Angular quantities
Angular position
|
|
Since the object moves around along a circle, the radial distance \(r\) stays constant throughout the motion (it is the radius of the circle). A single coordinate is enough to specify the position: the angle \(\phi\) between a fixed direction (eg, a horizontal vector pointing to the right) and the position vector. We call this angle the angular position. The angular position is measured in radians. If we use radians, the length of the arc \(s\) swept by the particle can be expressed using the angular position: \[s = r \phi.\] |
Period and frequency
In addition to the kinematic quantities describing linear motion, we can also define quantities that are related to the periodicity of circular motion.
Period
The period (\(T\)) is the time required for an object in circular motion to travel one full circle.
Frequency
Frequency (\(f\) or \(\nu\)) is the number of revolutions in a second.
Frequency can be obtained as the reciprocal of the period:
\[f := \frac{1}{T}.\]
The SI unit of frequency is the hertz:
\[\left[f\right] = 1\,\frac{1}{\textrm{s}} = 1\,\textrm{Hz}.\]
Angular kinematic quantities
We have seen that is convenient to use the angular position to specify where the particle is along the circular path. Just as we could define further linear quantities – velocity and acceleration – that describe the rate at which the position (and then the velocity) changes, we can define these rates of change in connexion with the angular position as well.
Angular frequency | angular speed
Angular frequency or angular speed is the rate of change of the angular position:
\[\omega := \frac{\mathrm{d}\phi}{\mathrm{d}t}.\]
The SI unit of the angular frequency is the one per second (sometimes referred to as the rad per second), which cannot be called hertz in connexion with angular frequency:
\[\left[\omega\right] = 1\,\mathrm{s}^{-1} = 1\,\frac{\mathrm{rad}}{\mathrm{s}}.\]
Angular acceleration
Angular acceleration is the rate of change of the angular speed:
\[\alpha := \frac{\mathrm{d}\omega}{\mathrm{d}t} = \frac{\mathrm{d}^2\phi}{\mathrm{d}t^2}.\]
The SI unit of the angular frequency is the one per second squared (sometimes referred to as the rad per second squared):
\[\left[\alpha\right] = 1\,\mathrm{s}^{-2} = 1\,\frac{\mathrm{rad}}{\mathrm{s}^2}.\]