KAHIBARO
Table of Contents
Motion described by angle
In circular motion, position can be described by an angle instead of an ordinary straight line distance. As an object moves around a circle, its location changes from one angular position to another. Angular velocity tells us how fast that angle changes with time.
If an object rotates through a larger angle in a shorter time, it has a greater angular velocity. This idea is the rotational version of ordinary velocity in straight line motion.
Definition of angular velocity
Angular velocity is the rate of change of angular position. If the angular position is $\theta$, then the average angular velocity over a time interval $\Delta t$ is
$$
\omega_{\text{avg}} = \frac{\Delta \theta}{\Delta t}
$$
If we want the angular velocity at an exact instant, we use the instantaneous angular velocity
$$
\omega = \frac{d\theta}{dt}
$$
The symbol for angular velocity is usually $\omega$, the Greek letter omega.
Important definition:
$$
\omega = \frac{d\theta}{dt}
$$
Angular velocity measures how quickly angular position changes with time.
Units of angular velocity
The SI unit of angular velocity is the radian per second, written as $\text{rad/s}$.
A radian is technically a ratio of lengths, so it is dimensionless in a strict mathematical sense, but in physics we keep the unit "rad" to remind us that the motion is angular.
Some common units are shown below.
| Quantity | Symbol | Unit |
|---|---|---|
| Angular position | $\theta$ | rad |
| Time | $t$ | s |
| Angular velocity | $\omega$ | rad/s |
Positive and negative angular velocity
Angular velocity can be positive or negative, depending on the chosen direction of rotation.
By common convention, counterclockwise rotation is taken as positive, and clockwise rotation is taken as negative.
For example, if the angle increases with time, then $\omega > 0$. If the angle decreases with time, then $\omega < 0$.
The sign of angular velocity depends on direction.
Counterclockwise is usually positive, clockwise is usually negative.
Average and instantaneous angular velocity
Average angular velocity describes the overall angular change during a time interval:
$$
\omega_{\text{avg}} = \frac{\theta_2 - \theta_1}{t_2 - t_1}
$$
Instantaneous angular velocity describes how fast the object is rotating at a particular moment:
$$
\omega = \lim_{\Delta t \to 0}\frac{\Delta \theta}{\Delta t}
$$
If the object rotates steadily, then the average and instantaneous angular velocities are the same. If the rotation speeds up or slows down, they are different.
Relation to period and frequency
For one complete revolution, the angular displacement is
$$
\Delta \theta = 2\pi \text{ rad}
$$
If the object takes time $T$ for one full revolution, where $T$ is the period, then
$$
\omega = \frac{2\pi}{T}
$$
If the frequency is $f$, meaning the number of revolutions per second, then since $f = 1/T$,
$$
\omega = 2\pi f
$$
These relations are very useful in circular and rotational motion.
For uniform circular motion:
$$
\omega = \frac{2\pi}{T} = 2\pi f
$$
where $T$ is the period and $f$ is the frequency.
Interpreting angular velocity
Angular velocity tells us how quickly an object sweeps out angle, not how far it travels along the circle in meters. Two objects can have the same angular velocity while moving on circles of different sizes.
If both objects complete one revolution in the same time, they have the same angular velocity. However, the object on the larger circle travels a greater distance along its path.
This means angular velocity describes rotational rate, not the actual linear distance covered.
Uniform angular velocity
When angular velocity remains constant, the object rotates through equal angles in equal time intervals. In that case, angular position changes linearly with time:
$$
\theta = \theta_0 + \omega t
$$
where $\theta_0$ is the initial angular position.
This equation is similar to the constant velocity equation in one dimensional motion, but here the motion is angular.
If angular velocity is constant:
$$
\theta = \theta_0 + \omega t
$$
This is the basic equation for uniform rotational motion.
Visual picture
The angle changes as the point moves around the circle. Angular velocity measures how fast that angle grows or shrinks.
A simple example
Suppose a wheel turns through an angle of $6\,\text{rad}$ in $2\,\text{s}$. Its average angular velocity is
$$
\omega_{\text{avg}} = \frac{6}{2} = 3\,\text{rad/s}
$$
Suppose instead a fan rotates at frequency $5\,\text{Hz}$. Then its angular velocity is
$$
\omega = 2\pi f = 2\pi(5) = 10\pi\,\text{rad/s}
$$
So the fan rotates with angular velocity $10\pi\,\text{rad/s}$.
Angular velocity as a vector idea
In basic circular motion, angular velocity is often treated as a signed scalar, positive or negative depending on rotation direction. In more advanced physics, angular velocity can also be represented as a vector pointing along the axis of rotation. That fuller vector treatment belongs to later study of rotation and vectors.
For now, the key idea is simple: angular velocity tells how fast an angle changes with time.
Key formulas
| Situation | Formula |
|---|---|
| Average angular velocity | $\omega_{\text{avg}} = \dfrac{\Delta \theta}{\Delta t}$ |
| Instantaneous angular velocity | $\omega = \dfrac{d\theta}{dt}$ |
| One revolution | $\Delta \theta = 2\pi$ |
| Relation to period | $\omega = \dfrac{2\pi}{T}$ |
| Relation to frequency | $\omega = 2\pi f$ |
| Constant angular velocity | $\theta = \theta_0 + \omega t$ |
Core ideas to remember:
$$
\omega_{\text{avg}} = \frac{\Delta \theta}{\Delta t}, \qquad \omega = \frac{d\theta}{dt}
$$
$$
\omega = \frac{2\pi}{T} = 2\pi f
$$
Angular velocity is the rate of change of angular position, measured in $\text{rad/s}$.