A block is falling in a lake at a depth of 100 m shows 0.2% decrease in its volume at the bottom. What is the bulk modulus of the block ?

Options

A. `4.9xx10^(-7) dyn//mm^(2)`
B. `4.9xx10^(8)N//m^(2)`
C. `4.9xx10^(8)dyn//cm^(2)`
D. `4.9xx10^(12)N//cm^(2)`

Answer

To find the bulk modulus of the block that is falling in a lake at a depth of 100 m, we can follow these steps:

### Step 1: Understand the given data
- Depth of the lake (h) = 100 m
- Percentage decrease in volume = 0.2%
- Initial volume (V) = 100 m³ (assumed for calculation)
- Change in volume (ΔV) = 0.2% of V = 0.002 × V = 0.002 × 100 = 0.2 m³

### Step 2: Calculate the pressure at the depth
The pressure at a depth in a fluid is given by the formula:
\[ P = \rho g h \]
Where:
- \( \rho \) (density of water) = \( 10^3 \, \text{kg/m}^3 \)
- \( g \) (acceleration due to gravity) = \( 9.8 \, \text{m/s}^2 \)
- \( h \) (depth) = \( 100 \, \text{m} \)

Substituting the values:
\[ P = (10^3 \, \text{kg/m}^3)(9.8 \, \text{m/s}^2)(100 \, \text{m}) \]
\[ P = 980000 \, \text{Pa} \, \text{(or N/m}^2\text{)} \]

### Step 3: Calculate the bulk modulus (K)
The bulk modulus is defined as:
\[ K = \frac{P}{\frac{\Delta V}{V}} \]
Where:
- \( \Delta V = 0.2 \, \text{m}^3 \)
- \( V = 100 \, \text{m}^3 \)

Now, calculate \( \frac{\Delta V}{V} \):
\[ \frac{\Delta V}{V} = \frac{0.2}{100} = 0.002 \]

Now substitute the values into the bulk modulus formula:
\[ K = \frac{980000 \, \text{Pa}}{0.002} \]
\[ K = 490000000 \, \text{Pa} \]
\[ K = 4.9 \times 10^8 \, \text{Pa} \]

### Step 4: Convert units if necessary
1. To convert from \( \text{Pa} \) to \( \text{N/cm}^2 \):
\[ 1 \, \text{Pa} = 10^{-4} \, \text{N/cm}^2 \]
\[ K = 4.9 \times 10^8 \, \text{Pa} = 4.9 \times 10^8 \times 10^{-4} \, \text{N/cm}^2 = 4.9 \times 10^4 \, \text{N/cm}^2 \]

2. To convert from \( \text{Pa} \) to \( \text{N/mm}^2 \):
\[ 1 \, \text{Pa} = 10^{-6} \, \text{N/mm}^2 \]
\[ K = 4.9 \times 10^8 \, \text{Pa} = 4.9 \times 10^8 \times 10^{-6} \, \text{N/mm}^2 = 4.9 \times 10^2 \, \text{N/mm}^2 \]

### Final Answer
The bulk modulus of the block is:
\[ K = 4.9 \times 10^8 \, \text{Pa} \, \text{or} \, 4.9 \times 10^4 \, \text{N/cm}^2 \, \text{or} \, 4.9 \times 10^2 \, \text{N/mm}^2 \]

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