Question

The volume contraction of a solid copper cube of edge length 10 cm, when subjected to a hydraulic pressure of \( 7 \times 10^6 \) Pa, would be ________________________ mm\(^3\). (Given bulk modulus of copper = \( 1.4 \times 10^{11} \) N m\(^{-2}\))

Hint:

For bulk modulus calculations, ensure proper unit conversions between volume in cm\(^3\), m\(^3\), and mm\(^3\).

Answer 1

Correct Answer: 10

The bulk modulus \(B\) is defined by the equation: \[ B = - \frac{\Delta P}{\frac{\Delta V}{V}} \] This can be rearranged to solve for the change in volume: \[ \Delta V = \frac{\Delta P}{B} V \] The initial volume of the cube is: \[ V = (10 \, { cm})^3 = 1000 \, { cm}^3 \] Converting the volume to cubic meters: \[ V = 10^{-3} \, { m}^3 \] Substituting the given values for pressure change (\( \Delta P = 7 \times 10^6 \)) and bulk modulus (\( B = 1.4 \times 10^{11} \)) into the equation: \[ \Delta V = \frac{(7 \times 10^6)}{1.4 \times 10^{11}} \times 10^{-3} \] This calculation yields a change in volume of: \[ \Delta V = 5 \times 10^{-8} \, { m}^3 \] Converting the change in volume to cubic millimeters: \[ \Delta V = 10.0 \, { mm}^3 \]