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

Please wait for the animation to completely load.

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 90^{o}), and x_{max} and v_{max} 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.

- Given the information above and the information depicted in the animation, when does the pendulum motion approximate simple harmonic motion?
- Determine the maximum angle for approximate simple harmonic motion from the animation.
- 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?