Uniform circular motion

Suppose that an object executes a circular orbit of radius $r$ with uniform tangential speed $v$. The instantaneous position of the object is most conveniently specified in terms of an angle $\theta$. See Fig. 57. For instance, we could decide that $\theta=0^\circ$ corresponds to the object's location at $t=0$, in which case we would write

$$\theta(t) = \omega t,$$ (245)

where $\omega$ is termed the angular velocity of the object. For a uniformly rotating object, the angular velocity is simply the angle through which the object turns in one second.

Figure 57: Circular motion.

Consider the motion of the object in the time interval between $t=0$ and $t=t$. In this interval, the object rotates through an angle $\theta$, and traces out a circular arc of length $s$. See Fig. 57. It is fairly obvious that the arc length $s$ is directly proportional to the angle $\theta$: but, what is the constant of proportionality? Well, an angle of $360^\circ$ corresponds to an arc length of $2\pi r$. Hence, an angle $\theta$ must correspond to an arc length of

$$s = \frac{2\pi}{360^\circ} r \theta(^\circ).$$ (246)

At this stage, it is convenient to define a new angular unit known as a radian (symbol rad.). An angle measured in radians is related to an angle measured in degrees via the following simple formula:

$$\theta({\rm rad.}) = \frac{2\pi}{360^\circ} \theta(^\circ).$$ (247)

Thus, $360^\circ$ corresponds to $2 \pi$ radians, $180^\circ$ corresponds to $\pi$ radians, $90^\circ$ corresponds to $\pi/2$ radians, and $57.296^\circ$ corresponds to 1 radian. When $\theta$ is measured in radians, Eq. (246) simplifies greatly to give

$$s = r \theta.$$ (248)

Henceforth, in this course, all angles are measured in radians by default.

Consider the motion of the object in the short interval between times $t$ and $t+\delta t$. In this interval, the object turns through a small angle $\delta\theta$ and traces out a short arc of length $\delta s$, where

$$\delta s = r \delta\theta.$$ (249)

Now $\delta s/\delta t$ (i.e., distance moved per unit time) is simply the tangential velocity $v$, whereas $\delta\theta /\delta t$ (i.e., angle turned through per unit time) is simply the angular velocity $\omega$. Thus, dividing Eq. (249) by $\delta t$, we obtain

$$v = r \omega.$$ (250)

Note, however, that this formula is only valid if the angular velocity $\omega$ is measured in radians per second. From now on, in this course, all angular velocities are measured in radians per second by default.

An object that rotates with uniform angular velocity $\omega$ turns through $\omega$ radians in 1 second. Hence, the object turns through $2 \pi$ radians (i.e., it executes a complete circle) in

$$T = \frac{2 \pi}{\omega}$$ (251)

seconds. Here, $T$ is the repetition period of the circular motion. If the object executes a complete cycle (i.e., turns through $360^\circ$) in $T$ seconds, then the number of cycles executed per second is

$$f = \frac{1}{T} = \frac{\omega}{2 \pi}.$$ (252)

Here, the repetition frequency, $f$, of the motion is measured in cycles per second--otherwise known as hertz (symbol Hz).

As an example, suppose that an object executes uniform circular motion, radius $r=1.2 {\rm m}$, at a frequency of $f=50 {\rm Hz}$ (i.e., the object executes a complete rotation 50 times a second). The repetition period of this motion is simply

$$T = \frac{1}{f} = 0.02 {\rm s}.$$ (253)

Furthermore, the angular frequency of the motion is given by

$$\omega = 2\pi f = 314.16 {\rm rad. / s}.$$ (254)

Finally, the tangential velocity of the object is

$$v = r \omega = 1.2\times 314.16 =376.99 {\rm m/s}.$$ (255)