A wooden log of mass `M` and length `L` is hinged by a frictionless nail at `O`. A bullet of mass `m` strikes with velocity `v` and sticks to it. Find the angular velocity of the system, immediately after the collision about `O`. Also, calculate the minimum value of `v` so that log becomes horizontal after collision.

Explanation

By the conservation of angular momentum about `O` between (`1`) and (`2`)
`mvL=I_(0)omega=(mL^(2)+(ML^(2))/(3))omega`
`omega=(3mv)/((M+3m)L)`
Takin `O` as reference level, applying energy conservation between (`2`) and (`3`)
`-mgl-Mg((L)/(2))+(1)/(2)I_(0)omega^(2)=0`
`((M+2m)gL)/(2)=(1)/(2)((M+3m)L^(2))/(3)xx[(3mv)/((M+3m)L)]^(2)`
`(M+2m)gL=(3m^(2)v^(2))/((M+3m))`
`v=((M+3m)(M+2m)gL)/(3m^(2))=v_(min)`