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 3.76 kg·m2 about its central axis, is set spinning counterclockwise (which may be taken as the positive direction) at 189 rev/min. The second disk, with rotational inertia 9.39 kg·m2 about its central axis, is set spinning counterclockwise at 893 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 893 rev/min, what are their (b) angular velocity (using the correct sign for direction) and (c) direction of rotation after they couple together?

Answer 1

Sign convention:

Clockwise: - ve

Counter clockwise: +ve

Data:

Rotational inertia of the first disk, I₁ = 3.76 kg m^2

Rotational inertia of the second disk, I₂ = 9.39 kg.m^2

Angular speed of the first disk, ω₁ = 189 rev/min

                                                           = 189 * (2π / 60) rad/s

                                                           = 19.79 rad/s

Angular speed of the second disk, ω₂ = 893 rev/min

                                                                = 893 * (2π / 60)

                                                                = 93.51 rad/s

(a)

From the law of conservation of angular momentum,

ω = [I₁ ω₁ + I₂ ω₂] / (I₁ + I₂)

    = [3.76 * 19.79 + 9.39 * 93.51] / (3.76 + 9.39)

    = 72.43 rad/s

(b)

From the law of conservation of angular momentum,

ω = [I₁ ω₁ - I₂ ω₂] / (I₁ + I₂)

    = [3.76 * 19.79 – 9.39 * 93.51] / (3.76 + 9.39)

    = - 61.11 rad/s

Ans:

Resulting angular speed = 61.11 rad/s

(c) direction of rotation after they couple together is clockwise