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**1. Homework Statement**

A flywheek with a cavity in its upper surface is constrained to rotate about a fixed vertical axis. Its moment of intertia about the axis is I and the cvity is a distance a from the axis. It is initially rotating with angular velocity w1. Into the cavity is dropped a sphere of mass M and moment of inertia i which is spinning with angular velocity w2 about a vertical axis. At the moment it is dropped into the cavity it has no horizontal motion. Derive an expression for the angular velocity of the composite system when the sphere and the flywheel are at relative rest.

**2. Homework Equations**

Parallel axis theorem.

L=Iw

**3. The Attempt at a Solution**

I think I'm struggling with this problem because the sphere and the flywheel are rotating around different axis.

The moment of inertia of the sphere about the flwheel's axis of rotation= i + Ma^2, so using the conservation of angular momentum if the sphere wasn't rotating initially the systems angular velocity would be straightforward to find.

However the sphere is rotating, and also the flywheel is constrained so I'm not sure how this affects the system.

A push in the right direction would be appreciated.

Thanks