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  4. Two discs rotating about their respective axis of rotation with angular speeds 2 rad s-1 and 5 rad s-1 are brought into contact such that their axes of rotation coincide. Now, the angular speed of the system becomes 4 rad s-1. If the moment of inertia of the second disc is 1 × 10-3 kg m2, then the moment of inertia of the first disc (in kg m2) is

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Q. Two discs rotating about their respective axis of rotation with angular speeds $2\, rad\,s^{-1}$ and $5\, rad\,s^{-1}$ are brought into contact such that their axes of rotation coincide. Now, the angular speed of the system becomes $4\, rad\,s^{-1}$. If the moment of inertia of the second disc is $1 \times 10^{-3} \,kg \, m^2$, then the moment of inertia of the first disc (in $kg\, m^{2}$) is

KEAMKEAM 2014System of Particles and Rotational Motion

Solution:

Relation between angular momentum and moment of inertia
$J=I \omega$
where $I_{1}=?$
$ I_{2} =1 \times 10^{-3} \,kg - m ^{2} $
$\omega_{1} =2$ rad / s
$ \omega_{2} =5$ rad / s
$\omega =4$ rad / s
So, $ J=J_{1}+J_{2}$
$\left(I_{1}+1 \times 10^{-3}\right) \times 4 =I_{1} \times 2+5 \times 1 \times 10^{-3} $
$4\, I_{1}+4 \times 10^{-3} =2 I_{1}+5 \times 10^{-3}$
$ 2 \,I_{1} =1 \times 10^{-3} $
$I_{1} =\frac{1}{2} \times 10^{-3} $
$=0.5 \times 10^{-3} \,kg - m ^{2}$