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  4. Two discs are mounted on frictionless bearings on a common shaft. The first disc has rotational inertia I and is spinning with angular velocity ω. The second disc has rotational inertia 2I and is spinning in opposite direction with angular velocity 3ω, as shown in figure. The two discs are slowly forced towards each other along the shaft until they couple and have a final common angular velocity of <img class=img-fluid question-image alt=image src=https://cdn.tardigrade.in/img/question/physics/3642807f2cc5f237151044faef3938f5-.png />

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Q. Two discs are mounted on frictionless bearings on a common shaft. The first disc has rotational inertia $I$ and is spinning with angular velocity $\omega$. The second disc has rotational inertia $2I$ and is spinning in opposite direction with angular velocity $3\omega$, as shown in figure. The two discs are slowly forced towards each other along the shaft until they couple and have a final common angular velocity of
image

Solution:

Total initial angular momentum
= Total final angular momentum
$2l[+3ω] + l[-ω] = [2l + l]ω_c$
$\therefore $ Common angular velocity,
$ω_c = \frac{5ω }{3}$