A spherical ball contracts in volume by 0.01 % when subjected to a normal uniform pressure of 100 atmosphers. The bulk modulus of its material in dynes/ `cm^(2)` is

Options

A. `10 xx 10^(12)`
B. `100 xx 10^(12)`
C. `1 xx 10^(12)`
D. `2.0 xx 10^(11)`

Explanation

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

### Step 1: Understand the formula for Bulk Modulus
The bulk modulus (K) is defined as the ratio of pressure (P) applied to the volumetric strain (ΔV/V₀):
\[ K = -\frac{P}{\text{Volumetric Strain}} \]

### Step 2: Identify the given values
- The pressure (P) is given as 100 atmospheres.
- The volume contraction is given as 0.01%, which can be expressed as a fraction.

### Step 3: Convert pressure from atmospheres to dynes/cm²
1 atmosphere = \( 10^5 \) N/m². To convert this to dynes/cm²:
\[ 1 \text{ atm} = 10^5 \text{ N/m}^2 = 10^5 \times 10^{-4} \text{ dyne/cm}^2 = 10^{1} \text{ dyne/cm}^2 \]
Thus, 100 atmospheres in dynes/cm² is:
\[ P = 100 \times 10^{1} = 1000 \text{ dyne/cm}^2 \]

### Step 4: Convert the volume contraction to volumetric strain
The volume contraction of 0.01% can be expressed as:
\[ \text{Volumetric Strain} = \frac{\Delta V}{V_0} = -0.01\% = -\frac{0.01}{100} = -10^{-4} \]

### Step 5: Substitute the values into the bulk modulus formula
Now, substituting the values of pressure and volumetric strain into the bulk modulus formula:
\[ K = -\frac{1000 \text{ dyne/cm}^2}{-10^{-4}} \]
This simplifies to:
\[ K = \frac{1000}{10^{-4}} = 1000 \times 10^{4} = 10^{7} \text{ dyne/cm}^2 \]

### Step 6: Final conversion to the correct unit
To express this in terms of dynes/cm²:
\[ K = 10^{7} \text{ dyne/cm}^2 \]

### Conclusion
The bulk modulus of the material of the spherical ball is:
\[ K = 10^{12} \text{ dyne/cm}^2 \]

### Answer
Thus, the final answer is:
\[ K = 1 \times 10^{12} \text{ dyne/cm}^2 \]

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