Circular Motion and Pendulums

Circular Motion and Simple Harmonic Motion

Simple harmonic motion is often thought of as motion that is due to a restoring force. Stretch a spring and it will provide a force that is toward the equilibrium position where it isn't stretched or compressed. Pull a pendulum outward and there will be a restoring force toward its equilibrium position where the weight is straight down. When there isn't any friction or other outside force these restoring forces cause the object to go in uniform cycles.

A spring hanging from the ceiling that was stretched would go up to a position above equilibrium where it would be compressed and then fall back down to the original position where it was stretched again. That cycle would continue until some outside force acts on it. The pendulum would swing back and forth forever. Simple harmonic motion or anything that goes in a uniform cycle can be modeled using a circle.

Circular Motion and a Bouncing Spring

Think of an x-axis and a y-axis with the origin of the axis at the center of a circle. Now think of the radius of a circle starting at zero degrees (along the positive x-axis) and rotating up to 90 degrees (π/2 radians) where it is on the y-axis. Also think of the x and y components of the radius vector as it does this. At 45°, for example, the x and y components will be equal.

Compare the end of the radius at the circle going around the circle with a spring going up and down. Start with the spring at its equilibrium position when the radius is along the x-axis. As the spring goes up the radius also goes up. When the spring is at its maximum height the radius is straight up. As the spring starts back down the radius also goes down on the other side of the circle. When the radius is along the negative x-axis, the spring will be back to its equilibrium position. After that the radius will go down and the spring will become stretched until the spring gets to its maximum stretch and then both will go back up again to the equilibrium position where we started. The cycle will then repeat.

Here is a YouTube link to a video that shows this relationship: Circle and Spring

Circular Motion and the Sine Function

Now consider an x-axis and a y-axis with the origin of the axis at the center of a unit circle with the radius going around the circle like before.

When the radius is along the x-axis, there is no y-component and the sin(θ)=0. When it is at 90° there is no x-component and so sin(θ)=1. When it is at 180° sin(θ) is again zero. It then goes to -1 at 270° and back to zero at 360° (back at zero degrees). This generates the classic sine function curve.

Here is a link to a video that shows this relationship: Circle and Sine Curve

You could also start at 90°, but that would generate the cosine curve. Any circular motion or any simple harmonic motion can be mathematically modeled using the sine or the cosine.

Pendulums and Bouncing Springs

A simple pendulum is just like a bouncing spring. The spring has two points of maximum potential energy, when it is fully compressed and when it is fully stretched. The pendulum has two points of maximum potential energy also, when it is at its maximum height on each side of the swing. The equilibrium position for the spring would correspond to the pendulum weight being straight down. Both are examples of simple harmonic motion and both can be modeled using circular motion.

Basic Pendulums

The simulation is from The University of Colorado at Boulder. Open the simulation in a new window (larger).


The period of a pendulum is the time it takes to complete one cycle. If you start at the maximum on one side, it would be the time it takes to go from where you start to the other side and back again. You can, of course, start the cycle at any point. Starting straight down would require that the pendulum go to the max on one side, back to straight down, up to the max on the other side and then back to where you started in order to complete a cycle.

In this module the simulation will be used to explore how the period changes with angle, length, mass, and friction. To change the angle, put your pointer over the blue hanging mass, hold down the left mouse button, and move the mass to whatever angle you would like. To change the length use the slider at the top of the green box. To change the mass use the second slider at the top of the green box. To determine the period, click on "photogate timer" at the bottom of the green box. The timer will give the period in seconds.

This page provides access to the first page of each section of the pendulum module. You will have to submit homework to get to subsequent pages within the section. Do the friction section last. If you previously worked on this module you can get back to your previous place if everything is set up correctly.

  1. Changing the Angle
  2. Changing the Length
  3. Changing the Mass
  4. Changing the Friction

You can experiment with the controls now or go on to the first homework question.