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

Options

A. `10 xx 10^12`
B. `100 xx 10^2`
C. `1 xx 10^12` s
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 given data
- The volume contraction is given as 0.01%.
- The pressure applied is 100 atmospheres.

### Step 2: Convert the volume contraction percentage to a decimal
The volume contraction percentage is given as 0.01%. To convert this to a decimal, we divide by 100:
\[ \frac{\Delta V}{V} = \frac{0.01}{100} = 0.0001 \]

### Step 3: Convert the pressure from atmospheres to Newtons per square meter
1 atmosphere is approximately \(1.01 \times 10^5\) Newtons per square meter. Therefore, to convert 100 atmospheres:
\[ P = 100 \, \text{atm} \times 1.01 \times 10^5 \, \text{N/m}^2 = 1.01 \times 10^7 \, \text{N/m}^2 \]

### Step 4: Use the formula for bulk modulus
The bulk modulus \(K\) is defined as the ratio of the change in pressure to the relative change in volume:
\[ K = \frac{P}{\frac{\Delta V}{V}} \]
Substituting the values we have:
\[ K = \frac{1.01 \times 10^7 \, \text{N/m}^2}{0.0001} \]

### Step 5: Calculate the bulk modulus
Now, we can perform the calculation:
\[ K = 1.01 \times 10^7 \, \text{N/m}^2 \div 0.0001 = 1.01 \times 10^{11} \, \text{N/m}^2 \]

### Step 6: Convert the bulk modulus to dyne/cm²
To convert from N/m² to dyne/cm², we use the conversion factor \(1 \, \text{N/m}^2 = 10^4 \, \text{dyne/cm}^2\):
\[ K = 1.01 \times 10^{11} \, \text{N/m}^2 \times 10^4 \, \text{dyne/cm}^2 = 1.01 \times 10^{15} \, \text{dyne/cm}^2 \]

### Final Answer
The bulk modulus of the material of the spherical ball is:
\[ K = 1.01 \times 10^{15} \, \text{dyne/cm}^2 \]

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