A ring with radius R and mass M has a beam of mass Mb and length 2R fixed to it

Question

A ring with radius R and mass M has a beam of mass Mb and length 2R fixed to it across its diameter. The composite object rotates initially in a horizontal plane at angular velocity omega0 around an axis through the center of the ring (the axis is perpendicular to the plane of the page and we view the ring from above). Suddenly a piece of putty with mass m falls straight down onto the beam at a distance R/2 from the center. What is the angular velocity omega1 after the putty sticks to the beam? [I ring = MR2; I beam = 1/3MbR2 for indicated axis of rotation]

Explanation / Answer

IInitial = Mr^2 + 1/3 *MbR^2

I final = IInitial + m* (R/2)^2

Applying conservation of angular momentum

I final * t =IInitial *0

  t = IInitial *0/I final = 0 * ( MR2 +MbR2/3)/( MR2 +MbR2/3+mR2/4) --- > Answer

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