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Consider an object executing uniform circular motion of radius . Let us set
up a cartesian coordinate system whose origin coincides with the centre of the circle,
and which is such that the motion is confined to the  plane.
As illustrated in Fig. 99, the instantaneous position of the object can be conveniently
parameterized in terms of an angle .
Figure 99:
Uniform circular motion.

Since the object is executing uniform circular motion, we expect the angle to increase
linearly with time. In other words, we can write

(536) 
where is the angular rotation frequency (i.e., the number of radians through which the
object rotates per second). Here, it is assumed that at , for the sake of convenience.
From simple trigonometry, the  and coordinates of the object can be written
respectively. Hence, combining the previous equations, we obtain
Here, use has been made of the trigonometric identity
.
A comparison of the above two equations with the standard equation of simple harmonic motion,
Eq. (505), reveals that our object is executing simple harmonic motion simultaneously along both the
 and the axes. Note, however, that these two motions are (i.e., radians)
out of phase. Moreover, the amplitude of the motion equals the radius of the circle.
Clearly, there is a close relationship between simple harmonic motion
and circular motion.
Next: Worked example 11.1: Piston
Up: Oscillatory motion
Previous: The compound pendulum
Richard Fitzpatrick
20060202