Elastic potential energy from force-extension graphs

Steps:

• Energy stored equals the area under the F-x graph, which for linear springs is (1/2)kx².
• For the same extension x, energy ratio equals k_Y / k_X, as x is identical.
• The graph shows spring Y has a steeper slope, indicating k_Y = 8k_X (F_Y = 8F_X at that x).
• Thus, area under Y's curve is 8 times the area under X's curve.

Why B is correct:

• For equal extensions, E = (1/2)kx², so E_Y / E_X = k_Y / k_X = 8, matching the graph's slope ratio.

Why the others are wrong:

• A: Graph shows 8-fold energy ratio, not 4, for same extension.
• C: For same force F, x_Y = F/k_Y = x_X/8, so E_Y = F²/(2k_Y) = E_X/8, not 4 times.
• D: For same force, energy ratio is k_X/k_Y = 1/8, not 8 times.

Final answer: B