Question

The pressure on an object of bulk modulus $B$ undergoing hydraulic compression due to a stress exerted by surrounding fluid having volume strain $\frac{\Delta V}{V}$ is:

• A. $B^2\left(\frac{\Delta V}{V}\right)$
• B. $B\left(\frac{\Delta V}{V}\right)^2$
• C. $\frac{1}{B}\left(\frac{\Delta V}{V}\right)$
• D. $\frac{1}{B^2}\left(\frac{\Delta V}{V}\right)$
• E. $B\left(\frac{\Delta V}{V}\right)$

Hint:

Bulk modulus always relates pressure and fractional volume change. Units of $B$ are the same as pressure (Pascals or $N/m^2$) because strain is dimensionless.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Bulk modulus ($B$) measures the resistance of a substance to uniform compression. It is defined as the ratio of infinitesimal pressure increase to the resulting relative decrease of the volume.
Step 2: Key Formula or Approach:
The formula for Bulk Modulus is: \[ B = \frac{\text{Stress}}{\text{Strain}} = \frac{P}{\left(\frac{\Delta V}{V}\right)} \] where $P$ is the hydraulic pressure (stress).
Step 3: Detailed Explanation:
In the case of hydraulic compression, the stress applied is the pressure $P$ exerted by the fluid.
The volume strain is given as $\frac{\Delta V}{V}$.
By rearranging the definition of Bulk Modulus: \[ P = B \times \text{Volume Strain} \] Substituting the given strain: \[ P = B \times \left(\frac{\Delta V}{V}\right) \] Step 4: Final Answer:
The pressure is given by $B\left(\frac{\Delta V}{V}\right)$.