In this explainer, we will learn how to model the change of the position with time of objects that move along circular paths.
Imagine we release a cylinder from rest at the top of an incline, as shown in the diagram below.
As the cylinder rolls down, it will experience two types of motion: linear motion down the slope and rotational motion. To visualize that rotation, say that a red dot is fixed at the center of the cylinder. The rolling cylinder moves in circles—that is, it rotates—about the red dot, as shown in the following diagram.
Now imagine we print the letter on the surface of the cylinder and track this letter relative to the red dot while the cylinder rolls down the slope.
At one instant in time, the letter will be directly above the center of the cylinder. Then, some time later, the letter will again be directly above the red dot. In the following diagram, the rolling cylinder is shown at equal distances apart, but not at equal times—the cylinder accelerates as it rolls down the incline.
Observing the letter directly above the red dot at two different instants in time tells us that the cylinder has gone through one complete rotation, or revolution. The cylinder has rotated through 360 degrees.
Another way to describe rotation is to use units called radians. If we take a circle with radius and highlight an arc of the circle’s circumference that has a length of , then the angle of that arc length measured from the center of the circle is one radian.
A complete rotation of the circle equals an angle of exactly radians. This means that
This equation shows us how to convert an angle from degrees to radians or from radians to degrees. If we divide both sides by radians, we have that:
Given an angle in radians, if we multiply it by the fraction on the left, we get that angle in degrees.
For the opposite conversion, we divide both sides of the original equation by 360 degrees, giving us
Given an angle in degrees, if we multiply it by the fraction on the right, we get that same angle in units of radians.
When an object rotates, we say that it has gone through an angular displacement, measured in degrees or radians.
Example 1: Converting an Angular Displacement in Radians to Degrees
Fill in the blank: An angular displacement of radians is equal to an angular displacement of . Give your answer to two decimal places.
We want to fill in the blank with the angular displacement in radians that is equal to an angular displacement of 155 degrees.
We will need to convert the angle of 155 degrees to the equivalent angle in units of radians.
To do this, we can recall that 360 degrees equals radians, which means that
Because this fraction equals one, we can use it to multiply 155 degrees without changing the angle’s overall value.
However, multiplying by this ratio will change the unit in which the angle is expressed—from degrees to radians:
Rounding our result to three significant figures, we find that
Thinking back to our cylinder rolling downhill, we can say that the cylinder goes through linear as well as angular displacement. The symbol for linear displacement is , while the symbol that represents angular displacement is .
As the cylinder rolls along, we know it will move faster and faster down the incline. Its linear velocity increases, and we can say the same thing about the rate at which the cylinder rotates.
The cylinder’s rate of rotation is the rate at which its angular displacement changes. We call this its angular velocity, and it is completely analogous to the cylinder’s linear velocity. Angular velocity is represented by the symbol :
We can see that the rolling cylinder’s angular velocity starts out small and becomes larger over time. So, not only does the cylinder’s angular displacement change over time, but also its angular velocity does. The rate at which angular velocity changes over time is called angular acceleration. This quantity is represented by the symbol :
Example 2: Calculating Angular Acceleration from Angular Rotation Rate
A drill bit is initially at rest. When the drill is activated, the drill bit rotates 47.5 times per second. The drill bit reaches this speed in a time of 175 ms. What is the angular acceleration of the drill bit? Give your answer to the nearest radian per second squared.
Here, we are told of a drill bit with an operating angular speed of 47.5 rotations every second. Starting from rest, the drill bit reaches this speed in 175 ms.
In solving for the drill bit’s angular acceleration, we would like to give our answer in units of radians per second squared rather than in revolutions per second squared.
As a first step, then, let’s convert 47.5 rotations per second to units of radians per second.
There are radians in one complete revolution. That means this angular speed equals radians per second, or radians per second. Let’s call this speed :
The drill bit increases its angular speed from 0 to over 175 ms. This means the bit goes through an angular acceleration, which in general equals the rate at which angular velocity changes with time. If we call the angular acceleration , then
In this instance, is just since the drill started at rest and is 175 ms.
We can then go ahead in calculating :
Before we solve for this fraction, we want the units of time in the numerator and the denominator to agree. We can convert 175 ms to the equivalent time in seconds. Since
Knowing this, we can now write and solve for in units of radians per second squared (rad/s2).
To three significant figures, the angular acceleration of the drill bit is 1 710 rad/s2.
Note that these three quantities, angular displacement, angular velocity, and angular acceleration, all have linear analogues.
We know that linear displacement, velocity, and acceleration appear often in what are called the equations of motion. These equations describe how objects move under constant acceleration.
It turns out that we can write similar equations of motion using angular quantities. We can think of these as “equations of angular motion”:
These equations let us analyze and understand rotational motion just like we do linear motion.
Example 3: Computing Rotational Motion from Angular Velocity and Acceleration
The blades of a large wind turbine rotate fully in a time of 3.25 s when it is operating at its full speed. The angular acceleration of the turbine while increasing to its full speed is 0.124 rad/s2. How much time is required to bring an initially inactive turbine to its full operating speed? Give your answer to one decimal place.
In this example, we want to solve for the time needed for these turbine blades to accelerate from rest to full operating speed.
We can first note that the angular acceleration of the blades is constant. This means we can use the equations of angular motion to describe the blades’ rotation.
There are four such equations, and we will pick the one that uses information we are given and also includes the variable we want to solve for: time .
We know the turbine’s angular acceleration, and we also know how long it takes the blades of the turbine to go through one revolution at full speed.
Recall that angular velocity is defined as a change in angular displacement divided by a change in time. Since we know the time required for the turbine to rotate once at full speed and also that one revolution equals an angular displacement of radians, we can write the maximum angular speed of the turbine:
Since the turbine starts from rest, we can consider this its final speed. This guides our choice of which equation of angular motion to work with. We can use and rearrange it to solve for time .
First let’s note that is zero. If we then divide both sides of the equation by the angular acceleration, we find that
We are told that , and we have calculated that , or radians per second.
Units of radians per second cancel from the numerator and the denominator, and we find that
Since the values given in this example each have three significant figures, we will round our answer to that same precision. To three significant figures, the time needed for the turbine to reach its full operating speed is 15.6 seconds.
Example 4: Determining the Number of Rotations of a Rolling Object
A felled tree trunk rolls down a slope in a time of 7.2 s. The trunk is initially at rest at the top of the slope and has an angular velocity of 12 rad/s at the base of the slope. How many complete rotations does the trunk make as it rolls down the slope?
We can consider this tree trunk as a cylinder rolling down a slope as shown in the figure. As the trunk rolls, it will rotate with increasing angular speed. The rate at which its angular speed increases is constant, meaning that the trunk experiences a constant angular acceleration. Therefore, the equations of angular motion can be applied to the trunk’s motion.
These equations are as follows:
We want to determine the number of times the trunk rotates completely as it rolls down the slope. The number of complete revolutions an object makes correlates with its angular displacement, represented by the symbol . If we solve for , we will be able to convert this value to revolutions.
In this exercise, we know the trunk’s angular velocity at the start and end of its descent, and we also know how much time the descent requires. These values correspond to the variables in the fourth equation of angular motion,
In our case, the initial angular velocity of the trunk, , is zero. , the trunk’s final angular velocity, is 12 radians per second (12 rad/s). The trunk takes 7.2 seconds to reach its final angular velocity from rest; we symbolize that time as . Substituting these values into our chosen equation of angular motion, we get
One complete revolution of an object equals a rotation of exactly radians. Therefore, to calculate the number of revolutions made by the trunk, we divide our result in radians by radians per revolution:
The trunk completed slightly fewer than 7 complete revolutions as it rolled. Therefore, the number of complete rotations made by the trunk is 6.
We can summarize now what we have learned about rotational kinematics.
- Rotations can be measured in units of radians or degrees. There are radians in one full revolution.
- Since radians equals 360 degrees, we can convert an angle in degrees to radians by multiplying it by , while we can convert an angle in radians to degrees by multiplying it by .
- The angle through which an object has rotated is called its angular displacement and is symbolized by .
- The rate at which changes over time equals an object’s angular velocity. It is symbolized by and .
- The rate at which changes over time equals an object’s angular acceleration. It is symbolized by and .
- Angular displacement, velocity, and acceleration are used in equations of motion describing rotation under constant angular acceleration: