Question

The density and bulk modulus of a metal bar is \( \rho \) and \( K \) respectively. When pressure \( P \) is applied from all sides to that metal bar, the increase in its density is

• A. \( \dfrac{\rho P}{K+P} \)
• B. \( \dfrac{\rho P}{K-P} \)
• C. \( \dfrac{K-P}{\rho} \)
• D. \( \dfrac{K+P}{\rho} \)

Hint:

For solids under very high pressure, effective bulk modulus must be corrected as \( K-P \).

Answer 1

The Correct Option is B

Step 1: Write definition of bulk modulus.
Bulk modulus is defined as \[ K = -\frac{P}{\Delta V/V} \]
Step 2: Relate density with volume.
Density is \[ \rho = \frac{m}{V} \] For constant mass, \[ \frac{\Delta \rho}{\rho} = -\frac{\Delta V}{V} \]
Step 3: Substitute in bulk modulus relation.
\[ K = \frac{P}{\Delta \rho / \rho} \Rightarrow \Delta \rho = \frac{\rho P}{K} \]
Step 4: Consider effective pressure.
Since pressure acts from all sides, effective bulk modulus becomes \( K-P \).
Step 5: Final expression.
\[ \Delta \rho = \frac{\rho P}{K-P} \]
Step 6: Conclusion.
The increase in density is \( \dfrac{\rho P}{K-P} \).