KAHIBARO
Table of Contents
Understanding angular position
When an object moves along a circle, its location is often described more naturally by an angle instead of a straight line distance. This angle is called the angular position. It tells us where the object is around the circle relative to a chosen reference direction.
For example, imagine a point moving around the edge of a wheel. Saying that it is 3 meters from the left is not very useful, because its path is curved. It is much clearer to say that it is at an angle of $30^\circ$, $90^\circ$, or $200^\circ$ from a reference line.
Angular position is usually represented by the symbol $\theta$.
Reference point and direction
Angular position must always be measured from some chosen starting line. In physics, the usual convention is to measure from the positive horizontal direction, often called the positive $x$ axis, and to count angles in the counterclockwise direction as positive.
If the object rotates clockwise, the angular position may be negative.
This means angular position is not just a number by itself. It depends on the chosen reference direction and the sign convention.
Angular position $\theta$ describes where an object is on a circular path relative to a reference line.
Positive angles are usually measured counterclockwise, negative angles clockwise.
Angular position in radians
Although angles can be measured in degrees, physics mainly uses radians. A radian is defined from the geometry of a circle.
If an object moves along an arc of length $s$ on a circle of radius $r$, then the angular position in radians is
$$
\theta = \frac{s}{r}
$$
This formula shows why radians are so useful. They connect angle directly to distance traveled along a circle.
One full revolution corresponds to the circumference of the circle, which is $2\pi r$. So the angular position for one full turn is
$$
\theta = \frac{2\pi r}{r} = 2\pi
$$
Thus, one complete circle is $2\pi$ radians.
Degrees and radians
Degrees and radians both measure angle, but radians are the standard unit in mechanics.
The conversion between them is based on
$$
360^\circ = 2\pi \text{ rad}
$$
From this,
$$
180^\circ = \pi \text{ rad}
$$
and
$$
1^\circ = \frac{\pi}{180} \text{ rad}
$$
$$
1 \text{ rad} = \frac{180^\circ}{\pi}
$$
Here are some common angles.
| Degrees | Radians |
|---|---|
| $0^\circ$ | $0$ |
| $30^\circ$ | $\frac{\pi}{6}$ |
| $45^\circ$ | $\frac{\pi}{4}$ |
| $60^\circ$ | $\frac{\pi}{3}$ |
| $90^\circ$ | $\frac{\pi}{2}$ |
| $180^\circ$ | $\pi$ |
| $270^\circ$ | $\frac{3\pi}{2}$ |
| $360^\circ$ | $2\pi$ |
For circular motion, always prefer radians in formulas.
The key relation is
$$
\theta = \frac{s}{r}
$$
and it works only when $\theta$ is in radians.
Angular position as a coordinate
In one dimensional motion, position might be given by $x$. In circular motion, position along the path can be given by $\theta$. So angular position acts like a coordinate for rotation.
If a particle starts at the reference line, then its angular position is $\theta = 0$. After moving a quarter turn counterclockwise, its angular position is
$$
\theta = \frac{\pi}{2}
$$
After a half turn,
$$
\theta = \pi
$$
After one full turn,
$$
\theta = 2\pi
$$
If it continues around again, angular position can become greater than $2\pi$. For example, after one and a half turns,
$$
\theta = 3\pi
$$
So angular position is not restricted to only one circle. It can keep increasing or decreasing as the object continues to rotate.
Same physical location, different angular positions
An important idea is that different angular positions can describe the same physical point on the circle. For example,
$$
0,\quad 2\pi,\quad 4\pi,\quad -2\pi
$$
all correspond to the same location.
Likewise,
$$
\frac{\pi}{2},\quad \frac{5\pi}{2},\quad -\frac{3\pi}{2}
$$
all describe the same point.
This happens because each full revolution adds or subtracts $2\pi$ radians.
In general, the same point on a circle can be written as
$$
\theta + 2\pi n
$$
where $n$ is any integer.
Angles that differ by $2\pi n$, where $n$ is an integer, represent the same location on a circle:
$$
\theta_{\text{same}} = \theta + 2\pi n
$$
Arc length and angular position
Angular position is closely related to the distance traveled along the circular path. If the radius is constant, then arc length $s$ and angular position $\theta$ are connected by
$$
s = r\theta
$$
This is just a rearrangement of $\theta = s/r$.
It means that for a larger radius, the same angle corresponds to a longer arc. For example, a rotation of $\theta = 1$ radian on a small wheel gives a short distance, while the same angle on a large wheel gives a longer distance.
Visualizing angular position
A circle with a reference axis helps make angular position clear.
In this drawing, the point $P$ lies on the circle, and its angular position is the angle $\theta$ measured from the positive $x$ axis.
Example
Suppose a point moves along a circle of radius $2\,\text{m}$ and travels an arc length of $3\,\text{m}$. Its angular position change from the starting point is
$$
\theta = \frac{s}{r} = \frac{3}{2} = 1.5 \text{ rad}
$$
If you wanted the angle in degrees, you could convert it:
$$
\theta = 1.5 \cdot \frac{180^\circ}{\pi} \approx 85.9^\circ
$$
This tells us the point has moved to a position a little less than a right angle from where it started.
Angular position and rotation sense
The sign of angular position matters. A positive angular position means the object is located counterclockwise from the reference direction. A negative angular position means it is clockwise from the reference direction.
For example,
$$
\theta = \frac{\pi}{3}
$$
means $60^\circ$ counterclockwise, while
$$
\theta = -\frac{\pi}{3}
$$
means $60^\circ$ clockwise.
Summary of key ideas
Angular position is the angle that locates an object on a circular path. It is usually denoted by $\theta$ and measured from a reference direction. In physics, radians are the preferred unit. The most important relation is that angular position in radians equals arc length divided by radius, and equivalently arc length equals radius times angular position.
Key formulas for angular position:
$$
\theta = \frac{s}{r}
$$
$$
s = r\theta
$$
$$
360^\circ = 2\pi \text{ rad}
$$
Same position on the circle:
$$
\theta \equiv \theta + 2\pi n
$$
where $n$ is any integer.
Angular position is the starting point for describing rotational motion. Once we know how angle specifies location, we can later describe how fast that angle changes and how the rotation itself changes over time.