Question

Two discs of moment of inertia I₁ and I₂ and angular speeds $\omega _ { 1 }$ and $\omega _ { 2 }$ are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate together along the same axis the rotational KE of system will be

• A. $\frac { I _ { 1 } \omega _ { 1 } + I _ { 2 } \omega _ { 2 } } { 2 \left( I _ { 1 } + I _ { 2 } \right) }$
• B. $\frac { \left( I _ { 1 } + I _ { 2 } \right) \left( \omega _ { 1 } + \omega _ { 2 } \right) ^ { 2 } } { 2 }$
• C. $\frac { \left( I _ { 1 } \omega _ { 1 } + I _ { 2 } \omega _ { 2 } \right) ^ { 2 } } { 2 \left( I _ { 1 } + I _ { 2 } \right) }$
• D. None of these

Answer 1

Answer: (C)

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

Answer 2

By the law of conservation of angular momentum

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

Angular velocity of system $\omega = \frac { I _ { 1 } \omega _ { 1 } + I _ { 2 } \omega _ { 2 } } { I _ { 1 } + I _ { 2 } }$

Rotational kinetic energy $= \frac { 1 } { 2 } \left( I _ { 1 } + I _ { 2 } \right) \omega ^ { 2 }$

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