If pressure P, velocity V and time T are taken as fundamental physical quantities, the dimensional formula of force if

Options

A. `PV^2T^2`
B. `P^-1V^2T^-2`
C. `PVT^2`
D. `P^-1VT^2`

Explanation

To find the dimensional formula of force (F) when pressure (P), velocity (V), and time (T) are taken as fundamental quantities, we can follow these steps:

### Step 1: Write the relationship for force
We start by expressing force in terms of the fundamental quantities:
\[ F \propto P^X V^Y T^Z \]
where \( X, Y, Z \) are unknown exponents that we need to determine.

### Step 2: Write the dimensional formulas
We know the following dimensional formulas:
- Force (F): \( [F] = M L T^{-2} \)
- Pressure (P): \( [P] = \frac{F}{A} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \)
- Velocity (V): \( [V] = \frac{L}{T} = L T^{-1} \)

### Step 3: Substitute the dimensional formulas
Now we substitute the dimensional formulas into our expression:
\[ F = K \cdot (M L^{-1} T^{-2})^X \cdot (L T^{-1})^Y \cdot (T)^Z \]

### Step 4: Expand the right-hand side
Expanding the right-hand side gives:
\[ F = K \cdot M^X \cdot L^{-X} \cdot T^{-2X} \cdot L^Y \cdot T^{-Y} \cdot T^Z \]
Combining the terms:
\[ F = K \cdot M^X \cdot L^{Y - X} \cdot T^{-2X - Y + Z} \]

### Step 5: Set up equations by comparing dimensions
Now we can compare the dimensions of both sides:
1. For mass (M): \( X = 1 \)
2. For length (L): \( Y - X = 1 \)
3. For time (T): \( -2X - Y + Z = -2 \)

### Step 6: Solve the equations
From the first equation, we have:
- \( X = 1 \)

Substituting \( X = 1 \) into the second equation:
- \( Y - 1 = 1 \) → \( Y = 2 \)

Substituting \( X = 1 \) and \( Y = 2 \) into the third equation:
- \( -2(1) - 2 + Z = -2 \)
- \( -2 - 2 + Z = -2 \)
- \( Z = 2 \)

### Step 7: Write the final expression for force
Now we can write the expression for force:
\[ F = K \cdot P^1 \cdot V^2 \cdot T^2 \]
Thus, the dimensional formula for force is:
\[ F = P V^2 T^2 \]

### Conclusion
The dimensional formula of force when pressure, velocity, and time are taken as fundamental quantities is:
\[ F = P V^2 T^2 \]