### 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:

#### F = -kr

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:

#### mx'' = -k_{1}x

#### my'' = -k_{2}y

where k_{1} and k_{2} represent the different spring constants. In this case, there are two different frequencies of oscillation, ω_{x} = (k_{1} / m)^{1/2} and ω_{y} = (k_{2} / m)^{1/2}. Solving the equations, the motion of the object is described by:

#### x = A cos(ω_{x}t + α)

#### y = B cos(ω_{y}t + 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 ω_{x} and ω_{y} are commensurate:

#### ω_{x} / n_{1} = ω_{y} / n_{2}

where n_{1} and n_{2} 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 3ω_{x} = 2ω_{y}).

**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 ω
_{x}and ω_{y}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.