Question

1. in stress-strain experiments on metal wires, the stress axis is often marked in units of \\(10^8\text{ pa}\\) and the strain axis is marked as a percentage. this is shown for a particular wire in the diagram.

what is the value of the young modulus for the material of the wire?

a \\(6.0 \times 10^7\text{ pa}\\)
b \\(7.5 \times 10^8\text{ pa}\\)
c \\(1.5 \times 10^9\text{ pa}\\)
d \\(6.0 \times 10^9\text{ pa}\\)

Explanation:

Identify the formula for Young modulus

The Young modulus \(E\) of a material is defined as the ratio of tensile stress \(\sigma\) to tensile strain \(\varepsilon\):

$$E = \frac{\sigma}{\varepsilon}$$

Extract values from the stress-strain graph

From the given linear graph, we can select a point to calculate the gradient. At the maximum point of the line:

• Stress \(\sigma = 3 \times 10^8\text{ Pa}\)
• Strain \(\varepsilon = 5\%\)

Convert strain from percentage to a decimal value

The strain is given as a percentage and must be converted to a dimensionless ratio:

$$\varepsilon = 5\% = \frac{5}{100} = 0.05$$

Calculate the Young modulus

Substitute the values into the Young modulus formula:

$$E = \frac{3 \times 10^8\text{ Pa}}{0.05}$$

$$E = 6.0 \times 10^9\text{ Pa}$$

Match the result with the given options

The calculated value of \(6.0 \times 10^9\text{ Pa}\) corresponds to option D.

Answer:

• (A) \(6.0 \times 10^7\text{ Pa}\)
• (B) \(7.5 \times 10^8\text{ Pa}\)
• (C) \(1.5 \times 10^9\text{ Pa}\)
• (D) \(6.0 \times 10^9\text{ Pa}\) (Correct answer)