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

Options

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

Answer

To find the dimensional formula of force in terms of pressure (P), velocity (V), and time (T), we can follow these steps:

### Step 1: Write the dimensional formula for force
The dimensional formula for force (F) is given by:
\[ F = m \cdot a \]
Where \( a \) (acceleration) is \( \frac{L}{T^2} \). Therefore, the dimensional formula for force is:
\[ [F] = [M][L][T^{-2}] = M^1 L^1 T^{-2} \]

### Step 2: Write the dimensional formulas for pressure and velocity
1. **Pressure (P)** is defined as force per unit area:
\[ P = \frac{F}{A} \]
The dimensional formula for area (A) is \( L^2 \). Thus, the dimensional formula for pressure is:
\[ [P] = \frac{[F]}{[A]} = \frac{M^1 L^1 T^{-2}}{L^2} = M^1 L^{-1} T^{-2} \]

2. **Velocity (V)** is defined as distance per unit time:
\[ V = \frac{L}{T} \]
Therefore, its dimensional formula is:
\[ [V] = L^1 T^{-1} \]

### Step 3: Express the force in terms of P, V, and T
Assume the dimensional formula for force can be expressed as:
\[ [F] = P^a V^b T^c \]
Substituting the dimensional formulas we found:
\[ [F] = (M^1 L^{-1} T^{-2})^a (L^1 T^{-1})^b (T^1)^c \]

### Step 4: Expand and combine the dimensions
Expanding the expression:
\[ [F] = M^a L^{-a + b} T^{-2a - b + c} \]

### Step 5: Set up equations by equating dimensions
Now, we equate the dimensions of force:
- For mass (M): \( a = 1 \)
- For length (L): \( -a + b = 1 \)
- For time (T): \( -2a - b + c = -2 \)

### Step 6: Solve the equations
1. From \( a = 1 \), we substitute into the second equation:
\[ -1 + b = 1 \Rightarrow b = 2 \]

2. Substitute \( a = 1 \) and \( b = 2 \) into the third equation:
\[ -2(1) - 2 + c = -2 \]
\[ -2 - 2 + c = -2 \Rightarrow c = 2 \]

### Step 7: Write the final expression for force
Now we can express the dimensional formula for force:
\[ [F] = P^1 V^2 T^2 \]

### Conclusion
Thus, the dimensional formula of force in terms of pressure, velocity, and time is:
\[ F = P^1 V^2 T^2 \]