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)

To solve this problem, we need to understand the relationship between pressure, bulk modulus, and volume strain in the context of hydraulic compression.

The bulk modulus \(B\) of a material is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. Mathematically, it can be represented as:

\(B = -\frac{\Delta P}{\frac{\Delta V}{V}}\)

where:

• \(\Delta P\) is the change in pressure.
• \(\frac{\Delta V}{V}\) is the volume strain (relative change in volume).

Rearranging this equation to solve for the pressure \(\Delta P\), we get:

\(\Delta P = -B \cdot \frac{\Delta V}{V}\)

The negative sign indicates that an increase in pressure leads to a decrease in volume (compression).

However, since we are typically interested in the magnitude of pressure change in such problems, we consider the absolute value:

\(\Delta P = B \cdot \frac{\Delta V}{V}\)

Therefore, the pressure on an object of bulk modulus \(B\) undergoing hydraulic compression due to a stress exerted by a surrounding fluid is given by:

\(B\left(\frac{\Delta V}{V}\right)\)

Based on this analysis, the correct option is:

• \(B\left(\frac{\Delta V}{V}\right)\)