Question

When a liquid of volume \[4\] litre is subjected to an additional pressure of \[5\times {{10}^{7}}\] \[\dfrac{N}{{{m}^{2}}}\] , the change in volume is found to be \[4\]ml. Calculate the bulk modulus of the liquid.

Answer 1

Hint: When pressure is applied on a body then normal stress will be there. If the pressure increases then the volume decreases. To find the bulk modulus first we have to find the stress and strain of the object. If the temperature of the liquid is low then it will be hard to compress it.

Complete step-by-step answer:
Bulk modulus is defined as the ratio of normal stress or pressure to the volumetric strain. Bulk modulus shows us how the body will be deformed when it is subjected to external pressure. It helps us in finding the compressibility of the liquid. It is present in solid, liquid as well as gases. The reciprocal of the Bulk modulus is known as compressibility. Bulk modulus is used in hydraulic systems and also for designing hydraulic components such as motor, valves, cylinders and pumps.
Bulk modulus is given by the formula –
Bulk modulus \[K=\dfrac{stress}{strain}\]
Stress will be equal to the pressure applied .Stress is defined as the force per unit area and its unit is\[\dfrac{N}{{{m}^{2}}}\]. Stress can also be defined as the force which tends to change the shape of the substance.
If the body is deformed due to external pressure then it is known as strain. Volumetric Strain is defined as the ratio of change in volume of the body to its original volume. Its value will be negative if the volume decreases. Volumetric strain has no unit and no dimension.
Stress \[=\dfrac{force}{area}\]
Strain \[=\dfrac{\Delta V}{V}\]
So, Bulk modulus
\[K=\dfrac{p}{\begin{align}
  & \dfrac{-\Delta V}{V} \\
 & \\
\end{align}}\]
(Volume is negative because as the pressure is applied, then volume is decreasing)
Bulk modulus \[=\dfrac{p}{\Delta V}\times V\] ……. Eq (1)
It is given that,
Original volume of the liquid \[=4\]litre
Pressure applied on the liquid or we can say, the stress applied\[=5\times {{10}^{7}}\]\[N/{{m}^{2}}\]
Change in volume of the liquid when the pressure is applied \[=4\times {{10}^{-3}}\]litre
Now putting all these values in Eq (1) ,we get
Bulk modulus \[=\dfrac{5\times {{10}^{7}}}{4\times {{10}^{-3}}}\times 4\]
On simplifying we get
Bulk modulus \[=5\times {{10}^{10}}\] \[\dfrac{N}{{{m}^{2}}}\]

Note: If the Bulk modulus of the material is greater , then the substance will be incompressible and if the value of Bulk modulus is smaller, then the substance will be more compressible . If the change is in length then it is known as tensile stress, if the change is in volume then it is bulk stress and if the change is in geometry then it is shear stress.