Hooke's law extension ratio

Steps:

• Extension δ = (F L) / (A Y); F and Y same for both wires.
• L_Q = (1/2) L_P, so L_Q / L_P = 1/2.
• Diameter_Q = 2 × diameter_P, so A_Q = 4 A_P (area ∝ diameter²), thus A_P / A_Q = 1/4.
• Ratio δ_Q / δ_P = (L_Q / L_P) × (A_P / A_Q) = (1/2) × (1/4) = 1/8.

Why A is correct:

• δ ∝ L / A; halving L and quadrupling A multiplies ratio by 1/8 per Hooke's law.

Why the others are wrong:

• B: Accounts for length but ignores area quadrupling.
• C: Ignores both length halving and area quadrupling.
• D: Considers only length halving, forgets area effect.

Final answer: A