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Suppose that an object, mass , is attached to the end of a light inextensible
string whose other end is attached to a rigid beam. Suppose, further, that the
object is given an initial horizontal velocity such that it executes a
horizontal circular orbit of radius with angular velocity . See
Fig. 60. Let be the vertical distance between the beam and the plane of
the circular orbit, and let be the angle subtended by the string with
the downward vertical.
Figure 60:
A conical pendulum.

The object is subject to two forces: the gravitational force which acts vertically
downwards, and the tension force which acts upwards along the string. The tension
force can be resolved into a component which acts vertically upwards, and
a component which acts towards the centre of the circle. Force balance
in the vertical direction yields

(262) 
In other words, the vertical component of the tension force balances the weight of the object.
Since the object is executing a circular orbit, radius , with angular velocity ,
it experiences a centripetal acceleration . Hence, it is subject to
a centripetal force
. This force is provided by the component of
the string tension which acts towards the centre of the circle. In other words,

(263) 
Taking the ratio of Eqs. (262) and (263), we obtain

(264) 
However, by simple trigonometry,

(265) 
Hence, we find

(266) 
Note that if is the length of the string then
. It follows that

(267) 
For instance, if the length of the string is
and the conical angle is
then the angular velocity of rotation is given by

(268) 
This translates to a rotation frequency in cycles per second of

(269) 
Next: Nonuniform circular motion
Up: Circular motion
Previous: Centripetal acceleration
Richard Fitzpatrick
20060202