If frequency `F`, velocity `V` , and density `D` are considered fundamental units , the dimensional formula for momentum will be

Options

A. `DVF^(2)`
B. `DV^(2)F^(-1)`
C. `D^(2)V^(2)F^(2)`
D. `DV^(4)F^(-3)`

Explanation

To find the dimensional formula for momentum in terms of frequency (F), velocity (V), and density (D), we can follow these steps:

### Step 1: Understand the dimensional formula for momentum
Momentum (p) is defined as the product of mass (m) and velocity (v). The dimensional formula for momentum is given by:
\[ [p] = [M][L][T^{-1}] \]
where:
- M = mass
- L = length
- T = time

### Step 2: Write the relationship in terms of F, V, and D
We express momentum in terms of the fundamental units F, V, and D:
\[ p = F^a V^b D^c \]
where \( a, b, c \) are the powers we need to determine.

### Step 3: Write the dimensional formulas for F, V, and D
1. **Frequency (F)**: The dimensional formula for frequency is:
\[ [F] = [T^{-1}] \]
2. **Velocity (V)**: The dimensional formula for velocity is:
\[ [V] = [L][T^{-1}] \]
3. **Density (D)**: The dimensional formula for density is:
\[ [D] = [M][L^{-3}] \]

### Step 4: Substitute the dimensional formulas into the equation
Substituting the dimensional formulas into the equation for momentum, we have:
\[ [M][L][T^{-1}] = [F^a][V^b][D^c] \]
This expands to:
\[ [M][L][T^{-1}] = [T^{-a}][L^{b}][M^{c}][L^{-3c}] \]
Combining the terms gives:
\[ [M^{c}][L^{b - 3c}][T^{-a}] \]

### Step 5: Equate the powers of M, L, and T
Now we equate the powers of M, L, and T from both sides:
1. For mass (M):
\[ c = 1 \]
2. For length (L):
\[ b - 3c = 1 \]
Substituting \( c = 1 \):
\[ b - 3(1) = 1 \implies b - 3 = 1 \implies b = 4 \]
3. For time (T):
\[ -a = -1 - b \implies -a = -1 - 4 \implies -a = -5 \implies a = 5 \]

### Step 6: Write the final expression for momentum
Now we have:
- \( a = 5 \)
- \( b = 4 \)
- \( c = 1 \)

Thus, the dimensional formula for momentum in terms of F, V, and D is:
\[ p = F^5 V^4 D^1 \]

### Final Answer
The dimensional formula for momentum is:
\[ p = F^5 V^4 D^1 \]