A conical pendulum is a pendulum that consist of a
mass
hanging from
a string of length
that describes a circle when in motion, see diagram on the left .
The characteristic of this motion can be obtained in terms of the length
of the string and the angle
with respect
to the vertical. In order to start solving this motion, the first
selection is of a suitable frame of reference; in this case, a xy-coordinate
system that travels with the mass is the most appropriate, .
In this drawing, the radius of the circle is also shown. This radius can
be calculated from the triangle formed by the string, the vertical, and
the radius of the circle,
,
resulting in .
The next step is to identified the forces acting on the mass, in this
case, there are two forces acting on the mass, the weight of the object,
,
and the tension of the string,
.
In order to complete the vector addition of this vector, it is
convenient to express the tension of the string in terms of the
components along the x and y axis,
. The values of these components are calculated after the angle
has been
recognized in the lower triangle of the drawing (equivalent to the
larger triangle,
).
Thus, the components are
and
.The
last drawing of the sequence includes the velocity of the mass. The
velocity is perpendicular to the plane form by the x and y axis,
.

To solve the remaining part of this setting the standard procedure involving
Newton's laws of motion is applied.

y-component:
with
the acceleration in the y-direction being zero,
.
Newton's second law for this component is
from
where the tension of the string may be found,
.

x-component: In this case, there is only one force in
this direction,
with
the acceleration in the x-direction being the centripetal acceleration,
.
Thus, applying Newton's second law to this component, the following equation is
established,
.
Substituting the values of the centripetal acceleration and the tension of the
string,
.
Substituting in this relation the value of the radius,
,
the equation becomes,
. From
this last expression, if the angle
is given,
the value of the velocity may be obtained,