A force of 14 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work W is done in stretching it from its natural length to 9 in. beyond its natural length?

To solve this problem, we use **Hooke's Law** for springs, which states that the force \( F(x) \) needed to stretch or compress a spring a distance \( x \) from its natural length is proportional to \( x \):

\[
F(x) = kx
\]

where:
- \( k \) is the spring constant, and
- \( x \) is the displacement from the natural length of the spring.

### Step 1: Determine the spring constant \( k \)

We are given that a force of 14 lb is required to stretch the spring 4 inches beyond its natural length. This means:

\[
F(4) = 14 \, \text{lb}
\]

Using Hooke’s law \( F(x) = kx \), we can set up the equation:

\[
14 = k \cdot 4
\]

Solving for \( k \):

\[
k = \frac{14}{4} = 3.5 \, \text{lb/in}
\]

### Step 2: Set up the work integral

The work \( W \) required to stretch the spring from its natural length to 9 inches beyond its natural length is given by:

\[
W = \int_{0}^{9} F(x) \, dx = \int_{0}^{9} kx \, dx
\]

Substitute \( k = 3.5 \) lb/in:

\[
W = \int_{0}^{9} 3.5x \, dx
\]

### Step 3: Solve the integral

\[
W = 3.5 \int_{0}^{9} x \, dx = 3.5 \left[ \frac{x^2}{2} \right]_{0}^{9}
\]

Evaluating the integral:

\[
W = 3.5 \left( \frac{9^2}{2} - \frac{0^2}{2} \right)
\]
\[
W = 3.5 \cdot \frac{81}{2} = 3.5 \cdot 40.5 = 141.75 \, \text{in-lb}
\]

Thus, the work done in stretching the spring from its natural length to 9 inches beyond its natural length is:

\[
W = 141.75 \, \text{in-lb}
\]

### Final Answer:
The work done is **141.75 in-lb**.

Would you like further clarification or details?

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Here are 5 related questions:
1. How does Hooke's law apply in different units (e.g., metric system)?
2. What is the relationship between potential energy and work done on a spring?
3. How is the spring constant \( k \) affected if the spring is made stiffer?
4. What would be the work done if the spring was stretched only to 6 inches beyond its natural length?
5. How can this problem be extended to 3D elastic materials?

**Tip:** Always check the units carefully when calculating work and force to avoid conversion errors.

Learn how to calculate the work done in stretching a spring using Hooke's Law. This problem covers the determination of the spring constant and involves integrating to find the total work required to stretch the spring 9 inches beyond its natural length. Suitable for students in Grades 10-12.

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