Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ω1 and ω2. They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is

Initial angular momentum  Iω1+Iω2 Let ω be angular speed of the combined system. Final angular momentum 2Iω∴ According to conservation of angular momentum Iω1+Iω2=2Iω   or   ω=ω1+ω22 initial rotational kinetic energy Ei=12Iω12+ω22Final rotational kinetic energy Ef=12(2I)ω2=12(2I)ω1+ω222=14Iω1+ω22∴ Loss of energy ΔE=Ei-Ef=I2ω12+ω22-I4ω12+ω22+2ω1ω2=I2ω12+ω22-ω12+ω22+2ω1ω22=I22ω12+2ω22-ω12-ω22-2ω1ω22=I4ω12+ω22-2ω1ω2=I4ω1-ω22