The volume of a solid rubber ball, when it is carried from the surface to the bottom of a 200 m deep lake decreases by 1%. Find the bulk modulus of rubber ball .

Answer

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

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

### Step 2: Determine the change in volume
The problem states that the volume of the rubber ball decreases by 1%. This means:
\[ \Delta V = -0.01 V \]
where \( V \) is the original volume of the rubber ball.

### Step 3: Calculate the change in pressure
The change in pressure when the ball is taken to a depth of 200 m in water can be calculated using the formula:
\[ \Delta P = \rho g h \]
where:
- \( \rho \) is the density of water (approximately \( 1000 \, \text{kg/m}^3 \)),
- \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)),
- \( h \) is the depth (200 m).

Substituting the values:
\[ \Delta P = 1000 \, \text{kg/m}^3 \times 10 \, \text{m/s}^2 \times 200 \, \text{m} \]
\[ \Delta P = 2000000 \, \text{Pa} \]
or \( 2 \times 10^6 \, \text{Pa} \).

### Step 4: Substitute values into the bulk modulus formula
Now we can substitute \( \Delta P \) and \( \Delta V/V \) into the bulk modulus formula:
\[ K = -\frac{2000000 \, \text{Pa}}{-0.01} \]
\[ K = \frac{2000000}{0.01} \]
\[ K = 200000000 \, \text{Pa} \]
or \( 2 \times 10^9 \, \text{Pa} \).

### Final Answer
The bulk modulus of the rubber ball is:
\[ K = 2 \times 10^9 \, \text{Pa} \]

---