KAHIBARO
Table of Contents
Meaning and Definition
Angular velocity describes how fast an object rotates and in which sense it rotates. In rotational motion, instead of tracking how far an object moves along a straight line, we track how its angular position changes.
If the angular position is $\theta$, then angular velocity tells us how quickly $\theta$ changes with time.
The average angular velocity over a time interval $\Delta t$ is
$$
\omega_{\text{avg}} = \frac{\Delta \theta}{\Delta t}
$$
where $\Delta \theta = \theta_f - \theta_i$.
The instantaneous angular velocity is the limit for a very small time interval:
$$
\omega = \frac{d\theta}{dt}
$$
This is the rotational version of linear velocity.
Important definition:
$$
\omega = \frac{d\theta}{dt}
$$
Angular velocity is the rate of change of angular position with respect to time.
Direction and Sign
Angular velocity has a sign that depends on the chosen direction of rotation. In a two dimensional description, it is common to take counterclockwise rotation as positive and clockwise rotation as negative.
So, if $\theta$ increases with time, then $\omega > 0$. If $\theta$ decreases with time, then $\omega < 0$.
This sign convention helps us describe rotational motion consistently.
SI Unit
Angular velocity is measured in radians per second.
$$
[\omega] = \text{rad/s}
$$
The radian is technically dimensionless because it is a ratio of arc length to radius, but in physics we write rad/s to make the meaning clear.
Relation to Angle Traversed
Suppose an object rotates through an angle $\Delta \theta$ in time $\Delta t$. Then its average angular velocity is just that angle change divided by the time.
For example, if a wheel turns through $2\pi$ radians in $4\,\text{s}$, then
$$
\omega_{\text{avg}} = \frac{2\pi}{4} = \frac{\pi}{2}\,\text{rad/s}
$$
A full revolution corresponds to
$$
2\pi \text{ radians}
$$
so angular velocity is often connected to revolutions as well.
Angular Velocity and Frequency
If an object completes one full rotation in a period $T$, then in that time it turns through $2\pi$ radians. Therefore,
$$
\omega = \frac{2\pi}{T}
$$
If the rotation frequency is $f$, meaning the number of revolutions per second, then
$$
f = \frac{1}{T}
$$
and so
$$
\omega = 2\pi f
$$
This is a very useful connection between angular motion and periodic motion.
Key relationships:
$$
\omega = \frac{2\pi}{T}
$$
$$
\omega = 2\pi f
$$
where $T$ is the period and $f$ is the frequency.
Angular Velocity and Linear Speed
For a point moving in a circle of radius $r$, the distance traveled along the circular path is related to the angle by
$$
s = r\theta
$$
Differentiating with respect to time gives the tangential or linear speed:
$$
v = r\omega
$$
This means points farther from the axis move faster in a linear sense, even though they share the same angular velocity if they belong to the same rigidly rotating body.
For example, on a spinning disk, every point completes each turn in the same time, so every point has the same $\omega$, but a point near the edge has a larger linear speed than a point near the center.
For circular motion,
$$
v = r\omega
$$
Same rotation rate does not mean same linear speed. Linear speed increases with distance from the axis.
Constant Angular Velocity
If angular velocity is constant, then the angular position changes uniformly with time. In that case,
$$
\theta = \theta_0 + \omega t
$$
where $\theta_0$ is the angular position at $t = 0$.
This is the rotational analogue of uniform motion in a straight line.
Comparison with Linear Velocity
Angular velocity and linear velocity are related, but they are not the same quantity. The table below highlights the parallel ideas.
| Linear motion | Rotational motion |
|---|---|
| Position $x$ | Angular position $\theta$ |
| Velocity $v = \frac{dx}{dt}$ | Angular velocity $\omega = \frac{d\theta}{dt}$ |
| Unit m/s | Unit rad/s |
This comparison helps show that rotational kinematics uses ideas very similar to ordinary kinematics, but with angular quantities.
Visualizing Angular Velocity
A rotating wheel is a good example. If the wheel rotates by larger angles in equal times, it has greater angular velocity.
In this picture, the radius line changes its angle by $\Delta \theta$ during some time interval $\Delta t$. The angular velocity is the ratio $\Delta \theta / \Delta t$.
Common Quantities and Formulas
| Quantity | Symbol | Formula | SI unit |
|---|---|---|---|
| Angular position | $\theta$ | angular coordinate | rad |
| Average angular velocity | $\omega_{\text{avg}}$ | $\frac{\Delta\theta}{\Delta t}$ | rad/s |
| Instantaneous angular velocity | $\omega$ | $\frac{d\theta}{dt}$ | rad/s |
| Relation to period | $\omega$ | $\frac{2\pi}{T}$ | rad/s |
| Relation to frequency | $\omega$ | $2\pi f$ | rad/s |
| Relation to linear speed | $v$ | $r\omega$ | m/s |
Final Idea
Angular velocity tells us how quickly rotation happens. It is one of the central quantities of rotational kinematics because it connects angular position, time, circular motion, and linear speed.
Core formulas for angular velocity:
$$
\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}
$$
$$
\omega = \frac{d\theta}{dt}
$$
$$
\omega = \frac{2\pi}{T} = 2\pi f
$$
$$
v = r\omega
$$