Question

Show that the angular momentum about any point of a single particle moving with constant velocity remains constant throughout the motion.

Answer 1

Refer to Fig. At any time t, suppose the particle moving with velocity `vec v` is at `P`. We have to calculate angular momentum of the particle about any orbitray point `O`. The angular momentum

`vecL = vecr xx vecp = vecr xx m vecv`
`|vecL| = r m v sin theta`
where `theta` is the smaller angle between `vecr` and `vecv`.
From Fig. `r sin theta = OK =` perpendicular distance of `O` from teh line of motion of particle.
As position of particle changes, `r` and `theta` both change, but `r sin theta` remains constant.
Therefore, magnitude of `vecL = m v r sin theta = m v (OK) =` constant
The direction of `vecL` is ` _|_ vecr` and `vecv` and inwards. This also does not change.
Hence `vecL` remains the same in magnitude and direction.