Question

A steel wire of length 4.7 m and cross-sectional area 3.0 × 10¯⁵ m² stretches by the same amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 × 10¯⁵ m² under a given load. What is the ratio of the Young’s modulus of steel to that of copper?

Answer 1

Given, the length and area of cross section of a steel wire is 4.7 m and 3.0× 10 −5   m 2 respectively. The corresponding values for a copper wire is 3.5 m and 4.0× 10 −5   m 2 respectively. The extension in both the wires is the same.

Let L 1 be the length of the steel wire, A 1 be the cross sectional area of the steel wire, ΔL be the change in the length of the wire and F be the given load.

Young’s modulus of the steel wire is,

Y 1 = F L 1 A 1 ΔL …… (1)

Let L 2 be the length of the copper wire, A 2 be the cross sectional area of the copper wire, ΔL be the change in the length of the wire and F be the given load.

Young’s modulus of the copper wire is,

Y 2 = F L 2 A 2 ΔL …… (2)

Dividing equation (1) by (2), we get:

Y 1 Y 2 = F L 1 A 1 ΔL F L 2 A 2 ΔL = L 1 A 2 L 2 A 1

Substituting the values in the above expression, we get:

Y 1 Y 2 = 4.7 m×4.0× 10 −5   m 2 3.5 m×3.0× 10 −5   m 2 =1.8

Hence, the ratio of Young’s modulus of steel to that of the copper is 1.8.