A spring with an unknown natural length and spring constant extends by 1.225 m when a 1 kg mass is attached to it. If the total length of the spring is then 2.49 m, what is its natural length and spri

Given:
Mass $m = 1 \text{ kg}$
Acceleration due to gravity $g = 9.8 \text{ m/s}^2$

The force ($F_2$) acting on the spring is the weight of the mass:

$$F_2 = mg$$ $$F_2 = 1 \text{ kg} \times 9.8 \text{ m/s}^2$$ $$F_2 = 9.8 \text{ N}$$

Using Hooke's Law ($F_2 = kx_2$) and the spring constant $k = 8 \text{ N/m}$ from Step 1:

$$x_2 = \frac{F_2}{k}$$ $$x_2 = \frac{9.8 \text{ N}}{8 \text{ N/m}}$$ $$x_2 = 1.225 \text{ m}$$

The total length of the spring when supporting the mass is given as $2.49 \text{ m}$. This total length is the sum of the natural length and the extension.
Total length $L = L_0 + x_2$

$$L_0 = L - x_2$$ $$L_0 = 2.49 \text{ m} - 1.225 \text{ m}$$ $$L_0 = 1.265 \text{ m}$$

The natural length of the spring is $\boxed{\text{1.265 m}}$ and the spring constant is $\boxed{\text{8 N/m}}$.