The problem discusses how a string stretches linearly with the mass applied. We are given the following information:

• No additional mass: the hanger is 30.6 cm off the ground.
• With 150 grams of mass: the hanger is 16.2 cm off the ground.

We are asked to find the mass of a wooden block that causes the hanger to be 24.2 cm off the ground.

Step-by-step solution:

1. Define the relationship: Since the stretch varies linearly with mass, we can assume the position of the hanger follows the equation of a line: $h = h_0 - k \cdot m$ where:

   • $h$ is the height of the hanger off the ground,
   • $h_0$ is the initial height (30.6 cm when no mass is added),
   • $k$ is the rate of change (slope) of the height with mass,
   • $m$ is the mass in grams.

2. Find the slope $k$: Using the two given points, $(0 \, \text{grams}, 30.6 \, \text{cm})$ and $(150 \, \text{grams}, 16.2 \, \text{cm})$, we can calculate the slope $k$: $k = \frac{30.6 - 16.2}{0 - 150} = \frac{14.4}{-150} = -0.096 \, \text{cm/gram}$

3. Write the equation: Now, we have the equation of the line: $h = 30.6 - 0.096 \cdot m$

4. Determine the mass of the wooden block: We are given that the wooden block stretches the spring to a height of 24.2 cm. Substitute $h = 24.2$ into the equation and solve for $m$: $24.2 = 30.6 - 0.096 \cdot m$ $0.096 \cdot m = 30.6 - 24.2 = 6.4$ $m = \frac{6.4}{0.096} = 66.67 \, \text{grams}$

So, the mass of the wooden block is approximately 66.7 grams.

Would you like further details or clarifications? Here are 5 questions you could explore:

1. How does the spring constant relate to the slope of the line in this problem?
2. What would happen to the equation if the mass varied non-linearly with height?
3. Can you graph the relationship between the mass and height for this spring system?
4. How would the system behave if you used a heavier block? Would the spring obey Hooke's law indefinitely?
5. Could the spring’s material impact the linearity of its stretch?

Tip: Linear relationships like this one can often be solved using the concept of proportionality, which simplifies many real-world physics problems.