### This two dimensional oscillator makes Lissajous figures!

Watch The Video:

Teachable Topics:
• Lissajous Figures
• Motion of a Particle
• Non-isotropic Oscillators

Theory:

Consider the motion of an object subjected to a linear restoring force that is always directed towards the origin of the coordinate system. The restoring force is represented by the equation:

Or:

#### mr'' = -kr

For a multidimensional oscillator where the spring constants are not all the same, we have a non-isotropic oscillator. In this case, the motion can be represented by the component equations:

#### my'' = -k2y

where k1 and k2 represent the different spring constants. In this case, there are two different frequencies of oscillation, ωx = (k1 / m)1/2 and ωy = (k2 / m)1/2. Solving the equations, the motion of the object is described by:

#### y = B cos(ωyt + β)

The constants of integration (A, B, α, β) are all determined by the initial conditions. For the most part, the object will follow a semi-chaotic path. However, for the case where ωx and ωy are commensurate:

#### ωx / n1 = ωy / n2

where n1 and n2 are some arbitrary integers, the object will follow a closed loop known as a Lissajous figure. The object will go around this loop, and after a certain amount of time, the object will return to its initial position and then repeat its path.

In the video above, the Lissajous figure is similar to a figure eight with half of one loop removed. The path the air puck traces out looks like: Figure 1: 3:2 Ratio Lissajous Figure

The air puck will follow this path when the phase difference is π / 2, and the ratio of frequencies is 3:2 (as in 3ωx = 2ωy).

Apparatus:

• Air table with air supply
• Air puck with hooks for springs
• Four springs (Note: The springs should be in two pairs, one pair for the x-direction and one pair for the y-direction. The spring constant of each pair must be so that ωx and ωy are commensurate.)
• Four clamps and supports for them. Figure 2: Apparatus Set-up (viewed from above)

Procedure:

• Assemble the apparatus, referring to figure 2 and the video.
• Pull the puck in one direction and let go. Observe the Lissajous figure that is traced out by the air puck.