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

Options

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

Answer

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

### Step-by-Step Solution:

1. **Assume the relationship**:
We start by assuming that force (F) can be expressed as a product of area (A), velocity (v), and density (ρ) raised to some powers:
\[ F = A^x \cdot v^y \cdot \rho^z \]

2. **Write the dimensions of each quantity**:
- The dimension of force (F) is given by:
\[ [F] = MLT^{-2} \]
- The dimension of area (A) is:
\[ [A] = L^2 \]
- The dimension of velocity (v) is:
\[ [v] = LT^{-1} \]
- The dimension of density (ρ) is:
\[ [\rho] = ML^{-3} \]

3. **Substitute the dimensions into the equation**:
Now we substitute the dimensions into our assumed equation:
\[ [F] = [A^x] \cdot [v^y] \cdot [\rho^z] \]
This gives us:
\[ MLT^{-2} = (L^2)^x \cdot (LT^{-1})^y \cdot (ML^{-3})^z \]

4. **Expand the right-hand side**:
Expanding the right-hand side, we get:
\[ MLT^{-2} = M^z \cdot L^{2x} \cdot L^y \cdot T^{-y} \cdot L^{-3z} \]
Combining the terms, we have:
\[ MLT^{-2} = M^z \cdot L^{2x + y - 3z} \cdot T^{-y} \]

5. **Compare the powers of M, L, and T**:
Now we compare the coefficients of M, L, and T on both sides:
- For mass (M):
\[ z = 1 \]
- For length (L):
\[ 2x + y - 3z = 1 \]
- For time (T):
\[ -y = -2 \implies y = 2 \]

6. **Substitute known values**:
We already found \( z = 1 \) and \( y = 2 \). Now substitute \( z \) into the length equation:
\[ 2x + 2 - 3(1) = 1 \implies 2x + 2 - 3 = 1 \implies 2x - 1 = 1 \implies 2x = 2 \implies x = 1 \]

7. **Final expression for force**:
Now that we have \( x = 1 \), \( y = 2 \), and \( z = 1 \), we can write the expression for force:
\[ F = A^1 \cdot v^2 \cdot \rho^1 = A \cdot v^2 \cdot \rho \]

### Conclusion:
Thus, the dimensional formula of force can be represented as:
\[ F = A \cdot v^2 \cdot \rho \]