Two coaxial discs, having moments of inertia I1 and I2, are rotating with respective angular velocities ω1 and ω2, about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If Ef and Ei are the final and initial total energies, then (Ef - Ei) is:

Ei = 1/2 I1ω1² + 1/2 I2ω2²
Let the common angular velocity be ω.
By the conservation of angular momentum,
I1ω1 + I2ω2 = (I1 + I2)ω
ω = (I1ω1 + I2ω2) / (I1 + I2)
Ef = 1/2 (I1 + I2)ω² = 1/2 (I1 + I2) [(I1ω1 + I2ω2) / (I1 + I2)]²
= (I1ω1 + I2ω2)² / 2(I1 + I2)
Assuming I1 = I2 = I
ω = (ω1 + ω2) / 2
Ef = 1/2 (2I) [(ω1 + ω2)/2]² = I (ω1 + ω2)² / 4
Ei = 1/2 I (ω1² + ω2²)
Ef - Ei = I (ω1 + ω2)² / 4 - 1/2 I (ω1² + ω2²) = I/4 (ω1² + 2ω1ω2 + ω2² - 2ω1² - 2ω2²)
= I/4 (-ω1² + 2ω1ω2 - ω2²) = -I/4 (ω1 - ω2)²
Let ω1 = ω and ω2 = 0
Then Ef - Ei = -Iω²/4
Let I1 = I, I2 = I
ω = (Iω1 + Iω2)/(2I) = (ω1 + ω2)/2
Ef = 1/2(2I)[(ω1+ω2)/2]² = I(ω1+ω2)²/4
Ei = 1/2Iω1² + 1/2Iω2²
Ef - Ei = I(ω1+ω2)²/4 - I/2(ω1²+ω2²) = I/4(ω1²+2ω1ω2+ω2²-2ω1²-2ω2²) = I/4(-ω1²+2ω1ω2-ω2²) = -I/4(ω1-ω2)²
If ω2 = 0
Ef - Ei = -Iω1²/4
In the given options, I1 = I, ω1 = ω
Therefore, Ef - Ei = -Iω²/4 = -I1ω1²/4
None of the given options match this result. There might be an error in the options or the problem statement.