Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are 0.1 kg-m² and 10 rad s –1 respectively while those for the second one are 0.2 kg-m² and 5 rad s –1 respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is :

Solution

Physics sir · 3 steps

When the discs stick, kinetic energy is lost, but angular momentum is not. Start from L.

1. Conserve angular momentum. Sticking together involves only internal forces, so there is no external torque about the axis. Total angular momentum stays the same: L = I₁ω₁ + I₂ω₂ = 0.1 × 10 + 0.2 × 5 = 2 kg m² s⁻¹.
2. Common angular speed. The combined moment of inertia is I = 0.1 + 0.2 = 0.3 kg m². So ω = L/I = 2/0.3 = 20/3 rad s⁻¹.

   K = L²/2I = 2²/(2 × 0.3) = 4/0.6 = 20/3 J

3. So. The combined system has kinetic energy 20/3 J, which is option (a). This is less than the starting 5 + 2.5 = 7.5 J, because sticking is like a perfectly inelastic collision.

The idea to keep. When rotating bodies stick together, conserve angular momentum, not kinetic energy. Then K = L²/2I gives the final energy in one line.

Answer from the NTA final answer key. Solution by PrepWiser.