A spherical ball contracts in volume by 0.0098% when subjected to a pressure of 100 atm. Calculate its bulk modulus. Given 1 atm `=1.01xx10^(5) Nm^(-2)`.

Answer

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

### Step 1: Understand the given data
- The volume contraction percentage is given as \(0.0098\%\).
- The pressure applied is \(100 \, \text{atm}\).
- The conversion factor for pressure is \(1 \, \text{atm} = 1.01 \times 10^5 \, \text{N/m}^2\).

### Step 2: Convert the volume contraction percentage to a decimal
The volume contraction in decimal form can be calculated as:
\[ \Delta V / V = 0.0098\% = \frac{0.0098}{100} = 0.000098 \]

### Step 3: Convert the pressure from atm to N/m²
To find the pressure in N/m², we multiply the pressure in atm by the conversion factor:
\[ P = 100 \, \text{atm} \times 1.01 \times 10^5 \, \text{N/m}^2/\text{atm} = 1.01 \times 10^7 \, \text{N/m}^2 \]

### Step 4: Calculate the bulk modulus
The bulk modulus \(K\) is defined as the ratio of the pressure applied to the relative change in volume:
\[ K = -\frac{P}{\Delta V/V} \]
Substituting the values we have:
\[ K = -\frac{1.01 \times 10^7 \, \text{N/m}^2}{0.000098} \]

### Step 5: Perform the calculation
Calculating the bulk modulus:
\[ K = -\frac{1.01 \times 10^7}{0.000098} \approx -1.0306122449 \times 10^9 \, \text{N/m}^2 \]
Since bulk modulus is a positive quantity, we take the absolute value:
\[ K \approx 1.03 \times 10^9 \, \text{N/m}^2 \]

### Final Answer
The bulk modulus of the spherical ball is approximately \(1.03 \times 10^9 \, \text{N/m}^2\).

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