To solve this problem, we will use the work-energy principle applied to Hooke's law.

Given:

• The force required to stretch the spring by $x$ meters is $f(x) = kx$.
• Work done to stretch the spring from $x_1$ to $x_2$ is given by: $W = \int_{x_1}^{x_2} f(x) \, dx = \int_{x_1}^{x_2} kx \, dx$ This integral will give the work done in stretching the spring between the two positions.

Step 1: Calculate $k$

We know the work required to stretch the spring from $13 \, \text{cm}$ to $15 \, \text{cm}$ (which is 0.13 m to 0.15 m in SI units) is 3 J, and the work done to stretch it from $15 \, \text{cm}$ to $17 \, \text{cm}$ (which is 0.15 m to 0.17 m) is 5 J.

1. Work done from $0.13 \, \text{m}$ to $0.15 \, \text{m}$: $W_1 = \int_{0.13}^{0.15} kx \, dx$ This gives: $W_1 = k \left[ \frac{x^2}{2} \right]_{0.13}^{0.15} = k \left( \frac{(0.15)^2}{2} - \frac{(0.13)^2}{2} \right)$ Simplifying: $W_1 = k \left( \frac{0.0225}{2} - \frac{0.0169}{2} \right) = k \left( \frac{0.0056}{2} \right) = k \cdot 0.0028$ Given $W_1 = 3 \, \text{J}$, we can solve for $k$: $3 = k \cdot 0.0028 \implies k = \frac{3}{0.0028} = 1071.43 \, \text{N/m}$

Step 2: Verify the second work value

We can check the second part to verify the consistency of $k$.

2. Work done from $0.15 \, \text{m}$ to $0.17 \, \text{m}$: $W_2 = \int_{0.15}^{0.17} kx \, dx$ This gives: $W_2 = k \left[ \frac{x^2}{2} \right]_{0.15}^{0.17} = k \left( \frac{(0.17)^2}{2} - \frac{(0.15)^2}{2} \right)$ Simplifying: $W_2 = k \left( \frac{0.0289}{2} - \frac{0.0225}{2} \right) = k \left( \frac{0.0064}{2} \right) = k \cdot 0.0032$ Given $W_2 = 5 \, \text{J}$, we can check: $5 = k \cdot 0.0032 \implies k = \frac{5}{0.0032} = 1562.5 \, \text{N/m}$

It looks like the spring constant $k$ is likely between these values. Given that both work values must align for a single $k$, the value we found for $k$ may be averaged or refined based on better integration approaches.

Step 3: Determine the natural length of the spring

The natural length corresponds to the point where no work is done to stretch the spring. Therefore, using the work-energy principle, we need to determine the constant $k$ that describes stretching force for distance. Given that forces