Question

Calculus-based physics rotational inertia:

a) 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 2.56 kg·m² about its central axis, is set spinning counterclockwise (which may be taken as the positive direction) at 309 rev/min. The second disk, with rotational inertia 7.07 kg·m² about its central axis, is set spinning counterclockwise at 760 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 760 rev/min, what are their (b) angular velocity (using the correct sign for direction) and (c) direction of rotation after they couple together?

b) The uniform rod (length 0.309 m) in the figure rotates in the plane of the figure about an axis through one end, with a rotational inertia of 0.275 kg·m². As the rod swings through its lowest position, it collides with a 0.242 kg putty wad that sticks to the end of the rod. If the rod's angular speed just before the collision is 3.21 rad/s, what is the angular speed of the rod-putty system immediately after the collision?

Answer 1

(a) Suppose I₁ and I₂ are the moment of inertia of disk1 and disk2 respectively, and let omega1 and omega2 are the initial angular speeds then according to law of conservation of angular momentum

I₁₁ + I₂₂ = I

where I is the moment of inertia after two disk coupled and is the angular speed of disks after they coupled

Hence

2.56x309 + 7.07x760 = (2.56+7.07)

9.63 = 6164.24
or
= 6164.24/9.63 = 640 rev/min (counter clockwise )

(b) Here second disk spins clockwise hence

I_{11 -} I₂₂ = I

where I is the moment of inertia after two disk coupled and is the angular speed of disks after they coupled

Hence

2.56x309 - 7.07x760 = (2.56+7.07)

9.63 = - 4582.16
or
= -4582.16/9.63 = - 475.8 rev/min

(c) negative sign indicates that after coupling, disk rotate clockwise