Question 220: Properties of Matter

Question

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

Accepted Answer

To find the volume contraction of the copper cube, we need to use the formula for volumetric strain under pressure, which involves the bulk modulus (K). The volumetric strain is given by the formula:
$ 	ext{Volumetric strain} = \frac{\Delta V}{V} = -\frac{P}{K} $
Where:
$ \Delta V $ is the change in volume or volume contraction,
$ V $ is the original volume of the cube,
$ P $ is the hydraulic pressure applied,
$ K $ is the bulk modulus of the material.
Given:
Edge length of the cube, $ a = 10 \, 	ext{cm} = 0.1 \, 	ext{m} $
Hydraulic pressure, $ P = 7 	imes 10^6 \, 	ext{Pa} $
Bulk modulus of copper, $ K = 1.4 	imes 10^{11} \, 	ext{Pa} $
Calculate the original volume ($ V $) of the cube:
$ V = a^3 = (0.1 \, 	ext{m})^3 = 0.001 \, 	ext{m}^3 $
Calculate the volumetric strain:
$ 	ext{Volumetric strain} = -\frac{P}{K} = -\frac{7 	imes 10^6}{1.4 	imes 10^{11}} = -5 	imes 10^{-5} $
Calculate the volume contraction ($ \Delta V $):
$ \Delta V = V 	imes 	ext{Volumetric strain} = 0.001 \, 	ext{m}^3 	imes (-5 	imes 10^{-5}) = -5 	imes 10^{-8} \, 	ext{m}^3 $
Convert $\Delta V$ from cubic meters to cubic millimeters:
1 cubic meter = $10^9$ cubic millimeters. Therefore,
$ \Delta V = -5 	imes 10^{-8} \, 	ext{m}^3 	imes 10^9 \, 	ext{mm}^3/	ext{m}^3 = -50 \, 	ext{mm}^3 $
Thus, the volume contraction of the copper cube, when subjected to the given hydraulic pressure, is $50 \, 	ext{mm}^3$.

Properties of Matter

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

Choose the correct answer: