Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the vector angular momentum of the two particle system is the same whatever be the point about which the angular momentum is taken.

Explanation

Suppose each particle of mass m is moving along two parallel lines at distance d.

Each particle has speed v in opposite direction. Suppose at any instant particles are at point `P_(1)andP_(2)`.
Suppose the angular moment of both particle with respect to point O is `vec(L_(1)) and vec(L_(2))`.
`:.vec(L_(1))=vec(r_(1))xxmvecv=andvec(L_(2))=vec(r_(2))xxmvecv`
Now suppose the angular momentum of the system is `vecL`
`:.vecL=vec(L_(1))-vec(L_(2))`
`=vec(r_(1))xxmvecv-vec(r_(2))xxmvecv`
Negative sign is present due to both particle moves in opposite direction
`:.|vecL|=|vec(L_(1))|-|vec(L_(2))|`
`=mvr_(1)sintheta_(1)-mvr_(2)sintheta_(2)`
`=mv[r_(1)sintheta_(1)-r_(2)sintheta_(2)|....(1)`
where `theta_(1)` angle is between `vec(r_(1)) and vecv and theta_(2)` angle is between `vec(r_(2)) and vecv`
`:.OM=r_(1)sintheta_(1) and ON=r_(2)sintheta_(2)` and `OM-ON=d`
`r_(1)sintheta_(1)-r_(2)sintheta_(2)=d`
`:.` From eqn. (1)
`L=mvd`
Direction L is perpendicular to the plane form by `vecrandvecv` and it toward inside end does not change it direction with line and hence the magnitude and direction of angular momentum remains constant, so L is unchanged.