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^(-1) V^2T^(-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 will follow these steps:

### Step 1: Write the dimensional formula for force.
The dimensional formula for force (F) is given by:
\[ F = M \cdot L \cdot T^{-2} \]
where:
- \( M \) is mass,
- \( L \) is length,
- \( T \) is time.

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

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

3. **Time (T)** has the dimensional formula:
\[ T = T \]

### Step 3: Set up the equation for dimensional analysis.
We express force (F) in terms of pressure (P), velocity (V), and time (T):
\[ F = P^x \cdot V^y \cdot T^z \]

### Step 4: Substitute the dimensional formulas into the equation.
Substituting the dimensional formulas we have:
\[ M \cdot L \cdot T^{-2} = (M \cdot L^{-1} \cdot T^{-2})^x \cdot (L \cdot T^{-1})^y \cdot (T)^z \]

### Step 5: Expand and equate the dimensions.
Expanding the right-hand side, we get:
\[ M^x \cdot L^{-x} \cdot T^{-2x} \cdot L^y \cdot T^{-y} \cdot T^z \]
This simplifies to:
\[ M^x \cdot L^{y-x} \cdot T^{-2x-y+z} \]

### Step 6: Equate the powers of M, L, and T.
Now we equate the coefficients of \( M \), \( L \), and \( T \):
1. For \( M \): \( x = 1 \)
2. For \( L \): \( y - x = 1 \)
3. For \( T \): \( -2x - y + z = -2 \)

### Step 7: Solve the equations.
From \( x = 1 \):
- Substitute \( x \) into the second equation:
\[ y - 1 = 1 \implies y = 2 \]

- Substitute \( x \) and \( y \) into the third equation:
\[ -2(1) - 2 + z = -2 \]
\[ -2 - 2 + z = -2 \implies z = 2 \]

### Step 8: Write the final dimensional formula.
Now we have:
- \( x = 1 \)
- \( y = 2 \)
- \( z = 2 \)

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

### Final Answer:
The dimensional formula of force is:
\[ F = P^1 \cdot V^2 \cdot T^2 \]

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