Question

Two discs of moments of inertia I1 and I2 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.
• B.
• C.
• D.

Answer 1

Answer: (A)

Answer 2

Total initial angular momentum of the two discs is $\mathrm { L } _ { \mathrm { i } } = \mathrm { I } _ { 1 } \omega _ { 1 } + \mathrm { I } _ { 2 } \omega _ { 2 }$

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., $\mathrm { I } = \mathrm { I } _ { 1 } + \mathrm { I } _ { 2 }$

Let $\omega$ be the final angular speed of the system.

The final angular momentum of the system is

$\mathrm { L } _ { \mathrm { f } } = \mathrm { I } \omega = \left( \mathrm { I } _ { 1 } + \mathrm { I } _ { 2 } \right) \omega$

As no external torque acts on the system, therefore according to law of conservation of angular momentum, we get

$\mathrm { I } _ { 1 } \omega _ { 1 } + \mathrm { I } _ { 2 } \omega _ { 2 } = \left( \mathrm { I } _ { 1 } + \mathrm { I } _ { 2 } \right) \omega$