Spinning disk and cylinder collision with friction

[HappyFlower]

Homework Statement

A solid disk of radius 23.4 cm and mass 1.45 kg is spinning at 43.1 radians per second. A solid cylinder of radius 12.1 cm and mass 3.33 kg is not spinning. The cylinder is dropped into the center of the spinning disk. After a short time friction has caused both objects to engage and turn in the same direction and rate.
a) Determine final rotational direction and angular velocity of the system?
b) Determine the change in mechanical energy of the system?

Homework Equations

The Attempt at a Solution

i found a) 26.7 rad/s
I am having trouble finding the mechanical energy I don't know which equation I should use. I tried using
K=Iwf^2-Iwi^2

 

[Orodruin]

Please show your work.

 

[HappyFlower]

Please show your work.

I fixed it.

 

[jbriggs444]

I fixed it.

Rather than just writing down equations, you should explain what those equations are for. For instance:

You have a cylinder sitting on a disk such that both are rotating at the same angular velocity. You have determined that angular velocity. What is the total rotational kinetic energy of the assembly?