Pressure is dimensionally

Options

A. Force per unit area
B. Energy per unit volume
C. Momentum per unit area per second
D. Momentum per unit Volume

Explanation

To determine the dimensional formula for pressure, we will analyze the given options and derive the dimensions step by step.

### Step 1: Understanding Pressure
Pressure is defined as force per unit area. Mathematically, it can be expressed as:
\[ \text{Pressure} = \frac{\text{Force}}{\text{Area}} \]

### Step 2: Finding the Dimension of Force
The dimension of force can be derived from Newton's second law of motion, which states that force is the product of mass and acceleration:
\[ \text{Force} = \text{Mass} \times \text{Acceleration} \]
The dimension of mass is \( [M] \), and acceleration is defined as the change in velocity per unit time. The dimension of velocity is \( [L][T^{-1}] \), so:
\[ \text{Acceleration} = \frac{\text{Velocity}}{\text{Time}} = \frac{[L][T^{-1}]}{[T]} = [L][T^{-2}] \]
Thus, the dimension of force is:
\[ \text{Force} = [M][L][T^{-2}] \]

### Step 3: Finding the Dimension of Area
Area is defined as length squared:
\[ \text{Area} = \text{Length}^2 = [L]^2 \]

### Step 4: Finding the Dimension of Pressure
Now substituting the dimensions of force and area into the formula for pressure:
\[ \text{Pressure} = \frac{[M][L][T^{-2}]}{[L]^2} = [M][L^{-1}][T^{-2}] \]
Thus, the dimension of pressure is:
\[ \text{Pressure} = [M][L^{-1}][T^{-2}] \]

### Step 5: Analyzing the Given Options
Now, we will check which of the given options matches the derived dimension of pressure:

1. **Force per unit area**:
- Dimension: \( \frac{[M][L][T^{-2}]}{[L]^2} = [M][L^{-1}][T^{-2}] \) (Matches)

2. **Energy per unit volume**:
- Energy has dimension \( [M][L^2][T^{-2}] \) and volume has dimension \( [L^3] \):
- Dimension: \( \frac{[M][L^2][T^{-2}]}{[L^3]} = [M][L^{-1}][T^{-2}] \) (Matches)

3. **Momentum per unit area per second**:
- Momentum has dimension \( [M][L][T^{-1}] \) and area has dimension \( [L^2] \):
- Dimension: \( \frac{[M][L][T^{-1}]}{[L^2][T]} = [M][L^{-1}][T^{-2}] \) (Matches)

4. **Momentum per unit volume**:
- Dimension: \( \frac{[M][L][T^{-1}]}{[L^3]} = [M][L^{-2}][T^{-1}] \) (Does not match)

### Conclusion
From the analysis, we conclude that pressure is dimensionally equal to:
- Force per unit area
- Energy per unit volume
- Momentum per unit area per second

However, it is not dimensionally equal to momentum per unit volume.