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!