A spring has a natural length of 3 m. If a 12-N force is required to keep it stretched to a length of 5 m, how much work W is required to stretch it from 3 m to 9 m? (Round your answer to two decimal places.)


We are tasked with finding the work \( W \) required to stretch a spring from its natural length of 3 m to 9 m, given that a 12-N force is required to stretch it to 5 m. We can solve this problem using **Hooke's Law** and the **work-energy principle**.

### Step 1: Hooke's Law
Hooke's Law states that the force required to stretch or compress a spring is proportional to the displacement from its natural length:
\[
F(x) = kx
\]
where:
- \( F(x) \) is the force applied,
- \( k \) is the spring constant, and
- \( x \) is the displacement from the natural length of the spring.

### Step 2: Find the Spring Constant \( k \)
We are told that a 12-N force is required to stretch the spring to 5 m. The displacement from the natural length of 3 m to 5 m is \( x = 5 - 3 = 2 \) m. Using Hooke's Law:
\[
12 = k \times 2
\]
Solving for \( k \):
\[
k = \frac{12}{2} = 6 \, \text{N/m}
\]

### Step 3: Work Done to Stretch the Spring
The work required to stretch a spring from its natural length to a distance \( x \) is given by the formula:
\[
W = \int_{x_1}^{x_2} F(x) \, dx = \int_{x_1}^{x_2} kx \, dx
\]
where \( x_1 \) is the initial displacement, and \( x_2 \) is the final displacement.

In this case:
- \( x_1 = 0 \) m (since the spring is at its natural length),
- \( x_2 = 9 - 3 = 6 \) m (displacement from natural length to 9 m),
- \( k = 6 \, \text{N/m} \).

Now, calculate the work:
\[
W = \int_0^6 6x \, dx
\]
\[
W = 6 \int_0^6 x \, dx = 6 \left[ \frac{x^2}{2} \right]_0^6
\]
\[
W = 6 \times \frac{6^2}{2} = 6 \times \frac{36}{2} = 6 \times 18 = 108 \, \text{J}
\]

Thus, the work required to stretch the spring from 3 m to 9 m is **108 joules**.

Would you like more details on this problem? Here are five related questions to explore further:

1. How does Hooke's Law apply to spring systems in real-world applications?
2. What happens to the work required if the spring is non-linear?
3. Can you calculate the work required to compress the same spring by a certain amount?
4. How does the spring constant \( k \) affect the behavior of different materials?
5. What are the units of work in other unit systems (like CGS or imperial)?

**Tip:** Always double-check the limits of integration when calculating work in variable force problems, as the limits can significantly impact the result.

A spring has a natural length of 3 m. If a 12-N force is required to keep it stretched to a length of 5 m, how much work W is required to stretch it from 3 m to 9 m? (Round your answer to two decimal places.)

Discover how to calculate the work required to stretch a spring from its natural length of 3 m to 9 m using Hooke's Law and integration. This example involves calculating the spring constant and applying calculus to find the work done. Ideal for advanced physics or calculus students.

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