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

Explanation

Let the particle with velocity v be at point P at some instant t. We want to caluclate tha angular momentum of the particle about an arbitary point O.

The angular momentum is `1=rxxmv`. Its magnitude is mvr `sin theta`, where `theta` is the angle between r and v as shown in Fig. 7.19. Although the particle changes position with time, the line of direction of v remains the same and hence OM= r sin `theta` is a constant.
Further, the direction of 1 is perpendicular to the plane of r and v. It is into the page of the figure. this direction does not change with time.
Thus, 1 remains the same in magnitude and direction and is therefore conserved. Is there any external torque on the particle ?