A disk with moment of inertia I1 rotates about a frictionless, vertical axle with angular speed ωi.  A second disk, this one having moment of inertia I2 and initially not rotating, drops onto the first disk (Fig.). Because of friction between the surfaces, the two eventually reach the same angular speed ωf The value of ωf is

# A disk with moment of inertia ${\mathrm{I}}_{1}$ rotates about a frictionless, vertical axle with angular speed ${\mathrm{\omega }}_{\mathrm{i}}$.  A second disk, this one having moment of inertia ${\mathrm{I}}_{2}$ and initially not rotating, drops onto the first disk (Fig.). Because of friction between the surfaces, the two eventually reach the same angular speed ${\mathrm{\omega }}_{\mathrm{f}}$ The value of ${\mathrm{\omega }}_{\mathrm{f}}$ is

1. A

$\frac{{\mathrm{I}}_{1}+{\mathrm{I}}_{2}}{{\mathrm{I}}_{1}}{\mathrm{\omega }}_{\mathrm{i}}$

2. B

$\frac{{\mathrm{I}}_{1}}{{\mathrm{I}}_{1}+{\mathrm{I}}_{2}}{\mathrm{\omega }}_{\mathrm{i}}$

3. C

$\frac{{\mathrm{I}}_{1}+{\mathrm{I}}_{2}}{{\mathrm{I}}_{2}}{\mathrm{\omega }}_{\mathrm{i}}$

4. D

$\frac{{\mathrm{I}}_{1}}{{\mathrm{I}}_{2}}{\mathrm{\omega }}_{\mathrm{i}}$

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### Solution:

From conservation of angular momentum for the isolated system of two disks:

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