KAHIBARO
Table of Contents
Change of Angular Velocity
Angular acceleration describes how quickly angular velocity changes with time. In rotational motion, it plays the same role that linear acceleration plays in straight line motion.
If an object spins faster, slows down, or changes the direction of its rotation rate, it has angular acceleration. The symbol for angular acceleration is usually $\alpha$.
Mathematically, average angular acceleration is
$$
\alpha_{\text{avg}} = \frac{\Delta \omega}{\Delta t}
$$
where $\Delta \omega$ is the change in angular velocity and $\Delta t$ is the time interval.
Instantaneous angular acceleration is
$$
\alpha = \frac{d\omega}{dt}
$$
Since angular velocity itself is the rate of change of angular position $\theta$, angular acceleration can also be written as
$$
\alpha = \frac{d^2 \theta}{dt^2}
$$
Important relationships:
$$
\alpha_{\text{avg}} = \frac{\Delta \omega}{\Delta t}
$$
$$
\alpha = \frac{d\omega}{dt}
$$
$$
\alpha = \frac{d^2\theta}{dt^2}
$$
Angular acceleration tells you how fast the rotational speed is changing.
Units
Angular acceleration is measured in radians per second squared, written as
$$
\text{rad/s}^2
$$
Because the radian is a ratio, it is technically dimensionless, but in physics we keep the word radian to remind ourselves that the motion is rotational.
| Quantity | Symbol | SI unit |
|---|---|---|
| Angular position | $\theta$ | rad |
| Angular velocity | $\omega$ | rad/s |
| Angular acceleration | $\alpha$ | rad/s$^2$ |
Positive and Negative Angular Acceleration
The sign of angular acceleration depends on the chosen direction of rotation. If counterclockwise is taken as positive, then a counterclockwise increase in angular velocity gives positive $\alpha$, while a clockwise change gives negative $\alpha$.
A negative angular acceleration does not always mean the object is slowing down. It only tells you the direction of the change in angular velocity. For example, if an object is rotating clockwise and speeding up in that same clockwise direction, both $\omega$ and $\alpha$ may be negative.
This means the sign must always be interpreted together with the sign of angular velocity.
| Situation | $\omega$ | $\alpha$ | Meaning |
|---|---|---|---|
| Rotating counterclockwise and speeding up | $+$ | $+$ | angular speed increases |
| Rotating counterclockwise and slowing down | $+$ | $-$ | angular speed decreases |
| Rotating clockwise and speeding up | $-$ | $-$ | angular speed increases |
| Rotating clockwise and slowing down | $-$ | $+$ | angular speed decreases |
Connection to Tangential Motion
When a point on a rotating object moves in a circle of radius $r$, its tangential velocity changes if the angular velocity changes. This gives rise to tangential acceleration.
The relation is
$$
a_t = r\alpha
$$
where $a_t$ is the tangential acceleration.
This formula shows that for the same angular acceleration, points farther from the axis have greater tangential acceleration.
For a point at distance $r$ from the axis,
$$
a_t = r\alpha
$$
A larger radius means a larger tangential acceleration for the same $\alpha$.
Physical Meaning
Angular acceleration appears whenever a rotating object changes how fast it spins. A fan turning on, a bicycle wheel braking, and a washing machine speeding up are all examples.
If $\alpha = 0$, the angular velocity is constant. The object may still be rotating, but its rate of rotation is not changing.
If $\alpha$ is constant, then the angular velocity changes by equal amounts in equal times.
Simple Example
Suppose a wheel has angular velocity
$$
\omega_i = 2 \,\text{rad/s}
$$
and after $4\,\text{s}$ it has
$$
\omega_f = 10 \,\text{rad/s}
$$
Then the average angular acceleration is
$$
\alpha_{\text{avg}} = \frac{\omega_f - \omega_i}{\Delta t}
= \frac{10 - 2}{4}
= 2\,\text{rad/s}^2
$$
So the wheel’s angular velocity increases by $2\,\text{rad/s}$ every second.
Visualizing Angular Acceleration
If angular velocity is plotted against time, the slope of the graph gives angular acceleration.
A steep positive slope means large positive angular acceleration. A steep negative slope means large negative angular acceleration. A horizontal line means zero angular acceleration.
Rotational Analogy with Linear Motion
Angular acceleration is part of a very useful analogy between straight line motion and rotational motion.
| Linear motion | Rotational motion |
|---|---|
| Position $x$ | Angular position $\theta$ |
| Velocity $v$ | Angular velocity $\omega$ |
| Acceleration $a$ | Angular acceleration $\alpha$ |
This analogy helps you transfer ideas from ordinary motion to spinning motion.
Key Idea
Angular acceleration measures the rate at which angular velocity changes. It can be positive, negative, or zero, and it connects rotational motion to the tangential acceleration of points on a rotating object.
Angular acceleration is not the same as angular velocity.
$\omega$ tells how fast something is rotating.
$\alpha$ tells how fast that rotation rate is changing.