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Two discs of moment of inertia I₁ and I₂ and angular speeds $ω_{1}$ and $ω_{2}$ are rotating along the collinear axes passing through their plane in the same direction. If the two are made to rotate after combining along the same axis the rotational K.E of system will be

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$\frac{I_{1} ω_{1} + I_{2} ω_{2}}{2 (I_{1} + I_{2})}$

$\frac{(I_{1} + I_{2}) {(ω_{1} + ω_{2})}^{2}}{2}$

$\frac{{(I_{1} ω_{1} + I_{2} ω_{2})}^{2}}{2 (I_{1} + I_{2})}$

$\frac{I_{1} ω_{1} - I_{2} ω_{2}}{2 (I_{1} + I_{2})}$

answer is C.

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Detailed Solution

Applying conservation of angular momentum

$(I_{1} ω_{1} + I_{2} ω_{2}) = (I_{1} + I_{2}) ω^{/}$

$ω^{/} = \frac{I_{1} ω_{1} + I_{2} ω_{2}}{(I_{1} + I_{2})} = final angular velocity$

$K_{f} = \frac{1}{2} (I_{1} + I_{2}) ω^{/ 2}$

$K_{f} = \frac{{(I_{1} ω_{1} + I_{2} ω_{2})}^{2}}{2 (I_{1} + I_{2})}$

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