Question

1. a student uses a force probe to exert varying forces on the ends of vertical springs a, b, and c. the student then sketches the magnitude of the applied force as a function of the length of the spring for each spring, as shown.

chart: force applied to spring (y-axis) vs. length of spring (x-axis), with lines a, b, c

which of the following correctly compares the spring constants k of each spring?

a. ( k_b > k_c > k_a )
b. ( k_b > k_a > k_c )
c. ( k_c > k_b > k_a )
d. ( k_a > k_b > k_c )

Explanation:

Step1: Recall Hooke's Law

Hooke's Law states that the force \( F \) applied to a spring is related to the displacement \( x \) (change in length) by \( F = kx \), where \( k \) is the spring constant. In a \( F \)-vs-\( L \) (force vs length) graph, the slope of the line represents the spring constant \( k \) (since \( F = k(L - L_0) \), where \( L_0 \) is the natural length, so the slope is \( k \)).

Step2: Analyze the slopes of the graphs

To compare the spring constants, we compare the slopes of the lines for springs A, B, and C. A steeper slope means a larger spring constant.

• Spring A has the steepest slope (rises the most over a small change in length).
• Spring B has a slope less steep than A but steeper than C? Wait, no—wait, let's look at the graph. Wait, the x-axis is length, y-axis is force. So the slope is \( \frac{\Delta F}{\Delta L} = k \). So a steeper line (more vertical) has a larger \( k \).
• Looking at the lines: Spring A's line is the steepest (so \( k_A \) is largest), then Spring B, then Spring C? Wait, no—wait, maybe I got the axes reversed. Wait, no: \( F = kx \), where \( x \) is the extension (change in length). So if we plot \( F \) vs \( L \) (length), then \( F = k(L - L_0) \), so the slope is \( k \). So the steeper the line, the larger \( k \).
• From the graph: Spring A's line is the steepest (so \( k_A \) is largest), then Spring B, then Spring C? Wait, no—wait the options are:
• a. \( k_B > k_C > k_A \)
• b. \( k_B > k_A > k_C \)
• c. \( k_C > k_B > k_A \)
• d. \( k_A > k_B > k_C \)
• Wait, maybe I misread the graph. Let's think again. The steeper the line (higher slope), the larger \( k \). So if Spring A's line is the steepest (most vertical), then \( k_A \) is largest. Then Spring B's line is less steep than A but steeper than C? No, wait the options have d as \( k_A > k_B > k_C \). Let's check the slopes:
• For a given change in length (\( \Delta L \)), the change in force (\( \Delta F \)) is larger for A than B than C. So \( \Delta F_A > \Delta F_B > \Delta F_C \) for the same \( \Delta L \). Since \( k = \frac{\Delta F}{\Delta L} \), then \( k_A > k_B > k_C \).

Answer:

d. \( k_A > k_B > k_C \)