Question

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

Short answer

Applied stress is,$\text{p}=2.4\times {10}^9{\text{\hspace{0.17em}N/m}}^{\text{2}}$ .

Step-by-step solution

Understanding the given information

The length of the edge of copper is$85.5\text{\hspace{0.17em}cm}$.

The desired length of copper is$85.0\text{\hspace{0.17em}cm}$.

The bulk modulus for copper is $1.4x{10}^{11}{\text{\hspace{0.17em}N/m}}^{\text{2}}$.

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.

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

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

$\begin{array}{c}\frac{\Delta V}{V}=\frac{\Delta L^3}{L^3}\\ =\frac{{(L+\Delta L)}^3-l^3}{L^3}\\ =\frac{3L^2\Delta L}{L^3}\\ =\frac{3\Delta L}{L}\end{array}$

Now, by the formula, pressure is

$\begin{array}{c}p=\frac{B\Delta V}{V}\\ =\frac{3B\Delta L}{L}\\ =\frac{3(1.4\times {10}^{11}N/m^2)(85.5m-85.0m)}{85.5m}\\ =2.4\times {10}^9{\text{\hspace{0.17em}N/m}}^{\text{2}}\end{array}$