An object is taken to a depth of 2km inside an ocean. Percentage change in volume is 1.39%. Find bulk modulus

Options

A. ` 1.47 xx 10^9 N/m^2`
B. ` 1.08 xx 10^9 N/m^2`
C. ` 1.75 xx 10^9 N/m^2`
D. ` 2.34 xx 10^9 N/m^2`

Answer

To find the bulk modulus of the object submerged in the ocean, we can use the formula for bulk modulus (K) in relation to the change in volume (ΔV) and the original volume (V):

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

Where:
- \( K \) is the bulk modulus,
- \( \Delta P \) is the change in pressure,
- \( \frac{\Delta V}{V} \) is the fractional change in volume.

### Step 1: Calculate the change in pressure (ΔP)

The change in pressure at a depth \( h \) in a fluid is given by:

\[ \Delta P = \rho g h \]

Where:
- \( \rho \) is the density of seawater (approximately \( 1025 \, \text{kg/m}^3 \)),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( h \) is the depth (in this case, \( 2000 \, \text{m} \)).

Substituting the values:

\[ \Delta P = 1025 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 2000 \, \text{m} \]

Calculating this gives:

\[ \Delta P = 1025 \times 9.81 \times 2000 = 20000000.5 \, \text{Pa} \approx 2.0 \times 10^7 \, \text{Pa} \]

### Step 2: Calculate the fractional change in volume (ΔV/V)

The percentage change in volume is given as \( 1.39\% \). To convert this percentage into a fraction:

\[ \frac{\Delta V}{V} = \frac{1.39}{100} = 0.0139 \]

### Step 3: Substitute values into the bulk modulus formula

Now we can substitute \( \Delta P \) and \( \frac{\Delta V}{V} \) into the bulk modulus formula:

\[ K = -\frac{\Delta P}{\frac{\Delta V}{V}} = -\frac{2.0 \times 10^7 \, \text{Pa}}{0.0139} \]

Calculating this gives:

\[ K = -\frac{2.0 \times 10^7}{0.0139} \approx -1.44 \times 10^9 \, \text{Pa} \]

Since bulk modulus is a positive quantity, we take the absolute value:

\[ K \approx 1.44 \times 10^9 \, \text{Pa} \]

### Final Answer

The bulk modulus of the object is approximately \( 1.44 \times 10^9 \, \text{Pa} \).

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