Next: Centripetal acceleration
Up: Circular motion
Previous: Introduction
Suppose that an object executes a circular orbit of radius
with uniform tangential speed . The instantaneous
position of the object is most conveniently specified in terms of an
angle . See Fig. 57. For instance, we could decide that
corresponds
to the object's location at , in which case we would write

(245) 
where 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
and . In this interval, the object rotates through an angle
, and traces out a circular arc of length . See Fig. 57.
It is fairly obvious that the arc length is directly proportional to the angle
: but, what is the constant of proportionality? Well, an angle of
corresponds to an arc length of . Hence, an angle
must correspond to an arc length of

(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:

(247) 
Thus, corresponds to radians, corresponds to radians,
corresponds to radians, and corresponds
to 1 radian. When is measured in radians, Eq. (246) simplifies
greatly to give

(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 and .
In this interval, the object turns through a small angle and
traces out a short arc of length , where

(249) 
Now
(i.e., distance moved per unit time)
is simply the tangential velocity , whereas
(i.e., angle turned through per unit time) is simply the angular velocity
. Thus, dividing Eq. (249) by , we obtain

(250) 
Note, however, that this formula is only valid if the angular velocity
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 turns through
radians in 1 second. Hence, the object turns through radians
(i.e., it executes a complete circle) in

(251) 
seconds. Here, is the repetition period of the circular motion. If the object
executes a complete cycle (i.e., turns through ) in seconds,
then the number of cycles executed per second is

(252) 
Here, the repetition frequency, , of the motion is measured in
cycles per secondotherwise known as hertz (symbol Hz).
As an example, suppose that an object executes uniform circular motion, radius
, at a frequency of
(i.e., the object executes a complete
rotation 50 times a second). The repetition period of this motion is simply

(253) 
Furthermore, the angular frequency of the motion is given by

(254) 
Finally, the tangential velocity of the object is

(255) 
Next: Centripetal acceleration
Up: Circular motion
Previous: Introduction
Richard Fitzpatrick
20060202