Find the increase in pressure required to decrease the volume of a water sample of 0.01%. Bulk modulus of water `=2.1xx10^9Nm^-2`.

Explanation

To find the increase in pressure required to decrease the volume of a water sample by 0.01%, we can use the concept of bulk modulus. Here’s a step-by-step solution:

### Step 1: Understand the given values
- We are given the bulk modulus of water \( B = 2.1 \times 10^9 \, \text{N/m}^2 \).
- The decrease in volume is \( 0.01\% \).

### Step 2: Convert the percentage decrease in volume to a decimal
The decrease in volume \( \Delta V \) can be expressed as a fraction of the original volume \( V \):
\[ \Delta V = 0.01\% \times V = \frac{0.01}{100} \times V = 0.0001 \times V \]
Thus, we can express the change in volume as:
\[ \frac{\Delta V}{V} = 0.0001 = 10^{-4} \]

### Step 3: Use the formula for bulk modulus
The bulk modulus \( B \) is defined as:
\[ B = -\frac{P \cdot V}{\Delta V} \]
Where \( P \) is the increase in pressure. Rearranging this formula gives us:
\[ P = -B \cdot \frac{\Delta V}{V} \]

### Step 4: Substitute the known values into the equation
Substituting the values we have:
\[ P = -B \cdot \frac{\Delta V}{V} = - (2.1 \times 10^9) \cdot (10^{-4}) \]

### Step 5: Calculate the increase in pressure
Calculating the above expression:
\[ P = - (2.1 \times 10^9) \cdot (10^{-4}) = -2.1 \times 10^5 \, \text{N/m}^2 \]
Since pressure cannot be negative in this context, we take the absolute value:
\[ P = 2.1 \times 10^5 \, \text{N/m}^2 \]

### Final Answer
The increase in pressure required to decrease the volume of the water sample by 0.01% is:
\[ \boxed{2.1 \times 10^5 \, \text{N/m}^2} \]