A disk is rotating with angular speed ω1=2.0 rad/s about axle. The moment of inertia of disk & axle is 0.47 kg m2. A second disk of moment of inertia 0.31 kg m2 is dropped onto first. If dropped disk is rotating in opposite direction of ω1 with angular velocity ω2=1.0 rad/s, find magnitude of angular velocity of combination of two disks. Express your answer in rad/s.

As indicated by the moment of inertia t the angular velocity of the disks we use the concept of conservation of the angular momentum, for this we define the system as formed by the two discs, therefore the torque during the crash is internal and the angular momentum is conserved

Let's write in angular momentum

Initial. Before impact

L₀ = I₁ w₁ + I₂ w₂

Final. After the rock has stuck

[tex]L_{f}[/tex] = (I₁ + I₂) w

The two discs are rotating in opposite directions, we consider the rotation of the first positive disc, so the angular velocity of the second is negative

L₀ =[tex]L_{f}[/tex]

I₁ w₁ - I₂ w₂ = (I₁ + I₂) w

w = (I₁ w₁ - I₂ w₂) / (I₁ + I₂)

Let's calculate

w = (0.47 2.0 - 0.31 1.0) / (0.47+ 0.31)

w = 0.63 / 0.78

w = 0.808 rad / s

in the direction of disc rotation 1