Problem Statement
The pressure of a liquid is increased from 60 N/cm² to 100 N/cm² and the volume decreases by 0.2 per cent. Determine the bulk modulus of elasticity.
Given Data
- Initial Pressure, \(p_1 = 60 \, \text{N/cm}^2\)
- Final Pressure, \(p_2 = 100 \, \text{N/cm}^2\)
- Percentage Decrease in Volume = 0.2%
Solution
1. Calculate the Change in Pressure (\(dp\))
The change in pressure is the difference between the final and initial pressures.
2. Determine the Volumetric Strain (\(-dV/V\))
The volume decreases by 0.2%. The volumetric strain is the fractional change in volume.
3. Calculate the Bulk Modulus (\(K\))
The bulk modulus of elasticity is defined as the ratio of the change in pressure to the volumetric strain.
Converting to SI units (N/m² or Pascals):
The bulk modulus of elasticity is \( K = 20,000 \, \text{N/cm}^2 \) or \( 2 \times 10^8 \, \text{N/m}^2 \).
Explanation of Bulk Modulus
The Bulk Modulus of Elasticity (\(K\)) is a measure of a substance’s resistance to being compressed. It quantifies how much pressure is required to cause a specific fractional decrease in volume.
The formula \( K = \frac{dp}{-dV/V} \) shows this relationship directly. A substance with a high bulk modulus requires a large change in pressure to cause even a small change in volume, meaning it is difficult to compress.
Physical Meaning
A high bulk modulus, like the value calculated here, indicates that the liquid is nearly incompressible. This is a characteristic property of most liquids and is the foundational principle behind hydraulic systems.
Because the liquid strongly resists compression, any pressure applied to it in an enclosed container is transmitted almost undiminished throughout the fluid (Pascal’s Principle). This allows hydraulic systems (like car brakes, lifts, and heavy machinery) to use a small initial force to generate a much larger output force, effectively multiplying the force.
