## Exploration 16.4: Pendulum Motion, Forces, and Phase Space

A 1-kg pendulum bob is attached to a 9.8-m massless string to form a pendulum (position is given in meters and time is given in seconds). A graph of angular velocity (rad/s) vs. angle (rad) is shown. This graph is sometimes called a "phase space" representation of the motion. Restart. In addition,

• the red arrow represents the total force
• the blue arrow represents the force of gravity
• the green arrow represents the velocity

The phase-space representation of motion is just another way to describe an object's motion (like a position vs. time graph). For example, when would the phase-space representation of the motion be circular? Well, x and v would have to have the same frequency, be out of phase with each other by π/2 radians (or 90o), and xmax and vmax would have to have the same magnitude. This occurs with simple harmonic motion when ω = 1rad/s.

You must first select the "drag pendulum" button, drag the pendulum bob into place, and then press "play" to begin the animation for a different initial angle.

1. Given the information above and the information depicted in the animation, when does the pendulum motion approximate simple harmonic motion?
2. Determine the maximum angle for approximate simple harmonic motion from the animation.
3. We have considered a special case of simple harmonic motion, ω = 1rad/s. What would the phase-space diagram look like for simple harmonic motion with a general ω?

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.

The OSP Network: