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 3.88 kg·m² about its central axis, is set spinning counterclockwise (which may be taken as the positive direction) at 176 rev/min. The second disk, with rotational inertia 7.15 kg·m² about its central axis, is set spinning counterclockwise at 823 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 823 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

Given: First Disk, Rotational inertia I₁ = 3.88 kgm²

Angular velocity (₁) = 176 rev./min

Second Disk, Rotational Inertia I₂ = 7.15 kg m²

Angular velocity (₂) = 823 rev/min

We will use conservation of angular momentum for this problem. So, change in angular momentum will be zero.

a)

Angular momentum of first disk + Angular momentum of second disk = Angular momentum of combined disks

I₁₁ + I₂₂ = (I₁+I₂) _f

(3.88×176 ) + (7.15×823) = ( 3.88+7.15)×_f

_f = 595.406 rev/min.

b) as we assume counter clockwise to be positive. So here , in this case second disk is rotating in negative direction .

₂ = - 823 rev/min.

So , (3.88×176)-(7.15×823) = (3.88+7.15) _f

_f = - 471.58 rev/min.

c) hence direction of rotation after they coupled is clockwise .