## 3-Way Teeter-Totter

• 3 people, weights W1, W2, W3
• 3-armed teeter-totter, with arms 120° apart
• Total arm length = LT (a board we plan to cut up)
• Find arm lengths L1, L2, L3 so that system balances.

## Equations

1. Sum of lengths constraint:
L1 + L2 + L3 = LT
2. First balance equation, using the fact that sin(30°) = 1/2):
W1 * L1 / 2 + W2 * L2 / 2 - W3 * L3 = 0
3. Second balance equation:
W1 * L1 / 2 - W2 * L2 + W3 * L3 / 2 = 0
4. Third balance equation:
W1 * L1 - W2 * L2 / 2 - W3 * L3 / 2 = 0

Note that we have 4 equations for 3 unknowns. The third balance equation is the sum of the first two, and hence it represents redundant information. This is known as a linear dependency

In matrix form we have:

```  [ 1     1     1   ] [L1]   [LT]
[ W1/2  W2/2 -W3  ] [L2] = [ 0]
[ W1/2 -W2    W3/2] [L3]   [ 0]
```

Also note that (most of) the same equations can be used to solve for different unknowns. For example, if we replace the sum-of-lengths constraint with a sum-of-weights, and start with known lengths, we have linear equations for the weights required to make the system balance.

1. Sum of weights constraint:
W1 + W2 + W3 = WT
2. First balance equation, written with the weights as the unknowns:
W1 * L1 / 2 + W2 * L2 / 2 - W3 * L3 = 0
3. Second balance equation:
W1 * L1 / 2 - W2 * L2 + W3 * L3 / 2 = 0

In matrix form:

```  [ 1     1     1   ] [W1]   [WT]
[ L1/2  L2/2 -L3  ] [W2] = [ 0]
[ L1/2 -L2    L3/2] [W3]   [ 0]
```