Problem Statement
What is the bulk modulus of elasticity of a liquid which is compressed in a cylinder from a volume of \(0.0125 \, \text{m}^3\) at \(80 \, \text{N/cm}^2\) pressure to a volume of \(0.0124 \, \text{m}^3\) at \(150 \, \text{N/cm}^2\) pressure?
Given Data
- Initial volume, \(V = 0.0125 \, \text{m}^3\)
- Final volume, \(V_{final} = 0.0124 \, \text{m}^3\)
- Initial pressure, \(P = 80 \, \text{N/cm}^2\)
- Final pressure, \(P_{final} = 150 \, \text{N/cm}^2\)
Solution
1. Calculate the Decrease in Volume (\(dV\))
2. Calculate the Increase in Pressure (\(dP\))
3. Apply the Formula for Bulk Modulus of Elasticity (K)
The bulk modulus of elasticity is defined as the ratio of the increase in pressure to the volumetric strain (decrease in volume divided by the original volume):
The negative sign indicates that an increase in pressure leads to a decrease in volume.
$$ K = \frac{70 \, \text{N/cm}^2}{\left( -\frac{0.0001 \, \text{m}^3}{0.0125 \, \text{m}^3} \right)} $$ $$ K = \frac{70 \, \text{N/cm}^2}{-0.008} $$ $$ K = 8750 \, \text{N/cm}^2 $$ $$ K = 8.75 \times 10^3 \, \text{N/cm}^2 $$The bulk modulus of elasticity of the liquid is \(8.75 \times 10^3 \, \text{N/cm}^2\).
Explanation
1. Bulk Modulus Definition:
The bulk modulus of elasticity (\(K\)) is a measure of a substance's resistance to compression. It quantifies how much pressure is needed to cause a given fractional decrease in volume. A higher bulk modulus indicates a less compressible substance.
2. Volumetric Strain:
Volumetric strain is the fractional change in volume (\(\frac{dV}{V}\)). In this case, it's the decrease in volume divided by the original volume. The negative sign in the formula for bulk modulus accounts for the fact that an increase in pressure (positive \(dP\)) causes a decrease in volume (negative \(dV\)).
3. Calculation Steps:
First, the change in volume and change in pressure are calculated. Then, these values are substituted into the bulk modulus formula. The units of pressure (N/cm\(^2\)) are maintained throughout the calculation, as the volume units cancel out.
Physical Meaning
1. Compressibility of Liquids:
While liquids are often considered incompressible for many practical purposes, they do exhibit a small degree of compressibility. The bulk modulus quantifies this property. A high value of \(K\) for a liquid indicates that a very large pressure change is required to produce even a small change in its volume.
2. Sound Propagation:
The bulk modulus is directly related to the speed of sound in a fluid. Fluids with a higher bulk modulus (less compressible) will transmit sound waves faster because the pressure disturbances propagate more quickly through the stiff medium.
3. Hydraulic Systems:
In hydraulic systems, the incompressibility of hydraulic fluids is a desirable property, as it allows for efficient transmission of force. However, even small compressibilities, quantified by the bulk modulus, can become significant in systems operating at very high pressures or with large volumes, affecting response times and system stiffness.
4. Material Properties:
The bulk modulus is a fundamental material property, alongside Young's modulus (for tensile/compressive stiffness) and shear modulus (for shear stiffness). It provides insight into how a material responds to uniform pressure from all directions.