An application of the definition of angular momentum is to calculate the
angular momentum of a falling particle under the influence of gravity.

Let us select the vertical axis along the direction of
motion of the falling particle as shown in the graph of the
left, .
The angular momentum of the particle of mass
with
respect to point
is
where
the linear momentum of the particle. Point
is located at the coordinate
of the
positive
-axis. The direction of the corresponding angular
momentum vector is pointing out of the page,
.
However, if point
is located on the
negative side of the
-axis,
,
the corresponding direction of the vector angular momentum is
directed inside the page,
.
On the other hand, if the selected location for point
is at the origin
of coordinates,
,
the corresponding angular momentum is zero.

The magnitude of the angular momentum with respect to point
is given by

with

but
.
Therefore, the magnitude of the angular momentum is
but
from the diagram
and
the magnitude of the angular momentum becomes

In free fall, the velocity of the particle changes due to the
acceleration of gravity. Consequently, the angular momentum also
changes. Notices that, if the angular momentum is calculated with
respect to the origin, the angular momentum is zero at all times.

In all the other cases, the change in the magnitude of the
angular momentum is due to the torque acting on the particle. The torque is
with
. When point
is located in the
positive side of the
-axis, the
direction of the torque is out of the page, into the page when point
is in the negative
side of the
-axis, and
zero when placed at the origin of coordinates. The magnitude of the torque is

but the force acting on the particle is the weight of it,
. Thus, the torque is

This result corresponds to the change with the time
associated with the angular momentum,

where the
value
corresponding to the location of point
is independent of
the time.