Question

Given below are two statements:

Statement I: A spherical ball under the surrounding pressure of 6 $P_0$ is brought to the surrounding of pressure $P_0$. If bulk modulus of ball is 100 $P_0$ then fractional change in volume of ball is 0.05

Statement II: For volumetric strain of 0.06 in a spherical ball the change in radius of ball is of magnitude of 2%.

In the light of above statements, choose the correct answer from the options given below.

• A. Statement I is true and Statement II is true.
• B. Statement I is true and Statement II is false.
• C. Statement I is false and Statement II is true.
• D. Statement I is false and Statement II is false.

Answer 1

Answer: (A)

Both Statement I and Statement II are correct.

Answer 2

Statement I: The bulk modulus is $B = -\frac{\Delta P}{\Delta V/V}$. Given initial pressure $P_{initial} = 6P_0$, final pressure $P_{final} = P_0$, so $\Delta P = P_0 - 6P_0 = -5P_0$. The bulk modulus is $B = 100P_0$. Thus, $\frac{\Delta V}{V} = -\frac{\Delta P}{B} = -\frac{-5P_0}{100P_0} = 0.05$. Statement I is correct.

Statement II: For a sphere, $V = \frac{4}{3}\pi r^3$. For small changes, $\frac{\Delta V}{V} \approx 3\frac{\Delta r}{r}$. Given $\frac{\Delta V}{V} = 0.06$, then $0.06 \approx 3\frac{\Delta r}{r}$, which gives $\frac{\Delta r}{r} \approx 0.02$ or $2\%$. Statement II is correct.