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The problem shown asks for the work done in stretching a spring based on Hooke's Law, which governs the force exerted by a spring. Let's break it down step by step:

1. The natural length of the spring is 14 inches.
2. A force of 5 pounds is needed to stretch the spring 2 inches (to 16 inches).
3. The spring is further stretched to 18 inches, so the total stretch is \( 18 - 14 = 4 \) inches.
4. We need to calculate the work done to stretch the spring from 14 inches to 18 inches.

### Step 1: Hooke’s Law
Hooke's Law states that the force needed to stretch a spring is proportional to the amount of stretch, which can be written as:

\[
F(x) = kx
\]

Where:
- \( F(x) \) is the force at a stretch of \( x \) inches.
- \( k \) is the spring constant.
- \( x \) is the distance the spring is stretched from its natural length.

### Step 2: Determine Spring Constant \( k \)
We know that a force of 5 pounds is required to stretch the spring by 2 inches. Thus:

\[
F(2) = k(2) = 5
\]

Solving for \( k \):

\[
k = \frac{5}{2} = 2.5 \, \text{pounds per inch}
\]

### Step 3: Work Calculation
The work \( W \) done in stretching a spring from a stretch \( x_1 \) to a stretch \( x_2 \) is given by the integral:

\[
W = \int_{x_1}^{x_2} F(x) \, dx = \int_{x_1}^{x_2} kx \, dx
\]

In this case, \( x_1 = 0 \) inches (natural length) and \( x_2 = 4 \) inches (the stretch to 18 inches). The spring constant \( k \) is 2.5, so:

\[
W = \int_0^4 2.5x \, dx
\]

### Step 4: Solve the Integral
\[
W = 2.5 \int_0^4 x \, dx = 2.5 \left[ \frac{x^2}{2} \right]_0^4
\]

Evaluating the definite integral:

\[
W = 2.5 \left( \frac{4^2}{2} - \frac{0^2}{2} \right) = 2.5 \times \frac{16}{2} = 2.5 \times 8 = 20 \, \text{inch-pounds}
\]

### Final Answer:
The work done in stretching the spring from 14 inches to 18 inches is **20 inch-pounds**.

Would you like more details or have any questions? Here are 5 related questions you might find useful:

1. How does Hooke’s Law apply to real-life situations like trampolines or vehicle suspensions?
2. What happens when a spring is stretched beyond its elastic limit?
3. How can you calculate the potential energy stored in a spring?
4. What is the relationship between work and force in non-linear springs?
5. How does the spring constant \( k \) affect the behavior of the spring?

**Tip:** Always check units when working with physical formulas—misunderstanding units can lead to significant errors!

A spring has a natural length of 14 inches. If a force of 5 pounds is required to keep the spring stretched 2 inches, how much work is done in stretching the spring from its natural length to a length of 18 inches?

Calculate the work done in stretching a spring from 14 inches to 18 inches using Hooke's Law. This problem covers key physics concepts such as force, spring constant, and the integral of Hooke’s Law. Ideal for students in Grades 11-12 or introductory college-level physics.

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