Radians and Angular Displacement
Learning Objectives
1. Understand what angular displacement is.
2. Understand what a radian is and how to convert between radians, revolutions and degrees
3. Understand what an arc length means and how to calculate an arc length with angular displacement and radius.
Key Terms
| Definition | SI Units | Symbol and Equation | |
| Angular Displacement | The Angle through which an object spins | radians | \(\Delta\theta=\theta_{f}\, – \theta_{o}\) |
| Radian | Unit for an Angle. Defined by the angle marked out by an arc length of 1 radius. | ||
| Arc Length | The distance between two points along a section of a circle. | meters | \(\displaystyle S = \Delta\theta\, r \) |
What is a Radian?
Public Domain, Link
An angular displacement \(\Delta\theta\) defines how far an object has rotated. We measure angular displacement in radians. A radian is a measure of an Angle. It is defined by a rotating object moving a circular distance equal to the radius. The simulation to the left illustrates how we get the measure of one radian.
Arc Length
In physics we use arc length to measure the distance an object travels while rotating around a fixed axis. If we use radians to measure degrees then the distance traveled in meters is equal to the angular displacement in radians times the radius. \(\\\displaystyle S = \Delta\theta\, r \).
Watch the video to learn how to calculate a distance or arc length from radians and and the radius.
Converting Radians Revolutions and Degrees
\(2\pi r\) is the circumference of a circle and a radian is defined by an object traveling the distance of one radius around a circular path. That means there are\(2\pi\) radians in one complete revolution. Converting radians degrees and revolutions is like any other unit conversion. The trick is to multiply the value you want to convert by a fraction that equals one so that the units you don’t want cancel. The conversion factors are listed below.
\(\\\displaystyle2\pi\, radians = 360\deg = 1revolution\\\)Watch the video for an example if your not proficient in angle conversions.
Arc Length and Unit Conversion Practice From Flipping Physics
Rolling Without Slipping
One last important note before you take the quiz. If an object is rolling without slipping, like a bicycle with its wheel rolling on the ground, then the linear distance travelled by the bicycle is the same as the arc length for the wheel. In cases like this we can say that \(\displaystyle \Delta x = \Delta\theta\, r \).