Chapter 12: Q67P (page 352)

A solid copper cube has an edge length of $85 .5 cm$ . How much stress must be applied to the cube to reduce the edge length to $85 .0 cm$ ? The bulk modulus of copper is $1 .4 \times 10^{11} {N/m}^{2}$ .

Short Answer

Expert verified

Applied stress is, $p = 2.4 \times 10^{9} {N/m}^{2}$ .

Step by step solution

Understanding the given information

The length of the edge of copper is $85.5 cm$ .

The desired length of copper is $85.0 cm$ .

The bulk modulus for copper is $1.4 x 10^{11} {N/m}^{2}$ .

Concept and formula used in the given question

Using the formula for the bulk modulus, you can find the applied stress with the help of the given data.

$\begin{array}{l} V = L^{3} \\ p = \frac{B Δ V}{V} \end{array}$

Calculation for the stress that must be applied to the cube to reduce the edge length to 85.0 cm

You know that the cube has volume v and in form length,

V=L³

Now, consider

$\begin{matrix} \frac{Δ V}{V} = \frac{Δ L^{3}}{L^{3}} \\ = \frac{{(L + Δ L)}^{3} - l^{3}}{L^{3}} \\ = \frac{3 L^{2} Δ L}{L^{3}} \\ = \frac{3 Δ L}{L} \end{matrix}$

Now, by the formula, pressure is

$\begin{matrix} p = \frac{B Δ V}{V} \\ = \frac{3 B Δ L}{L} \\ = \frac{3 (1.4 \times 10^{11} N / m^{2}) (85.5 m - 85.0 m)}{85.5 m} \\ = 2.4 \times 10^{9} {N/m}^{2} \end{matrix}$