The mass of the liquid flowing per second per unit area of cross-section of the tube is proportional to (pressure difference across the ends)^(n) and (average velocity of the liquid)^(m) . Which of the following relations between m and n is correct?

Options

A. `m=n`
B. `m=-n`
C. `m^(2)=n`
D. `m=-n^(2)`

Explanation

To solve the problem, we need to establish a relationship between the mass flow rate of a liquid (per second per unit area) and the pressure difference and average velocity of the liquid using dimensional analysis.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that the mass flow rate \( \dot{m} \) (mass per unit time per unit area) is proportional to the pressure difference \( P \) raised to the power \( n \) and the average velocity \( v \) raised to the power \( m \). This can be expressed as:
\[ \dot{m} \propto P^n v^m \]

2. **Writing the Equation**:
We can write the equation as:
\[ \dot{m} = k P^n v^m \]
where \( k \) is a proportionality constant.

3. **Dimensional Analysis**:
The dimensions of the quantities involved are:
- Mass flow rate \( \dot{m} \): \([M][T^{-1}][L^{-2}]\)
- Pressure \( P \): \([M][L^{-1}][T^{-2}]\)
- Velocity \( v \): \([L][T^{-1}]\)

4. **Expressing Dimensions**:
The dimensional formula for each term can be expressed as:
\[ \text{Dimensions of } \dot{m} = [M][T^{-1}][L^{-2}] \]
\[ \text{Dimensions of } P^n = [M^n][L^{-n}][T^{-2n}] \]
\[ \text{Dimensions of } v^m = [L^m][T^{-m}] \]

5. **Combining Dimensions**:
Now, we combine the dimensions of pressure and velocity:
\[ P^n v^m = [M^n][L^{-n}][T^{-2n}] \cdot [L^m][T^{-m}] = [M^n][L^{m-n}][T^{-2n-m}] \]

6. **Setting Up the Equation**:
Now, we equate the dimensions of both sides:
\[ [M][T^{-1}][L^{-2}] = [M^n][L^{m-n}][T^{-2n-m}] \]

7. **Comparing Dimensions**:
- For mass (\( M \)):
\[ 1 = n \quad \text{(1)} \]
- For length (\( L \)):
\[ -2 = m - n \quad \text{(2)} \]
- For time (\( T \)):
\[ -1 = -2n - m \quad \text{(3)} \]

8. **Solving the Equations**:
From equation (1), we have:
\[ n = 1 \]
Substituting \( n = 1 \) into equation (2):
\[ -2 = m - 1 \implies m = -1 \]

9. **Finding the Relationship**:
Thus, we have:
\[ m = -n \]

10. **Conclusion**:
The correct relationship between \( m \) and \( n \) is:
\[ m = -n \]
Therefore, the answer is option 2.