Two cylinders of the same material and length, are held horizontally one atop th

Question

Two cylinders of the same material and length, are held horizontally one atop the other with space in between so they are not touching.. Viewed from the end they look like two circles on top of each other with space between(looks like a lower case o on top of an upper case O with space between). The upper cylinder has a smaller radius. Both can rotate freely about and axis through their centers. the lower cylinder rotates with a constant angular velocity. The upper cylinder(which is not rotating initially) is lowered onto the lower cylinder. The forces of friction will eventually cause the cylinders to rotate with constant angular velocity but in opposite directions. What is the final angular velocity of the smaller cylinder? Are any quantities conserved?. Please adequately explain any equations, thx.

Explanation / Answer

in this case,the angular velocity of the WHOLE SYSTEM is conserved since the frictional force which acts is an internal force between the two bodies(i.e. friction speeds up one of the body and slows down the other body)
so, applying law of conservation of angular momentum about the line of contact of the cylinders,(let the radius's be r and R and masses be m1 and m2 and initial angular velocity be o and assuming the cylinders to be hollow). so,

2m1r1^2o=2m1r1^2-2m2r2^2 (moment of inertia about the line of contact=2mr^2)

or =m1r1^2o/(m1r1^2-m2r2^2)

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