Two discs have moments of inertia I₁ and I₂ about their respective axes perpendicular to their planes and through their centres. They rotate with angular speeds ω₁ and ω₂, respectively, and are brought into contact face to face with their axes coaxial. The loss in kinetic energy of the system is:
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Question

Two discs have moments of inertia I₁ and I₂ about their respective axes perpendicular to their planes and through their centres. They rotate with angular speeds ω₁ and ω₂, respectively, and are brought into contact face to face with their axes coaxial. The loss in kinetic energy of the system is:

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G1 Physics · Accepted Answer

Write translation ΣF=Ma and rotation Στ about CM=I about CMα, together with the no-slip relation v of CM=ωR and a of CM=αR whenever pure rolling applies. Solving the equations for the shown geometry gives [I₁I₂/(2(I₁+I₂))](ω₁-ω₂)².