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 2.5 kg·m2 about its central axis, is set spinning counterclockwise at 450 rev/min. The second disk, with rotational inertia 5.4 kg·m2 about its central axis, is set spinning counterclockwise at 700 rev/min. They then couple together. (a) What is their angular speed after coupling? rev/min (b) If instead the second disk is set spinning clockwise at 700 rev/min, what is the angular velocity after coupling? (Include the sign. Take counterclockwise to be positive.) rev/min

Explanation / Answer

I1 = moment of inertia of first disk = 2.5 kgm2

I2 = moment of inertia of second disk = 5.4 kgm2

a)

consider the clockwise angular momentum as negative and counter clockwise momentum as positive

W1 = angular velocity of first disk = 450 rev/min

W2 = angular velocity of first disk = 700 rev/min

W = angular velocity of the combination after coupling

using conservation of angular momentum

I1 W1 + I2 W2 = (I1 + I2) W

(2.5) (450) + (5.4) (700) = (2.5 + 5.4) W

W = 620.886 rev/min

620.886 rev/min in counterclockwise direction

b)

using conservation of angular momentum

I1 W1 + I2 W2 = (I1 + I2) W

(2.5) (450) - (5.4) (700) = (2.5 + 5.4) W

W = - 336.1 rev/min

336.1 rev/min in clockwise direction

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