Uniform circular motion

Consider an object executing uniform circular motion of radius $a$. 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 $x$-$y$ plane. As illustrated in Fig. 99, the instantaneous position of the object can be conveniently parameterized in terms of an angle $\theta$.

Figure 99: Uniform circular motion.

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

$$\theta = \omega t,$$ (536)

where $\omega$ is the angular rotation frequency (i.e., the number of radians through which the object rotates per second). Here, it is assumed that $\theta=0$ at $t=0$, for the sake of convenience.

From simple trigonometry, the $x$- and $y$-coordinates of the object can be written

$$x = a \cos\theta,$$ (537)

$$y = a \sin\theta,$$ (538)

respectively. Hence, combining the previous equations, we obtain

$$x = a \cos(\omega t),$$ (539)

$$y = a \cos(\omega t - \pi/2).$$ (540)

Here, use has been made of the trigonometric identity $\sin\theta = \cos(\theta-\pi/2)$. 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 $x$- and the $y$ -axes. Note, however, that these two motions are $90^\circ$ (i.e., $\pi/2$ 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.