Angular Displacement

Radians vs Degrees

In the example given above, we used the unit of degrees \( (^\circ) \) when calculating the angular displacement. You may feel very familiar with this unit, having used it for most of your calculations so far. However, there is another unit used for angles and angular displacement, which is often much simpler and far more elegant. This unit is the radian \((\mathrm{rad}\)).

This definition may seem a little wordy so let's break it down. Consider a circle, we can draw two distinct radii from the circle's center to any two points on the circumference. The section of the circle's circumference between the two points on the radii is called an arc. An angle of one radian produces an arc with a length equal to the circle's radius. The image below illustrates this idea clearly.

Fig.3- The radian is a more suitable unit for angular displacement as it is intrinsically linked to the properties of a circle

Using this definition, we can calculate some simple conversions between degrees and radians. Consider a circle of radius \(r\), we know that there are \( 360\; ^{\circ} \) in a circle. We also know that the circumference (total arc length) of a circle is \( 2\pi r \), which means that a complete circumference defines an angle of \( 2\pi\;\mathrm{rad} \). Thus \( 360\;^{\circ} \) degrees must be equal to \(2\pi\) radians. Similarly \(180\;^{\circ}=\pi \;\mathrm{rad}\). We can convert from degrees to angles using the following handy equations.

Angular Displacement Formula

Radians are very useful for measuring angular displacements as it gives us a link to the linear displacement via the arc length. In fact, by measuring angular displacement in radians, we can immediately derive a formula for angular displacement in terms of the arc length of distance traveled.

\[\begin{aligned}\text{Angular displacement (radians)}&=\frac{\text{arc length (meters)}}{\text{radius of circular motion (meters)}}\\\Delta\theta&=\frac{s}{r}\end{aligned}\]

Here the arc length is the linear distance traveled by the object during it's circular motion, and the radius of circular motion is the shortest distance between the object and its axis of rotation.

It is important to keep in mind that a radian defines an angle measure, but it is a dimensionless unit! We can see this in the above formula, as both \( s \) and \( r \) have the same units of length. Then, the units cancel out when calculating their quotient.

Fig. 4 - A sector of a circle with radius \(r\), arc length \(s\) and angle \(\theta\)

Let's see how this applies in a real-world context.

Relationship between Angular and Linear Displacement

As we can see from the previous formula, angular and linear displacement are closely related.

\[\Delta\theta=\frac{s}{r}\]

In fact, they are directly proportional. This means that as the angular displacement increases or decreases, so does the linear displacement. However, the formula also states that the angular displacement is inversely proportional to the radial distance from the axis of rotation. This means that if two objects rotate through the same angular displacement, the one rotating further from the axis of rotation will cover a larger linear distance than the one rotating closer. We can see this in the graph below.

Fig.5 - Both segments subtend the same angle \(\theta\) but as \(r_a\) is smaller than \(r_b\), \(s_a\) is smaller than \(s_b\)

It's important to remember that the equation for angular displacement includes the radial distance. Two objects can cover the same linear distance but have very different angular displacements and vice versa.

Angular Displacement Graph

Much like regular linear displacement, we can use motion graphs to find the angular displacement of objects moving in circular motion. By plotting an angular position vs time graph of an object, we can find its angular displacement between two specific times by calculating the vertical change between the points with the corresponding time coordinates.

Fig.6 - Finding the angular displacement of an object between \(1\,\mathrm{s}\) and \(4\,\mathrm{s}\), StudySmarter Originals

We can use the graph above to find the angular velocity \(\omega\) of the object.

We measured angular velocity in \(\frac{\mathrm{rad}}{\mathrm{s}}\).

Therefore, we can find the angular velocity by calculating the slope of the graph.

The slope of a graph can be calculated using

\[ \text{slope} = \frac{\text{Change in vertical direction}}{\text{Change in horizontal direction}} \]

As we can see in the above graph, the angular displacement between \(1\,\mathrm{s}\) and \(4\,\mathrm{s}\) is \(\frac{3\pi}{2}\,\mathrm{rad}\). This implies that the slope of the graph is\[\text{Gradient}=\omega=\frac{\frac{3\pi}{2}\;\mathrm{rad}}{4\;\mathrm{s} -1\;\mathrm{s}}=\frac{\pi}{2}\,\frac{\mathrm{rad}}{s}.\]

Thus, the angular velocity for the object is \( \frac{\pi}{2}\,\frac{\mathrm{rad}}{s}. \)

Angular Displacement Example

Let's dive into some more practical and application-oriented examples.