2.5.1.3. Angular Acceleration

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.

Rendering scientific sketch…

Angular velocity increasing with time

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.

Comments

Please login to add a comment.

Don't have an account? Register now!