Considering a particle of mass
moving
in a circle of radius at a
constant rate. That is maintaining constant the magnitude of the velocity but
changing the direction. Such motion requires of a force,
, directed to the center of
the circle, the centripetal force.

In this motion, if the angular momentum is calculated
with respect to the center of the circle,
, the angular
momentum is unchanged, .
The angular momentum is given by

where
the
radius of the circle,
the
magnitude of the linear momentum of the particle, and
is
the unit vector pointing out of the page. Every one of the quantities
involved are constant and, consequently, the angular momentum is
unchanged or conserved (constant). Because
and
are anti-parallel
the torque with respect to point
is zero at all times, confirming that the angular momentum should be unchanged when
calculated respect to point
.

However, the same calculation with respect to another
point, such as point
, results in a
changing angular momentum. In this case,

where
is
given above and
is
calculated below.

From the diagram,
,
with
and .

Thus,
and
which implies that the angular momentum with respect to point
is

but
and the angular momentum becomes
.

This angular momentum correspond to a torque with respect to point
obtained from
where
as
discussed above. Therefore, the torque becomes
with ,
,
implying that
.
The result for the torque with respect to point
is