Question

Determine the volume contraction of a solid copper cube, 10 cm on an edge, when subjected to a hydraulic pressure of $7\times {{10}^{6}}Pa$ .( Bulk modulus of copper, $B=140\times {{10}^{9}}Pa$)

Answer 1

Hint: learn about hydraulic pressure. Understand the concept of bulk modulus and obtain the relation of volume contraction with bulk modulus. Put the given quantities on the equation to find the required value.

Formula used:

Bulk modulus of a material can be expressed as, $B=\dfrac{P}{\dfrac{\Delta v}{v}}$

Complete step by step answer:
Bulk modulus of a substance can be defined as the resistance of the substance to compression under pressure. It is given by the ration of the pressure on the substance to the relative change or decrease in volume of the substance.

Bulk modulus of a material can be expressed as, $B=\dfrac{P}{\dfrac{\Delta v}{v}}$

Where B is the bulk modulus, P is the pressure on the substance and v is the original volume and $\Delta v$ is the change in volume of the object.

Hydraulic pressure is the pressure which is obtained by a hydraulic system made using compressed fluid.

the length of the copper cube is given as, $l=10cm=.1m$

volume of the cube , $v={{l}^{3}}={{0.1}^{3}}{{m}^{3}}=0.001{{m}^{3}}$

The copper cube is subjected to a hydraulic pressure of $P=7\times {{10}^{6}}Pa$
Bulk modulus of copper is given as, $B=140\times {{10}^{9}}Pa$

Bulk modulus of a material can be expressed as, $B=\dfrac{P}{\dfrac{\Delta v}{v}}$

Where, $\vartriangle v$ is the volume contraction.

Now, putting the values in the equation above, we get that,

$\begin{aligned} & 140\times {{10}^{9}}Pa=\dfrac{7\times {{10}^{6}}Pa}{\dfrac{\Delta v}{.001{{m}^{3}}}} \\\ & \Delta v=\dfrac{7\times {{10}^{6}}Pa}{140\times {{10}^{9}}Pa}\times 0.001{{m}^{3}} \\\ & \Delta v=5\times {{10}^{-8}}{{m}^{3}} \\\ \end{aligned}$

So, the volume contraction of the cube after subjected to hydraulic pressure is $\Delta v=5\times {{10}^{-8}}{{m}^{3}}$

Note: Pascal is the SI unit of pressure. One pascal is equal to ${{10}^{-5}}$bar. Since this is in the SI unit, when solving the numerical we should change all the units to SI units.
Bulk modulus will be different objects. So for different elements we will get different answer