Question

Two discs of moments of inertia and I2 about their respective axes (normal to the disc and passing through the centre), and rotating with angular speed $\omega _ { 1 }$ and $\omega _ { 2 }$ are brought into contact face to face with their axes of rotation coincident.

What is the loss in kinetic energy of the system in the process?

• A.
• B.
• C.
• D.

Answer 1

Answer: (A)

Answer 2

Let the moment of inertial of disc I be and the angular speed of disc I be $\omega _ { 1 }$ .

let the moment of inertia of disc II be $\mathrm { I } _ { 2 }$ and the angular speed of disc II be $\omega _ { 2 }$

Angular momentum of disc I be $\mathrm { L } _ { 1 } = \mathrm { I } _ { 1 } \omega _ { 1 }$

and angular momentum of disc II be

$\therefore$ The initial angular momentum of two discsis given by $\mathrm { L } _ { \mathrm { i } } = \mathrm { I } _ { 1 } \omega _ { 1 } + \mathrm { I } _ { 2 } \omega _ { 2 }$

When two discs are brought into contact face to face (one on top of other) and their axis 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 = $\mathrm { I } _ { 1 } + \mathrm { I } _ { 2 }$

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

$\therefore$ The final angular momentum of the system is given by.

According to law of conservation of angular momentum, we get

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

$\omega = \frac { I _ { 1 } \omega _ { 1 } + I _ { 2 } \omega _ { 2 } } { I _ { 1 } + I _ { 2 } }$ …(i)

Kinetic energy of disc I is given by,

$K _ { 1 } = \frac { 1 } { 2 } I _ { 1 } \omega _ { 1 } ^ { 2 }$

Kinetic energy of disc II is given by,

$\mathrm { K } _ { 2 } = \frac { 1 } { 2 } \mathrm { I } _ { 2 } \omega _ { 2 } ^ { 2 }$

$\therefore$ Initial kinetic energy of two discs is

Final kinetic energy of the system is,

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

Loss in kinetic energy of the system.

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

]

$\left( \omega _ { 1 } ^ { 2 } + \omega _ { 2 } ^ { 2 } - 2 \omega _ { 1 } \omega _ { 2 } \right)$