Three particles are moving as shown in the figure, Calculate the angular momentum of the third particle `C`, so that the angular momentum of system about point `O` becomes zero.

Explanation

We define the co-ordinate system as shown.
Angular momentum of particle at point `A` about point `O` is
`vecL_(A)=vecr_(A)xxvecp_(A)`
`=2hatjxx6hati=-12hatk kgm^(2)//s`
Angular momentum of particle at point `B` about point `O` is
`vecL_(B)=vecr_(B)xxvecp_(B)`
`=4hatixx24(-hatj)=-96(hatk) kg m^(2)//s`
Total angular momentum
`vecL_("Net")=vecL_(A)+vecL_(B)+vecL_(C)`
`implies0=vecL_(A)+vecL_(B)+vecL_(C)`
`impliesvecL_(C)=-(vecL_(A)+vecL_(B))`
`=12hatk+96hatk=108(hatk) kg m^(2)//s`