KAHIBARO
Table of Contents
Rotation as a Change in Angle
Angular displacement tells us how much an object has rotated. In straight line motion, we describe change in position with a distance or displacement. In rotational motion, we describe change in orientation with an angle.
If a wheel, disk, or pointer turns from one position to another, the angular displacement is the angle swept out during that turn. It is usually represented by the symbol $\Delta \theta$.
If the initial angular position is $\theta_i$ and the final angular position is $\theta_f$, then
$$
\Delta \theta = \theta_f - \theta_i
$$
This is the rotational version of displacement in linear motion. It tells us the change in position around a circle, not how fast the turning happens. Speed of turning belongs to angular velocity, which is a separate topic.
Important definition:
$$
\Delta \theta = \theta_f - \theta_i
$$
Angular displacement is a change in angular position, not just a final angle.
Angular Position and Reference Line
To measure angular displacement, we first choose a reference direction. This is like choosing an origin in linear motion. Then we measure the angle from that reference line to the object’s current position.
For example, imagine a point on the rim of a wheel. As the wheel turns, that point moves around the center. The line from the center to the point changes direction. The angle of that line is the angular position $\theta$.
The change from one angular position to another is the angular displacement.
Direction Matters
Angular displacement has a sign. By convention, counterclockwise rotation is usually taken as positive, and clockwise rotation as negative.
This means that rotating $90^\circ$ counterclockwise gives a positive angular displacement, while rotating $90^\circ$ clockwise gives a negative angular displacement.
This sign convention is very important because rotation is not only about how much turning occurs, but also about which way the turning occurs.
Standard sign convention:
Counterclockwise rotation, positive
Clockwise rotation, negative
Units of Angular Displacement
Angles can be measured in degrees, but in physics the standard unit is the radian.
A full circle is
$$
360^\circ = 2\pi \text{ rad}
$$
So common angle conversions are:
| Degrees | Radians |
|---|---|
| $30^\circ$ | $\pi/6$ |
| $45^\circ$ | $\pi/4$ |
| $60^\circ$ | $\pi/3$ |
| $90^\circ$ | $\pi/2$ |
| $180^\circ$ | $\pi$ |
| $360^\circ$ | $2\pi$ |
Radians are especially useful because they connect angle directly to arc length.
Angular Displacement and Arc Length
When an object rotates, a point on it moves along a circular path. The distance traveled along the edge of that circle is called the arc length, denoted by $s$.
If the radius of the circle is $r$ and the angular displacement is $\Delta \theta$, then
$$
s = r \Delta \theta
$$
This formula works when the angle is measured in radians.
It shows that angular displacement describes the same rotation for every point on a rigid object, but points farther from the center travel a larger linear distance.
Key relation between angular displacement and arc length:
$$
s = r\Delta\theta
$$
Use radians in this formula.
Angular Displacement Is Not the Same as Distance Traveled
This is a common source of confusion. Suppose a wheel turns forward half a revolution and then backward a quarter of a revolution. The total turning that occurred is not the same as the net angular displacement.
For example, half a revolution is $\pi$ radians, and a quarter of a revolution backward is $-\pi/2$ radians. So the net angular displacement is
$$
\Delta\theta = \pi - \frac{\pi}{2} = \frac{\pi}{2}
$$
But the total amount of turning done is
$$
\pi + \frac{\pi}{2} = \frac{3\pi}{2}
$$
So angular displacement is like linear displacement, it depends on initial and final positions and includes direction.
Revolutions and Angular Displacement
Sometimes rotation is described in revolutions rather than radians. One full revolution equals $2\pi$ radians.
If an object rotates through $N$ revolutions, then the angular displacement is
$$
\Delta\theta = 2\pi N
$$
If the rotation is clockwise, the value is negative.
Examples:
| Rotation | Angular displacement |
|---|---|
| 1 revolution counterclockwise | $2\pi$ rad |
| 2 revolutions counterclockwise | $4\pi$ rad |
| 1 revolution clockwise | $-2\pi$ rad |
| 0.5 revolution counterclockwise | $\pi$ rad |
Angular Displacement for Different Points on a Rigid Body
If a rigid body rotates, every point on the body undergoes the same angular displacement in the same time interval. This is true even though different points may move different linear distances.
For example, a point near the center of a spinning disk moves only a short distance along a small circle, while a point near the rim moves much farther along a larger circle. But both points sweep out the same angle.
This is one of the main reasons angular quantities are useful in rotational motion.
Visualizing Positive and Negative Rotation
Summary
Angular displacement measures how much an object rotates from one angular position to another. It is given by the change in angle,
$$
\Delta\theta = \theta_f - \theta_i
$$
It has direction, so it can be positive or negative. In physics, it is usually measured in radians. It is related to arc length by
$$
s = r\Delta\theta
$$
and one full revolution equals
$$
2\pi \text{ radians}
$$
Essential facts to remember:
$$
\Delta\theta = \theta_f - \theta_i
$$
$$
360^\circ = 2\pi \text{ rad}
$$
$$
s = r\Delta\theta
$$
Angular displacement includes direction and is not the same as total angle turned.