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 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.
- 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?