A spring is 1 in. long when loaded with 5 lb and 8 in. long when loaded with 25 lb. Determine the spring constant in lb/in.

Round to the nearest two decimal places.

To determine the spring constant \( k \) of the spring in lb/in, we can use **Hooke's Law**, which is expressed as:

\[
F = k \cdot x
\]

where:
- \( F \) is the force applied to the spring,
- \( k \) is the spring constant,
- \( x \) is the displacement of the spring from its natural length.

### Step 1: Determine the displacement for each force
We are given two conditions for the spring:
- At 1 inch with a load of 5 lb,
- At 8 inches with a load of 25 lb.

First, we calculate the displacements (\( x \)) for each of these forces, relative to the natural length of the spring (which is unknown, so we'll denote it as \( L_0 \)).

- When \( F_1 = 5 \) lb, the spring is 1 inch long. The displacement is \( x_1 = 1 - L_0 \).
- When \( F_2 = 25 \) lb, the spring is 8 inches long. The displacement is \( x_2 = 8 - L_0 \).

### Step 2: Use Hooke's Law to set up equations
From Hooke's Law, we have the following equations for the two forces:

\[
5 = k \cdot (1 - L_0) \quad \text{(equation 1)}
\]
\[
25 = k \cdot (8 - L_0) \quad \text{(equation 2)}
\]

### Step 3: Solve the system of equations
We can solve these two equations simultaneously to find \( k \). Let's solve for \( k \).The solution to the system of equations gives:

- The spring constant \( k = \frac{20}{7} \approx 2.86 \, \text{lb/in} \).
- The natural length of the spring \( L_0 = -\frac{3}{4} \, \text{in} \).

Thus, the spring constant is approximately **2.86 lb/in**.

Would you like further details or explanations?

Here are 5 related questions for further practice:
1. How would the spring constant change if the forces were doubled?
2. What would happen to the displacement if the spring constant were larger?
3. How can you determine the natural length of the spring using different loads?
4. What units would you use for the spring constant if the forces were measured in newtons?
5. Can you derive Hooke's law from basic physical principles?

**Tip:** When solving systems of equations, always check your solutions by substituting them back into the original equations to verify accuracy.

A spring is 1 in. long when loaded with 5 lb and 8 in. long when loaded with 25 lb. Determine the spring constant in lb/in. Round to the nearest two decimal places.

Solve a problem where a spring is loaded with 5 lb and 25 lb, resulting in lengths of 1 inch and 8 inches, respectively. Learn how to calculate the spring constant using Hooke's Law. Includes detailed step-by-step calculations. Suitable for Grades 10-12 physics students.

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