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__Problem Statement__:Suppose you have a uniform disk of mass M and radius R that can rotate about its central axis. A particle with mass M and velocity V strikes the rim of the disk (along a path tangent to the disk), gets lodged into it, and causes the disk to spin. Show that the linear momentum of the system is conserved.

__Attempted Solution__:I start by using the conservation of angular momentum:

MVR = (0.5MR^2 + MR^2)ω

The left side of the equation is the total angular momentum before the collision (it is just the angular momentum of the particle since the disk is stationary) and the right side is the angular momentum after the collision. The expression in parenthesis is the moment of inertia of the disk-particle system. With a bit of algebra you can conclude that:

ω = (2V)/(3R)

To analyze the linear momentum after the collision I look at the linear velocity of the center of mass of the disk-particle system. The center of mass is in between the center of the disk and the lodged particle, so the linear momentum is given by:

(0.5Rω)(2M) = (V/3)(2M) = (2/3)MV

However, the linear momentum before the collision is MV. Does anyone know what I am doing wrong?