Question

A Spherical ball is subjected to a pressure of 100 atmosphere. If the bulk modulus of the ball is 10¹¹ N/m²,then change in the volume is:

• A. 10^-1%
• B. 10^-2%
• C. 10^-3%
• D. 10^-4%
• E. 10^-5%

Answer 1

The Correct Option is B

Step 1: Understand the Given Parameters

We are given:

• Applied pressure (P) = 100 atm
• Bulk modulus (B) = 10¹¹ N/m²

Step 2: Convert Pressure to SI Units

1 atm = 1.01325 × 10⁵ N/m²

\[ P = 100 \text{ atm} = 100 × 1.01325 × 10^5 \text{ N/m}^2 = 1.01325 × 10^7 \text{ N/m}^2 \]

Step 3: Apply Bulk Modulus Formula

The bulk modulus formula relates pressure change to volume change:

\[ B = -V \frac{\Delta P}{\Delta V} \]

For small changes, the fractional volume change is:

\[ \frac{\Delta V}{V} = -\frac{P}{B} \]

Step 4: Calculate Fractional Volume Change

\[ \frac{\Delta V}{V} = -\frac{1.01325 × 10^7}{10^{11}} = -1.01325 × 10^{-4} \]

The negative sign indicates volume decreases under pressure.

Step 5: Convert to Percentage

\[ \% \text{ change} = \left| \frac{\Delta V}{V} \right| × 100 = 1.01325 × 10^{-4} × 100 ≈ 10^{-2} \% \]

Answer 2

1. Recall the definition of bulk modulus:

Bulk modulus (B) is defined as the ratio of the change in pressure (ΔP) to the fractional change in volume (ΔV/V):

\[B = -V\frac{\Delta P}{\Delta V}\]

The negative sign indicates that the volume decreases as pressure increases.

2. Convert pressure to SI units:

1 atmosphere = \(1.013 \times 10^5 \, Pa\) (Pascals, which are equivalent to N/m²). Therefore:

\[\Delta P = 100 \, atm = 100 \times 1.013 \times 10^5 \, Pa = 1.013 \times 10^7 \, Pa\]

3. Solve for the fractional change in volume:

\[\frac{\Delta V}{V} = -\frac{\Delta P}{B} = -\frac{1.013 \times 10^7 \, Pa}{10^{11} \, Pa} = -1.013 \times 10^{-4}\]

4. Calculate the percentage change in volume:

Percentage change = \(\frac{\Delta V}{V} \times 100\%\)

Percentage change = \(-1.013 \times 10^{-4} \times 100\% = -1.013 \times 10^{-2} \% \approx -10^{-2}\%\)