This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
6. a) Step 1: Understand the Force-Extension graph for a single spring. For a spring obeying Hooke's law, the force (F) is directly proportional to the extension (e), so the graph of Force vs. Extension is a straight line passing through the origin, with a positive slope. Step 2: Consider two identical springs connected end to end (in series). When two identical springs are connected in series, the total extension for a given force is the sum of the extensions of each spring. If a force F causes an extension e in one spring, then the same force F will cause an extension of e in each of the two springs in series, resulting in a total extension of 2e. This means that for the same force, the total extension is doubled. Therefore, the effective spring constant of the series combination is half that of a single spring (k_series = k/2). Step 3: Sketch the graph. The new graph for two springs in series will also be a straight line passing through the origin. However, its slope will be half the slope of the graph for a single spring, indicating that it is less steep. [ xlabel=Extension, ylabel=Force (N), xmin=0, xmax=5, ymin=0, ymax=10, axis lines=middle, xtick=, ytick=, clip=false ] [thick, blue, domain=0:4] 2*x; % Original spring (steeper) [above right, blue] at (axis cs: 3, 6) Single spring; [thick, red, domain=0:4] x; % Two springs in series (less steep) [above right, red] at (axis cs: 3, 3) Two springs in series; (Description of sketch: Draw a graph with "Extension" on the x-axis and "Force (N)" on the y-axis. Draw two straight lines starting from the origin. The first line (representing a single spring) should have a steeper positive slope. The second line (representing two springs connected end to end) should have a less steep positive slope, approximately half the slope of the first line.) b) Determine the extension of the system. Step 1: Determine the extension of a single spring under a given force. Given that each spring extends by 2 cm when a force of 50 N is suspended from it. So, for one spring: F_0 = 50 N e_0 = 2 cm. Step 2: Analyze the series combination of springs C and D. The 50 N load is suspended from spring D, which is suspended from spring C. In a series connection, the force acting on each spring is the same as the total load. Force on spring C (F_C) = 50 N. Force on spring D (F_D) = 50 N. Extension of spring C (e_C) = 2 cm (since it's an identical spring under 50 N). Extension of spring D (e_D) = 2 cm (since it's an identical spring under 50 N). The total extension of the series combination (C and D) is the sum of their individual extensions: e_CD = e_C + e_D = 2 cm + 2 cm = 4 cm Step 3: Analyze the parallel combination of springs A and B. The horizontal bar supports the combination of C and D, which carries the 50 N load. Therefore, the total force acting downwards on the horizontal bar is 50 N. This force is distributed equally between springs A and B because they are identical and connected in parallel. Force on spring A (F_A) = 50 N2 = 25 N. Force on spring B (F_B) = 50 N2 = 25 N. Now, calculate the extension of spring A (or B) under a force of 25 N. Since extension is proportional to force: