A spring has a natural length of 6 m. If a 4-N force is required to keep it stretched to a length of 10 m, how much work W is required to stretch it from 6 m to 12 m? (Round your answer to two decimal places.)


To find the work required to stretch the spring from 6 m to 12 m, we will use **Hooke's Law** and the work-energy principle for springs.

### Step 1: Hooke’s Law
Hooke’s Law states that the force \( F \) needed to stretch or compress a spring is proportional to the displacement from its natural length. Mathematically, it is given by:

\[
F = kx
\]
where:
- \( k \) is the spring constant (N/m),
- \( x \) is the displacement from the spring's natural length.

The natural length of the spring is 6 m. The displacement when the spring is stretched to 10 m is:

\[
x = 10 - 6 = 4 \, \text{m}
\]

We are told that a 4-N force is required to stretch the spring to this length. Using Hooke's Law, we can solve for \( k \):

\[
4 = k \cdot 4
\]
\[
k = 1 \, \text{N/m}
\]

### Step 2: Work Formula for Springs
The work \( W \) required to stretch a spring from a length \( x_1 \) to \( x_2 \) is given by:

\[
W = \frac{1}{2} k (x_2^2 - x_1^2)
\]

Here, the spring constant \( k = 1 \, \text{N/m} \), \( x_1 = 0 \, \text{m} \) (natural length of 6 m), and \( x_2 = 6 \, \text{m} \) (since 12 m - 6 m = 6 m).

Substituting the values:

\[
W = \frac{1}{2} \times 1 \times (6^2 - 0^2)
\]
\[
W = \frac{1}{2} \times (36)
\]
\[
W = 18 \, \text{J}
\]

Thus, the work required to stretch the spring from 6 m to 12 m is **18 J**.

Would you like any further details or clarification?

Here are 5 related questions to deepen your understanding:
1. What would the spring constant \( k \) be if a 6-N force were required to stretch the spring to 10 m?
2. How does Hooke’s Law apply to real-world spring systems?
3. How much work is required to compress the same spring from 6 m to 4 m?
4. How would the work change if the spring constant were doubled?
5. What happens to the work done if the spring is stretched beyond its elastic limit?

**Tip:** Remember, the work done by a spring depends on the square of the displacement, so small changes in stretch can result in significant changes in the work required!

A spring has a natural length of 6 m. If a 4-N force is required to keep it stretched to a length of 10 m, how much work W is required to stretch it from 6 m to 12 m? (Round your answer to two decimal places.)

Find out how to calculate the work required to stretch a spring from 6 m to 12 m using Hooke's Law. This problem involves understanding spring constants, Hooke's Law, and applying the work-energy principle. Suitable for Grades 10-12.

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