Two disks are mounted (like a merry-go-round) on low-friction bearings on the sa

Question

Two disks are mounted (like a merry-go-round) on low-friction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with rotational inertia 4.2 sl middot ft^2 about its central axis, is spinning counterclockwise (which may be taken as the positive direction) at 480 rev/min. The second disk, with rotational inertia 2.8 sl middot ft^2 about its central axis, is spinning counterclockwise at 750 rev/min. They then couple together. (a) What is their angular velocity (rev/min) after coupling? (b) If instead the second disk is initially spinning clockwise at 750 rev/min, what is their angular velocity (in rev/min, using the correct sign for direction) after they couple together?

Explanation / Answer

4.2 slft^2 = 5.694 kgm^2
2.8 slft^2 = 3.796 kgm^2

(a) L0 = +(5.694 kg m2)(480 rev/min) + (3.796 kg m2)(750 rev/min) = 5580.12 kg m2 rev/min.
L0 = Lf = (If )f f = L0/If = (5580.12 kg m2 rev/min)/(5.694+3.796 kg m2)
= 588 rev/min.

(b) L0 = +(5.694 kg m2)(480 rev/min) - (3.796 kg m2)(750 rev/min) = 113.88 kg m2 rev/min.
L0 = Lf = (If )f f = L0/If = (113.88 kg m2 rev/min)/(5.694+3.796 kg m2) = 12 rev/min.

(c) clockwise

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