool
= I_s+p = 12 kgm²
I_w = 0.6 kgm²
ω₀ = -6π rad/s, ω_f = 6π rad/s

L_i = L_f
(I_w)(ω₀) = (I_w)(ω_f) + (I_s+p)(ω_s+p)
(0.6)(-6π) = (0.6)(6π) + (12)(ω_s+p)
ω_s+p = -1.88 rad/s

b.
W = (1/2)(I)(ω_f)2 - (1/2)(I)(ω_i)2
= (1/2)(12)(6π)2 - (1/2)(12)(-6π)2
= 0 J ?

I got these two answers above, but I think they will be most likely wrong
Please give me a correct solution, thank you very much
I may have written the question unclearly, if you have anything unclear, feel free to ask.

 

[Spinnor]

For part a I think the wheel and person will be rotating in opposite directions form your math it looks like you have them going the same way?

 

[Spinnor]

For part b the energy of the wheel does not change but the stool and the person have gained energy. That energy came from the work done flipping the wheels rotation? What is the energy of the rotating man and stool?