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

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Let's tackle the problem at hand. We're dealing with the concept of Young's modulus, which measures the stiffness or elasticity of a material. Young's modulus is given by the ratio of stress to strain within the elastic limits of the material.

Given: For steel wire:

• Length (L1) = 4.7 m
• Cross-sectional area (A1) = 3.0 x 10^(-5) m^2

For copper wire:

• Length (L2) = 3.5 m
• Cross-sectional area (A2) = 4.0 x 10^(-5) m^2

Now, to find the Young's modulus ratio, we need to calculate the stress and strain for both wires.

Stress (σ) = Force (F) / Area (A) Strain (ε) = Change in length (ΔL) / Original length (L)

The same load is applied to both wires, so the force is the same.

Let's denote:

• F: Applied force
• ΔL_steel: Change in length for steel wire
• ΔL_copper: Change in length for copper wire

Since the wires stretch by the same amount under the given load, we can equate the strains:

ε_steel = ΔL_steel / L1 = ε_copper = ΔL_copper / L2

Using the stress-strain relationship, Young's modulus (E) can be defined as:

E = σ / ε

Now, for both wires: E_steel = F / (A1 * ΔL_steel) E_copper = F / (A2 * ΔL_copper)

We know that ΔL_steel = ΔL_copper (given in the problem).

So, the ratio of Young’s modulus of steel to that of copper (E_steel / E_copper) can be simplified to:

(E_steel / E_copper) = (F / (A1 * ΔL_steel)) / (F / (A2 * ΔL_copper))

Since ΔL_steel = ΔL_copper, we can cancel out the terms:

(E_steel / E_copper) = (A2 * ΔL_copper) / (A1 * ΔL_steel)

Substituting the given values:

(E_steel / E_copper) = (4.0 x 10^(-5) * ΔL_copper) / (3.0 x 10^(-5) * ΔL_steel)

Now, we need numerical values to compute this ratio. If you have the values for the applied force and the change in length for both wires, we can proceed with the calculation. Once we have those values, we can find the ratio of Young’s modulus of steel to that of copper.