Say you had two fixed, non-identical objects rotating around one axis at different speeds relative to one another, and they can be coupled together by friction (like how a clutch in a car works). How would I calculate the frictional force required for them to stick together instantaneously?
Object A and B are two solid cylinders rotating about a single axis (imagine how they would rotate if they were rolling) Object A is rotating at 10000 RPM, and object B is rotating at 1000 RPM; Object B has a larger radius, is made of a denser material, and has more mass and a larger moment of inertia. The two object come into contact with each other (picture the clutch of a car).
If the frictional force is sufficient, they should "stick" and the angular momentum should be conserved (I1ω1+I2ω2=Inetω); but if it isn't, they should slip.
What would be the frictional force required to "stick" the two objects together instantaneously?