Question

chapter 6 - deformation of solids

1. (a) define the young modulus. 1

(b) the young modulus of steel is \\(1.9 \times 10^{11}\text{ pa}\\). the young modulus of copper is \\(1.2 \times 10^{11}\text{ pa}\\).

a steel wire and a copper wire each have the same cross-sectional area and length. the two wires are each extended by equal forces.

(i) use the definition of the young modulus to determine the ratio
\\\frac{\text{extension of the copper wire}}{\text{extension of the steel wire}}\\ 3

(ii) the two wires are each extended by a force. both wires obey hookes law.

on fig. 10.1, sketch a graph for each wire to show the variation with extension of the force.

label the line for steel with the letter s and the line for copper with the letter c. 1

Explanation:

Define the Young modulus

Using the Young Modulus knowledge point

$$ E = \frac{\sigma}{\varepsilon} = \frac{\text{stress}}{\text{strain}} $$

where stress is force per unit cross-sectional area and strain is extension per unit original length, provided the limit of proportionality is not exceeded.

Express extension in terms of Young modulus

Using the Young Modulus knowledge point

$$ E = \frac{F / A}{x / L} \implies x = \frac{F L}{A E} $$

Calculate the ratio of extensions

Using the Young Modulus knowledge point

$$
LATEXBLOCK0
$$

Determine the relationship between gradients

Using the Hookes Law knowledge point

$$
LATEXBLOCK1
$$

Both wires obey Hooke's law, so their force-extension graphs are straight lines passing through the origin. The gradient of the line for steel (S) is steeper than that for copper (C).

Answer:

Question 10(a)

The Young modulus is defined as the ratio of tensile stress to tensile strain.

Question 10(b)(i)

$$ \frac{\text{extension of the copper wire}}{\text{extension of the steel wire}} = 1.6 $$

Question 10(b)(ii)

The graph consists of two straight lines starting from the origin \((0,0)\):

• A straight line with a steeper gradient labeled S (representing steel).
• A straight line with a shallower gradient labeled C (representing copper).