CONSERVATION OF ANGULAR MOMENTUM

Explanation

Law (or principle) of conservation of angular momentum : The angular momentum of a body is conserved if the resultant external torque acting on the body is zero.
Proof : Consider a particle of mass m in motion. Let its position vector with respect to the origin at any instant be `vecr`.
Then, at this instant the linear velocity of this particle is `vecv=(vecdr)/(dt)`, its linear momentum is `vecp=mvecv` and its angular momentum about an axis through the origin is `vecl=vecrxxvecp`.
Consider a case where its angular momentum `vecl` changes with time due to a torque `vecoita` exerted on the particle.
The time rate of change of angular momentum,
`(dvecl)/(dt)=(d)/(dt)(vecrxxvecp)`
`=(dvecr)/(dt)xxvecp+vecr(dvecp)/(dt)`
`=vecvxxmvecv+vecrxxvecF," "` where `(dvecp)/(dt)=vecF`, the force on the particle
`=vecrxxvecF " " (:' vecvxxvecu=0)`
`=vecoita`
Hence, if `vecoita=0,(dvecl)/(dt)=0`
`:. vecl`= constant, i.e., `vecl` is conserved. This proves the law (or principle) of conservation of angular momentum as the result obtained here holds true for every particle belonging to a rotating body.