A disc is rotating with angular speed $ω_{1}$ . The combined moment of inertia of the disc and its axis is $l_{1}$ . A second disc of moment of inertia $l_{2}$ is dropped on to the first and ends up rotating with it. Find the angular velocity of the combination if the original angular velocity of the upper disc was $(a)$ zero $(b)$ $ω_{2}$ in the same direction as $ω_{1}$ and $(c)$ $ω_{2}$ in a direction opposite to $ω_{1}$ .

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Solution

The total angular momentum of the system before coupling $= | I_{1} {\to ω}_{1} + I_{2} {\to ω}_{2} |$

(The total angular momentum after coupling when they rotate with equal angular velocity) $= | I_{1} \to ω + I_{2} \to ω |$

Conservation of angular momentum
$| (I_{1} + I_{2}) \to ω | = | I_{1} {\to ω}_{1} + I_{2} {\to ω}_{2} | \Rightarrow ω = \frac{| I_{1} {\to ω}_{1} + I_{2} {\to ω}_{2} |}{(I_{1} + I_{2})}$

(a) When $ω_{2} = 0$ ,

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

(b) When $ω_{1}$ and $ω_{2}$ are unidirectional, $ω = \frac{I_{1} ω_{1} + I_{2} ω_{2}}{(I_{1} + I_{2})}$

(c) When $ω_{1}$ and $ω_{2}$ are anti-parallel, $ω = \frac{| I_{1} ω_{1} I_{2} - ω_{2} |}{I_{1} + I_{2}}$

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A uniform heavy disc of moment of inertia $I_{1}$ is rotating with constant angular velocity $ω_{1}$ . Then, a second nonrotating disc of moment of inertia $I_{2}$ is dropped on it. The two discs rotate together. The final angular velocity of the system becomes