A circular wooden hoop of mass `m` and radius `R` rests fiat on a frictionless surface. A bullet, also of mass `m` and moving with a velocity `v`, strikes the hoop and gets embedded in it. The thickness of the hoop is much smaller than `R`. Find the angular velocity with which the system rotates in after the bullet strikes the hoop.

Explanation

Let the velocity of the `CM` of the system after the strike be `v_(c)` By conservation of linear momentum

Position of `CM` `y=(m.O+m.R).((m+m))+m=R/2`
Using conservation of angular mometum about `CM` of the system (hoop`+`bullet).
`mvR/2I_(c)omega`
`=(I_("bullet")+I_("hoop")_(c)omega`
`=[m(R/2)^(2)+{mR^(2)+m(R/2)^(2)}omega]`
After solving we get `omega=v/R`