Question

The stress required to produce a fractional compression of 1.5% in a liquid having bulk modulus of \( 0.9 \times 10^9 \, {Nm}^{-2} \) is:

• A. \( 2.48 \times 10^7 \, {Nm}^{-2} \)
• B. \( 0.26 \times 10^7 \, {Nm}^{-2} \)
• C. \( 3.72 \times 10^7 \, {Nm}^{-2} \)
• D. \( 1.35 \times 10^7 \, {Nm}^{-2} \)
• E. \( 4.56 \times 10^7 \, {Nm}^{-2} \)

Hint:

To calculate the stress required for a given fractional compression, use the relation \( K = -\frac{{Stress}}{{Fractional compression}} \). Rearranging the equation will give you the stress.

Answer 1

The Correct Option is D

The bulk modulus \( K \) is related to stress (\( \sigma \)) and the fractional change in volume (\( \Delta V / V \)) by the equation: \[ K = -\frac{{Stress}}{{Fractional compression}}, \] where: - \( K = 0.9 \times 10^9 \, {Nm}^{-2} \) is the bulk modulus,
- The fractional compression is given as \( 1.5% = 0.015 \).
Rearranging the equation to solve for stress (\( \sigma \)): \[ {Stress} = - K \times {Fractional compression} = 0.9 \times 10^9 \times 0.015 = 1.35 \times 10^7 \, {Nm}^{-2}. \] Thus, the required stress is \( 1.35 \times 10^7 \, {Nm}^{-2} \), which corresponds to option (D).