A force of 3 pounds compresses a 15 inch spring a total of 3 inches. How much work (in ft-lbs) is done in compressing the spring 10 inches?

To solve this problem, we need to use **Hooke's Law** for springs and the work done to compress a spring. Hooke's Law states that the force required to compress or extend a spring is proportional to the displacement from its natural length:

\[
F = kx
\]

Where:
- \( F \) is the force applied,
- \( k \) is the spring constant (stiffness),
- \( x \) is the displacement (amount the spring is compressed or stretched).

### Step 1: Find the Spring Constant \(k\)

We are given:
- A force \( F = 3 \) pounds,
- A compression distance \( x = 3 \) inches.

Using Hooke’s Law:

\[
3 = k \times 3
\]

Solving for \( k \):

\[
k = \frac{3}{3} = 1 \text{ pound per inch}
\]

### Step 2: Set Up the Work Formula

The work done on a spring is given by the formula:

\[
W = \int_0^x F \, dx = \int_0^x kx \, dx
\]

Where:
- \( W \) is the work done,
- \( x \) is the final displacement.

For a spring with spring constant \( k \), the work to compress it from \( 0 \) to \( x \) is:

\[
W = \frac{1}{2} k x^2
\]

### Step 3: Calculate the Work for a 10-Inch Compression

We are asked to find the work required to compress the spring by 10 inches. Using the work formula:

\[
W = \frac{1}{2} \times 1 \times (10^2) = \frac{1}{2} \times 100 = 50 \text{ inch-pounds}
\]

### Step 4: Convert Inch-Pounds to Foot-Pounds

Since there are 12 inches in a foot, we convert 50 inch-pounds to foot-pounds:

\[
W = \frac{50}{12} \approx 4.17 \text{ foot-pounds}
\]

Thus, the work done in compressing the spring by 10 inches is approximately **4.17 foot-pounds**.

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Would you like more details on any part of the solution or have any further questions?

### Related Questions:
1. What is Hooke’s Law, and how is it applied in real-world examples?
2. How does the spring constant affect the amount of work done in compressing or stretching a spring?
3. What is the formula for the potential energy stored in a compressed spring?
4. How do you convert between different units of work, such as inch-pounds and foot-pounds?
5. What is the relationship between force and displacement for non-linear systems?

#### Tip: Always ensure unit consistency, especially when switching between systems like inches to feet!

Learn how to calculate the work done in compressing a spring using Hooke's Law. This problem involves determining the spring constant and applying the work-energy principle to find the work in foot-pounds for a 10-inch compression. Suitable for students in Grades 10-12.

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