If area `(A)` velocity `(v)` and density `(rho)` are base units, then the dimensional formula of force can be represented

Options

A. `Av rho`
B. `Av^(2) rho`
C. `Av rho^(2)`
D. `A^(2)v rho`

Explanation

To find the dimensional formula of force (F) in terms of area (A), velocity (v), and density (ρ) as base units, we can follow these steps:

### Step 1: Establish the relationship
We start with the assumption that force (F) can be expressed in terms of area (A), velocity (v), and density (ρ) as follows:
\[ F \propto A^X v^Y \rho^Z \]
Where \(X\), \(Y\), and \(Z\) are the powers we need to determine.

### Step 2: Write the dimensional formulas
Next, we need to write the dimensional formulas for each of the quantities involved:
- The dimensional formula of force (F) is:
\[ [F] = M^1 L^1 T^{-2} \]
- The dimensional formula of area (A) is:
\[ [A] = L^2 \]
- The dimensional formula of velocity (v) is:
\[ [v] = L^1 T^{-1} \]
- The dimensional formula of density (ρ) is:
\[ [\rho] = M^1 L^{-3} \]

### Step 3: Substitute the dimensional formulas
Substituting these dimensional formulas into our equation gives:
\[ M^1 L^1 T^{-2} = C \cdot (L^2)^X \cdot (L^1 T^{-1})^Y \cdot (M^1 L^{-3})^Z \]

### Step 4: Expand and combine the dimensions
Expanding the right-hand side:
\[ = C \cdot L^{2X} \cdot L^{Y} \cdot M^{Z} \cdot L^{-3Z} \cdot T^{-Y} \]
This simplifies to:
\[ = C \cdot M^{Z} \cdot L^{2X + Y - 3Z} \cdot T^{-Y} \]

### Step 5: Equate the dimensions
Now, we equate the dimensions on both sides:
1. For mass (M):
\[ 1 = Z \]
2. For length (L):
\[ 1 = 2X + Y - 3Z \]
3. For time (T):
\[ -2 = -Y \]

### Step 6: Solve the equations
From the first equation, we have:
\[ Z = 1 \]

Substituting \(Z = 1\) into the second equation:
\[ 1 = 2X + Y - 3(1) \]
\[ 1 = 2X + Y - 3 \]
\[ 2X + Y = 4 \quad \text{(Equation 1)} \]

From the third equation:
\[ Y = 2 \quad \text{(Equation 2)} \]

Substituting \(Y = 2\) into Equation 1:
\[ 2X + 2 = 4 \]
\[ 2X = 2 \]
\[ X = 1 \]

### Step 7: Conclusion
Now we have:
- \(X = 1\)
- \(Y = 2\)
- \(Z = 1\)

Thus, we can express force (F) as:
\[ F \propto A^1 v^2 \rho^1 \]
Or:
\[ F \propto A v^2 \rho \]

### Final Answer
The dimensional formula of force can be represented as:
\[ F \propto A v^2 \rho \]