# About the Coupled Harmonic Oscillators Animation

This animation demonstrates two pendulums coupled by a spring. We assume that dissipative forces such as air resistance are negligible. Each pendulum consists of a mass suspended by a massless rigid rod. The mass of the left-hand red ball is 1 kg. You may set the mass of the right-hand blue ball to values between 0.5 and 2.0 kg.

## Using the Animation

There are 3 playback controls for the animation:

 Plays the animation Pauses the animation Rewinds the animation and stops.

There are also a set of "radio buttons" that allows you to choose the mass of the right-hand blue ball.

Clicking on the Rewind button allows you to choose another mass for the blue ball.

When the masses of the two pendulums are almost the same, the energy of oscillation is transferred back and forth between them. This phenomenon is called resonance. When the masses are equal the energy is completely transferred back and forth between the two pendulums. Less energy is transferred as the difference between the masses increases.

When the mass of the right-hand blue ball is slightly greater than the left-hand red one, when the maximum energy has been transferred to it you can see that the two pendulums are oscillating out of phase with each other.

When the mass of the right-hand blue ball is slightly less than the left-hand red one, maximum energy transfer occurs when the two pendulums are oscillating in phase with each other.

## Details

The animation requires a Flash player on your computer of at least version 6. The latest version of the player is available free from http://www.macromedia.com/.

The natural frequencies of the pendulums, when not connected by the spring, depend only on their lengths and the acceleration due to gravity. Since the pendulums are of the same length, their natural frequencies are the same. In the animation these frequencies are 5 radians per second.

The initial amplitude of the left-hand red ball is 20 degrees, and the amplitude of the right-hand blue ball is zero. We assume that the maximum amplitude of the pendulums is sufficiently small that their motion when not connected by the spring is simple harmonic.

The spring that connects the pendulums is mounted two-thirds of the way down from the support point to the center of the masses. Its spring constant is 1 N/m.

The resolution of the animation is one pixel on the screen, and time flows at 12 frames per second. Within these constraints and small mathematical rounding errors, the animation is exact. We used Maple 8 to do the calculations of the motion, which are hard-coded into the animation.

Although the size of the animation file is fairly small (46k) the calculations of the motion of the balls requires a moderately powerful computer. For example, when the two masses are the same the angle in degrees for the left-hand red pendulum as a function of time is:

`theta = 10.*[ cos(5.196152425*time)+ cos(5.*time) ]`

This calculation must be performed in real time for each frame of the animation, with a similar calculation for the right-hand blue pendulum and a simpler calculation for the length of the spring. It is a credit to the developers for Flash that the calculations do not require great computational power.

## Author

The animation and this small page describing it were written by David M. Harrison, Dept. of Physics, Univ. of Toronto, harrison@physics.utoronto.ca in February 2003.