A spherical ball contracts in volume by 0.08% when subjected to a normal uniform pressure of 500atm. What is the bulk modulus of ball.

Options

A. `4.7 xx 10^(5)N//m^(2)`
B. `5 xx 10^(7) N//m^(2)`
C. `2 xx 10^(7) N//m^(2)`
D. `6.3 xx 10^(10)N//m^(2)`

Explanation

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

### Step 1: Identify the given data
- The volume contraction of the ball is 0.08%.
- The pressure applied is 500 atm.

### Step 2: Convert the percentage contraction into a decimal
The volume contraction in decimal form is:
\[ \Delta V/V = 0.08\% = \frac{0.08}{100} = 0.0008 \]

### Step 3: Calculate the change in volume
Let’s denote the original volume as \( V \). The change in volume \( \Delta V \) can be expressed as:
\[ \Delta V = V \times \left(\frac{\Delta V}{V}\right) = V \times 0.0008 \]

### Step 4: Convert pressure from atm to Newton per meter square
1 atm is equivalent to \( 1.01 \times 10^5 \) N/m². Therefore, the pressure in N/m² is:
\[ P = 500 \, \text{atm} \times 1.01 \times 10^5 \, \text{N/m}^2/\text{atm} = 500 \times 1.01 \times 10^5 = 5.05 \times 10^7 \, \text{N/m}^2 \]

### Step 5: Use the formula for bulk modulus
The bulk modulus \( B \) is defined as:
\[ B = -\frac{P}{\Delta V/V} \]
Substituting the values we have:
\[ B = -\frac{5.05 \times 10^7 \, \text{N/m}^2}{0.0008} \]

### Step 6: Calculate the bulk modulus
Calculating the above expression gives:
\[ B = -\frac{5.05 \times 10^7}{0.0008} = -6.3125 \times 10^{10} \, \text{N/m}^2 \]
Since bulk modulus is a positive quantity, we take the absolute value:
\[ B = 6.3125 \times 10^{10} \, \text{N/m}^2 \]

### Final Answer
The bulk modulus of the ball is:
\[ B = 6.3125 \times 10^{10} \, \text{N/m}^2 \]

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