KAHIBARO
Table of Contents
Understanding the Change in Angular Velocity
In circular motion, angular position tells us where an object is on its circular path, and angular velocity tells us how fast that angular position is changing. Angular acceleration describes how the angular velocity itself changes with time.
If an object spins faster and faster, it has angular acceleration. If it spins more slowly over time, it also has angular acceleration, but in the opposite direction. So angular acceleration is not only about speeding up. It is about any change in rotational rate.
Mathematically, average angular acceleration is defined as
$$
\alpha_{\text{avg}} = \frac{\Delta \omega}{\Delta t}
$$
where $\Delta \omega$ is the change in angular velocity and $\Delta t$ is the time interval.
The instantaneous angular acceleration is
$$
\alpha = \frac{d\omega}{dt}
$$
This is the rotational version of ordinary acceleration in straight line motion.
Important definition:
$$
\alpha = \frac{d\omega}{dt}
$$
Angular acceleration is the rate of change of angular velocity with time.
Units of Angular Acceleration
Angular velocity is measured in radians per second, so angular acceleration is measured in radians per second squared:
$$
\text{unit of } \alpha = \text{rad/s}^2
$$
Because the radian is technically a ratio, it is dimensionless in a strict mathematical sense, but in physics we keep the word radian to make the meaning clear.
| Quantity | Symbol | Common unit |
|---|---|---|
| Angular position | $\theta$ | rad |
| Angular velocity | $\omega$ | rad/s |
| Angular acceleration | $\alpha$ | rad/s$^2$ |
Physical Meaning
Angular acceleration tells us how quickly rotational motion changes.
If $\alpha > 0$, the angular velocity increases in the positive angular direction.
If $\alpha < 0$, the angular velocity decreases in the positive direction, or increases in the negative direction.
The sign of angular acceleration depends on the chosen positive direction of rotation. Usually, counterclockwise is taken as positive.
For example, if a wheel has angular velocity
$$
\omega = 4 \,\text{rad/s}
$$
and after $2\,\text{s}$ it has angular velocity
$$
\omega = 10 \,\text{rad/s}
$$
then its average angular acceleration is
$$
\alpha_{\text{avg}} = \frac{10 - 4}{2} = 3 \,\text{rad/s}^2
$$
So the wheel’s spin rate increases by $3 \,\text{rad/s}$ each second.
Relation to Linear Acceleration
For a point at distance $r$ from the axis of rotation, angular acceleration is connected to tangential acceleration. Tangential acceleration is the part of linear acceleration that changes the speed along the circular path.
The relation is
$$
a_t = r\alpha
$$
where $a_t$ is the tangential acceleration.
This means that points farther from the center experience greater tangential acceleration for the same angular acceleration.
Key connection between rotational and linear motion:
$$
a_t = r\alpha
$$
Tangential acceleration is proportional to distance from the axis.
This formula is very useful because it links rotation to ordinary motion along a path.
Sign and Direction
Angular acceleration is a vector quantity in full rotational physics, but in basic circular motion problems it is often treated as a signed scalar. The sign indicates whether the rotation rate is increasing or decreasing relative to the chosen positive direction.
Consider these cases:
| Angular velocity $\omega$ | Angular acceleration $\alpha$ | Meaning |
|---|---|---|
| Positive | Positive | Rotating in positive direction and speeding up |
| Positive | Negative | Rotating in positive direction and slowing down |
| Negative | Negative | Rotating in negative direction and speeding up |
| Negative | Positive | Rotating in negative direction and slowing down |
So you must look at both $\omega$ and $\alpha$ together to understand what the object is doing.
Constant Angular Acceleration
A very important special case is constant angular acceleration. This means $\alpha$ does not change with time. Then rotational motion behaves much like straight line motion with constant acceleration.
If angular acceleration is constant, then
$$
\omega = \omega_0 + \alpha t
$$
where $\omega_0$ is the initial angular velocity.
This equation says the angular velocity changes linearly with time.
For constant angular acceleration:
$$
\omega = \omega_0 + \alpha t
$$
This is the rotational counterpart of $v = v_0 + at$.
Other equations for constant angular acceleration belong with rotational kinematics as a broader topic, but this basic relation already shows how angular acceleration controls the change in spin rate.
Graphical Interpretation
On an angular velocity versus time graph, angular acceleration is the slope.
If the graph of $\omega$ against $t$ is steep, the angular acceleration is large. If the graph is flat, the angular acceleration is zero.
$$
\alpha = \frac{\Delta \omega}{\Delta t}
$$
for the slope of a straight line segment, or
$$
\alpha = \frac{d\omega}{dt}
$$
for the slope at a single instant on a curved graph.
A horizontal line on a $\omega$ versus $t$ graph means constant angular velocity and zero angular acceleration.
Everyday Examples
A ceiling fan that starts from rest and speeds up has positive angular acceleration if counterclockwise is positive.
A bicycle wheel that slows because of friction has angular acceleration opposite to its angular velocity.
A washing machine during spin cycle may speed up, rotate at nearly constant angular velocity, and then slow down. During these different stages, its angular acceleration changes.
These examples show that angular acceleration appears whenever rotational motion is not steady.
A Simple Example
Suppose a disk starts with angular velocity
$$
\omega_0 = 2\,\text{rad/s}
$$
and after $4\,\text{s}$ it reaches
$$
\omega = 14\,\text{rad/s}
$$
Then the average angular acceleration is
$$
\alpha = \frac{14 - 2}{4} = 3\,\text{rad/s}^2
$$
If a point on the rim is at radius
$$
r = 0.50\,\text{m}
$$
then its tangential acceleration is
$$
a_t = r\alpha = 0.50 \times 3 = 1.5\,\text{m/s}^2
$$
So the point on the rim speeds up along the circular path with tangential acceleration $1.5\,\text{m/s}^2$.
Key Idea to Remember
Angular acceleration measures how fast angular velocity changes.
It plays the same role in rotational motion that ordinary acceleration plays in straight line motion.
Summary:
$$
\alpha_{\text{avg}} = \frac{\Delta \omega}{\Delta t}, \qquad \alpha = \frac{d\omega}{dt}, \qquad a_t = r\alpha
$$
Angular acceleration tells us how rotational speed changes with time, and it determines the tangential acceleration of points on a rotating object.