To solve this problem, we need to find how long the bottom spring becomes when the mass is hanging from it. The extension of the spring is determined by Hooke's law, which relates the force applied to the spring and the resulting displacement (stretch or compression). The equation for Hooke's law is:

$F = k \cdot \Delta x$

Where:

• $F$ is the force applied to the spring.
• $k$ is the spring constant.
• $\Delta x$ is the change in length (stretch) of the spring.

Steps to solve:

1. Determine the force on the bottom spring: The force on the bottom spring is the weight of the mass attached to it, which is given by:

$F = m \cdot g$

Where:

• $m = 6.40 \, \text{kg}$ is the mass.
• $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity.

2. Apply Hooke’s Law: Use the force to find the displacement ($\Delta x$) of the bottom spring by rearranging the Hooke’s law equation:

$\Delta x = \frac{F}{k}$

Where:

• $k = 5.50 \, \text{kN/m} = 5500 \, \text{N/m}$.

3. Find the new length of the bottom spring: The original length of the spring was 12.0 cm (0.12 m). The final length of the bottom spring is:

$L_{\text{final}} = L_{\text{initial}} + \Delta x$

Now let’s calculate this step by step.

Step 1: Force on the bottom spring

$F = 6.40 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 62.72 \, \text{N}$

Step 2: Displacement of the bottom spring

$\Delta x = \frac{62.72 \, \text{N}}{5500 \, \text{N/m}} = 0.0114 \, \text{m} = 1.14 \, \text{cm}$

Step 3: Final length of the bottom spring

$L_{\text{final}} = 12.0 \, \text{cm} + 1.14 \, \text{cm} = 13.14 \, \text{cm}$

So, the length of the bottom spring when hanging as shown is 13.14 cm.

---

Would you like more details on how the forces interact with the other springs, or any further clarification on Hooke's law? Here are some related questions:

1. How would the length of the bottom spring change if the spring constant were higher?
2. How do you calculate the displacement for multiple springs in parallel?
3. How can you determine the total extension when multiple masses are attached to a spring system?
4. What effect does the gravitational acceleration change (e.g., on another planet) have on the spring's displacement?
5. How would the situation differ if the spring was compressed instead of stretched?

Tip: Remember, the force constant (spring constant) $k$ determines how stiff the spring is. A larger $k$ means less stretch for the same force!