Section 5.4 Angular Motion
An object moving around a circle is an especially common form of two-dimensional motion. Instead of using Cartesian coordinates, circular polar coordinates (usually labeled \(r\) for the radial position and \(\theta\) for the angular position) would make a natural alternate choice to describe such motion. Because the radius is constant for circular motion, this choice turns a two-dimensional context where both \(x\) and \(y\) are changing into a one-dimensional context where only \(\theta\) is changing.
Definition 5.4.1. Angular Position.
The angular position of an object moving around a circle measures the location of the object as an angle from some fixed axis (by default, the positive horizontal axis): measured in radians, it is given by \(\theta = \frac{s}{r}\text{,}\) where \(s\) is the arc length and \(r\) is the radius.
Using polar coordinates, \(\theta\) can be thought of as the angular position of an object moving around a circle; here, \(\theta\) is an analogue for the translational position \(\vec{r}\) with which you are already familiar. The other motion quantities you have studied—velocity and acceleration—also have angular equivalents as defined below.
Definition 5.4.2. Angular Velocity.
The angular velocity of an object moving around a circle is given by \(\vec{\omega} = \frac{d \vec{\theta}}{d t}\text{.}\) Combining this with the definition of angular position allows you to relate angular and tangential speed as \(\omega = \frac{v_t}{r}\text{.}\)
Definition 5.4.3. Angular Acceleration.
The angular acceleration of an object moving around a circle is given by \(\vec{\alpha} = \frac{d \vec{\omega}}{d t}\text{.}\) Combining this with the definition of angular position allows you to relate the magnitudes of the angular and tangential acceleration as \(\alpha = \frac{a_t}{r}\text{.}\)
Exercises Activities - Units
Angular position is virtually always measured using radians. Other choices, such as degrees, introduce additional units that typically need to be converted to radians when working with other quantities.
1.
Use the definition of angular position to find the units of radians in terms of other units you are familiar with.
Answer.
Both \(s\) and \(r\) are measured using the same units (probably meters), so when you divide them the units of \(\theta\) cancel! This means that radians have no units at all—which is why they are preferred for angular measurements.
2.
Determine the units of angular velocity and angular acceleration.
As with translational motion quantities, the angular motion quantities are vectors, with both magnitude and direction. For straightforward examples of angular motion in which the axis of rotation
1 does not change, the direction may often be summarized as clockwise or counterclockwise, but the true direction of the angular motion quantities is along the axis of rotation.
Representation 5.4.4. Direction of Angular Vectors.
The direction of an angular vector points along the axis of rotation. For an object moving in the \(xy\)-plane, the vector pointing outward (in the \(+z\)-direction) corresponds to counterclockwise while the vector pointing inward (in the \(-z\)-direction) corresponds to clockwise.
Exercises Activities - Circling Helicopter
A helicopter is flying in a horizontal circle of radius 40 m with an initial angular speed of 1.2 rad/s. The angular velocity vector points upward.
1.
When viewed from above, is the helicopter moving clockwise or counterclockwise?
2.
How long does it take for the helicopter to complete one full rotation? This time is often known as the period, and is denoted using the letter \(T\text{.}\)
3.
What is the helicopter’s tangential speed?
4.
Suppose the helicopter begins slowing down with an angular acceleration that has a magnitude of \(4 \frac{\text{m}}{\text{s}^2}\text{.}\) What is the helicopter’s angular acceleration (magnitude and direction!)?
