9. Two wires are made of the same material and have the same volume. However, wire 1 has a cross
sectional area A, and wire-2 has a cross-sectional area 3A. If the length of wire 1 increases by ∆x on
applying force F, how much force is needed to stretch wire 2 by the same amount?
a.9F
b.4F
c.6F
d.F
Problem Explanation
Two wires made of the same material share identical volume V. Wire 1 has cross-sectional area A and stretches by Δx under force F. Wire 2 has cross-sectional area 3A. Determine the force required for wire 2 to stretch by the same Δx.
Since volumes are equal, V = A · L₁ = 3A · L₂, so L₁ = 3 L₂.[web:3][web:10] Young’s modulus Y remains constant: Y = F L₁/(A Δx) for wire 1 and Y = F₂ L₂/(3A Δx) for wire 2.
Equating expressions yields F L₁/A = F₂ L₂/(3A), simplifying to F₂ = 9F.
Option Analysis
- a. 9F: Correct. Force scales with (A₂/A₁)² due to volume conservation and inverse length dependence in extension formula.
- b. 4F: Incorrect. Ignores squared area ratio; assumes linear scaling with length only.
- c. 6F: Incorrect. Might arise from partial factors like 3 × 2, but overlooks Y proportionality.
- d. F: Incorrect. Same force yields different extensions due to area and length difference.
