A ring (mass 4 M, radius 2 R) rotates in a CCW direction with an initial angular

Question

A ring (mass 4 M, radius 2 R) rotates in a CCW direction with an initial angular speed 2 . A disk (mass 4 M, radius 2 R) rotates in a CW direction with initial angular speed 2 . The ring and disk "collide" and eventually rotate together. Assume that positive angular momentum and angular velocity values correspond to rotation in the CCW direction.

What is the initial angular momentum Li of the ring+disk system? Write your answer in terms of MR2.

What is the final angular velocity f of the ring+disk system? Write your answer in terms of .

Explanation / Answer

angular momentum is given by the formula L, = I W

where I is moment of inertia and W is angular velocity

so I here differ for Dsk anf RIng

MOI of ring I = MR^2

MOI of Disk I = 0.5 MR^2

so

Intial Angular Momentum of Ring Ir = 4M * (2R)^2 * 2W = + 32 MR^2W

Intial angular Momentun of disk Id = 0.5* 4M *(2R)^2 * 2W = -16 MR^2W

total Intial angular Momentum Li = Ir +Id = (32-16) = 16 M R^2 W

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let W be the final angular velcoity

then final angular Momentum Id = ( MR^2+ 0.5 MR^2) W

Lf = 1.5 MR^2 W

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