A uniform disk, with radius R and mass M, is rotating with angular velocity ?o a

Question

A uniform disk, with radius R and mass M, is rotating with angular velocity ?o about a
vertical axis as shown in the left figure below. Then, sand is poured from above so that it
falls a distance b from the center of the disk, forming a ring of radius b. See right figure
below. When the sand forms a complete ring, the angular velocity both the disk and sand is
?o/4. Find m, the mass of sand added. Give answer in terms M, R, b, and/or ?o. Treat the
disk as a solid cylinder and the ring of sand as a thin-walled hollow cylinder.

                                          before                                                                                                       after

THE ANSER: M=3/2M (R/b)^2

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Explanation / Answer

I disk w = (Idisk + I sand) w0/4

I disk = 1/2 M r^2

I sand = msand b^2

1/2 M r^2 w = ( 1/2 M r^2 + msand b^2) w/4

2 M r^2 = 1/2 M r^2 + m sand b^2

3/2 M R^2 = M sand b^2

M sand = 3/2 M (R/b)^2

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