Two discs of moments of inertia `I_1` and `I_2` about their respective axes, rotating with angular frequencies, `omega_1` and `omega_2` respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be `A`.

Options

A. `(I_1 omega_1 + I_2 omega_2)/(I_1 + I_2)`
B. `(I_2 omega_1 + I_1 omega_2)/(I_1 + I_2)`
C. `(I_1 omega_1 + I_2 omega_2)/(I_1 - I_2)`
D. `(I_2 omega_1 + I_1 omega_2)/(I_1 - I_2)`

Answer

A

Explanation

(a) Total initial angular momentum of the two discs is `L_i = I_1 omega_1 + I_2 omega_2`
When two discs are brought into contact face to face (one on top of the other) and their axes of rotation coincide, the moment of inertia `I` of the system is equal to the sum of their individual moments of inertia.
i.e., `I = I_1 + I_2`
Let `omega` be the final angular speed of the system.
The final angular momentum of the system is.
`L_f = I omega = (I_1 + I_2) omega`
As no external torque acts on the system, therefore according to law of conservation of angular momentum, we get
`L_i = L_f`
`I_i omega_1 + I_2 omega_2 = (I_1 + I_2) omega`
`omega = (I_1 omega_1 + I_2 omega_2)/(I_1 + I_2)`.