Two disks are spinning freely about axes that run through their respective centres (see figure below). The larger disk

(R1 = 1.42 m)
has a moment of inertia of 1180 kg · m2 and an angular speed of 4.0 rad/s. The smaller disk
(R2 = 0.60 m)
has a moment of inertia of 906 kg · m2 and an angular speed of 8.0 rad/s. The smaller disk is rotating in a direction that is opposite to the larger disk. The edges of the two disks are brought into contact with each other while keeping their axes parallel. They initially slip against each other until the friction between the two disks eventually stops the slipping. How much energy is lost to friction? (Assume that the disks continue to spin after the disks stop slipping.)

The initial angular momentum of disc 1

is I1 w1
The initial angular momentum of disc 2
is I2 w2
Add those for total angular momentum, which DOES NOT CHANGE in this problem
because there are no external moments

Afterwards the no slip condition:
R1 w1 = R2 W2
calculate the angular momentum again and find new w s

Now
Initial KE = (1/2) I1 w1^2+(1/2)I2 w2^2

Final Ke = same formula, new w s

final better be less than initial :)

find difference

To solve this problem, we need to calculate the initial angular momentum of both disks and then compare it to the final angular momentum after the slipping stops. The energy lost to friction can then be calculated from the difference in angular momentum.

1. Calculate the initial angular momentum:
Angular momentum is given by the formula L = I * ω, where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
For the larger disk, L1 = I1 * ω1 = 1180 kg · m^2 * 4.0 rad/s
For the smaller disk, L2 = I2 * ω2 = 906 kg · m^2 * (-8.0 rad/s) [Note the negative sign due to opposite direction]

2. Calculate the final angular momentum:
When the disks stop slipping, they will have the same angular speed, ωf. Since their moments of inertia are different, we can express the final angular momentum as:
Lf = (I1 + I2) * ωf

3. Equate the initial and final angular momenta:
L1 + L2 = Lf
Substitute the values of L1 and L2 from step 1, and solve for ωf.

4. Calculate the energy lost to friction:
The energy lost to friction can be calculated as the difference between the initial and final kinetic energies.
Initial kinetic energy: KEi = 0.5 * I1 * ω1^2 + 0.5 * I2 * ω2^2
Final kinetic energy: KEf = 0.5 * (I1 + I2) * ωf^2
Energy lost to friction: ΔE = KEi - KEf

Now you can substitute the values given in the problem (R1, R2, I1, I2, ω1, ω2) into the equations to calculate the angular momentum and energy lost to friction.