A sphere contract in volume by 0.1% when taken to bottom of the sea 1 km deep. Find the bulk modulus of the material of the sphere. Density of sea water `= 10^(3) kg //m^(3)`.

Explanation

To find the bulk modulus of the material of the sphere, we can follow these steps:

### Step 1: Understand the Concept of Bulk Modulus
The bulk modulus (K) is defined as the ratio of pressure increase to the relative decrease in volume. Mathematically, it is given by:
\[ K = -\frac{P}{\frac{\Delta V}{V}} \]
where:
- \( P \) is the pressure applied,
- \( \Delta V \) is the change in volume,
- \( V \) is the original volume.

### Step 2: Determine the Change in Volume
The sphere contracts in volume by 0.1%. This can be expressed as:
\[ \frac{\Delta V}{V} = -0.1\% = -\frac{0.1}{100} = -0.001 \]

### Step 3: Calculate the Pressure at 1 km Depth
The pressure at a depth \( h \) in a fluid is given by:
\[ P = h \cdot \rho \cdot g \]
where:
- \( h = 1000 \, \text{m} \) (depth),
- \( \rho = 10^3 \, \text{kg/m}^3 \) (density of sea water),
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity).

Substituting the values:
\[ P = 1000 \, \text{m} \cdot 10^3 \, \text{kg/m}^3 \cdot 9.8 \, \text{m/s}^2 \]
\[ P = 1000 \cdot 1000 \cdot 9.8 \]
\[ P = 9.8 \times 10^6 \, \text{N/m}^2 \]

### Step 4: Substitute Values into the Bulk Modulus Formula
Now we can substitute the values of pressure and the relative change in volume into the bulk modulus formula:
\[ K = -\frac{P}{\frac{\Delta V}{V}} \]
\[ K = -\frac{9.8 \times 10^6 \, \text{N/m}^2}{-0.001} \]
\[ K = \frac{9.8 \times 10^6}{0.001} \]
\[ K = 9.8 \times 10^9 \, \text{N/m}^2 \]

### Step 5: Conclusion
The bulk modulus of the material of the sphere is:
\[ K = 9.8 \times 10^9 \, \text{N/m}^2 \]

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