Two discs of moment of inertia I1 and I2 and angular speeds ω1 and &#

Question

Two discs of moment of inertia

I_{1}

and

I_{2}

and angular speeds

ω_{1}

and

ω_{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

Options:

A . I1ω1+I2ω22(I1+I2)

B . (I1ω1+I2ω2)22(I1+I2)

C . (I1+I2)(ω1+ω2)22

D . None of these

Answer: Option B
:
B
By the law of conservation of angular momentum

I_{1} ω_{1} + I_{2} ω_{2} = (I_{1} + I_{2}) ω

Angular velocity of system

ω = \frac{I_{1} ω_{1} + I_{2} ω_{2}}{I_{1} + I_{2}}

Rotational kinetic energy

= \frac{1}{2} (I_{1} + I_{2}) ω^{2} = \frac{1}{2} (I_{1} + I_{2}) {(\frac{I_{1} ω_{1} + I_{2} ω_{2}}{I_{1} + I_{2}})}^{2} = \frac{(I_{1} ω_{1} + I_{2} ω_{2})^{2}}{2 (I_{1} + I_{2})} .

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