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 3.30 kg·m² about its central axis, is set spinning counterclockwise at 450 rev/min. The second disk, with rotational inertia 6.60 kg·m² about its central axis, is set spinning counterclockwise at 900 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 900 rev/min,
what are their:
(b) angular speed and
(c) direction of rotation after they couple together?

a) 750 rev/min

b) 450 rev/min

c) clockwise direction

Explanation:

We are given;

Rotational inertia of first disk;I_1 = 3.3 kg.m²

Angular velocity of first disk;ω_1 = 450 rev/min

Rotational Inertia of second disk;I_2 = 6.6 kg.m²

Angular velocity of second disk;ω_2 = 900 rev/min

Total rotational inertia;I_tot = 3.3 + 6.6 = 9.9 kg.m²

Since they couple together, we have;

I_tot × ω = (I_1 × ω_1) + (I_2 × ω_2)

Plugging in values, we have;

9.9ω = (3.3 × 450) + (6.6 × 900)

9.9ω = 7425

ω = 7425/9.9

ω = 750 rev/min

b) now, we are told the second disk is spinning clockwise, thus we have;

I_tot × ω = (I_1 × ω_1) - (I_2 × ω_2)

9.9ω = (3.3 × 450) - (6.6 × 900)

9.9ω = -4455

ω = -4455/9.9

ω = - 450 rev/min

The minus sign indicates a clockwise direction.

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