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The problem involves analyzing a suspension bridge with certain geometric and structural characteristics. Let's break down the steps to solve the given problem:

### Problem Breakdown

1. **Determine the equation of the parabola \(L_1\):**
   - Given distances:
     - \(OC = 100 \, \text{m}\)
     - \(AO = BC = 17 \, \text{m}\)
     - \(P\) is the lowest point on \(L_1\), and \(PD = 2 \, \text{m}\)
   
2. **Find the length of \(FO\):**
   - Given:
     - \(EF = 26 \, \text{m}\)
     - \(E\) is on \(L_1\)
     - \(EF \perp FF'\)
     - \(FO < OD\)

### Step-by-Step Solution

#### 1. Equation of the Parabola \(L_1\)

The parabola is symmetric with respect to the \(y\)-axis, so its vertex form can be expressed as:
\[ y = ax^2 + c \]

- The vertex \(O\) is at \( (0, 0) \)
- Distance \(OC = 100 \, \text{m}\) gives \(C = (100, y)\)
- Given \(AO = 17 \, \text{m}\), the height at \(A\) and \(B\) can be deduced from the bridge geometry.

Using the distance \(PD = 2 \, \text{m}\), the point \(P\) is on the parabola:
\[ P = (0, 2) \]

Thus, we have:
\[ 2 = a(0)^2 + c \implies c = 2 \]

Now, the parabola equation becomes:
\[ y = ax^2 + 2 \]

Given \(A = 17 \, \text{m}\) from the vertex:
\[ y_A = a(17)^2 + 2 \]

To find \(a\), we use the distance \(OC = 100 \, \text{m}\):
\[ y_C = a(100)^2 + 2 \]

#### 2. Length of \(FO\)

Given:
- \(EF = 26 \, \text{m}\)
- \(E\) on \(L_1\)

From \(EF \perp FF'\), the \(x\)-coordinate of \(E\) is 0 (as it is vertically aligned), thus:
\[ E = (0, 2) \]

To find \(FO\), we note that:
\[ FO = y_F \]
where \(y_F\) is the y-coordinate when \(x = 0\) on the parabola:
\[ y = 2 \]
So:
\[ FO = 2 \]

### Detailed Answers

1. **Equation of the parabola \(L_1\):**
   \[
   y = ax^2 + 2
   \]
   Determine \(a\) from given coordinates.

2. **Length of \(FO\):**
   \[
   FO = 2 \, \text{m}
   \]

Would you like more detailed steps or further assistance with a specific part of this problem?

### Questions for Further Exploration

1. How do we solve for the constant \(a\) in the parabola equation using another point?
2. What additional information is required to find the height of the tower \(A\) from the ground?
3. How do we verify if \(E\) lies on \(L_1\)?
4. Can we find the coordinates of point \(D\) from the given problem?
5. What geometric properties can help simplify the calculation of distances on the bridge? 

### Tip:
Always start by identifying known points and distances in geometric problems, as this helps in forming equations that relate these points.

Explore the analysis of a suspension bridge involving the parabola equation y = ax^2 + c and calculations of geometric distances. Understand how to determine key structural characteristics and solve related mathematical problems.

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