Which of the following pairs of physical quantites does not have same dimensional formula ?

Options

A. Work and torque
B. Angular momentum and Planck's constant.
C. Tension and surface tension.
D. Impulse and linear momentum.

Explanation

To solve the question of which pair of physical quantities does not have the same dimensional formula, we will analyze the dimensional formulas of the given quantities.

### Step 1: Identify the quantities
We need to identify the pairs of physical quantities provided in the question. For this example, we will consider:
1. Tension
2. Surface Tension

### Step 2: Determine the dimensional formula of Tension
Tension is a force, and the dimensional formula for force is given by:
\[ \text{Force} = \text{mass} \times \text{acceleration} \]
The dimensional formula for mass (m) is [M], and for acceleration (a), it is [L][T]^-2. Therefore, the dimensional formula for tension (T) is:
\[ \text{Tension} = [M][L][T]^{-2} = [M^1 L^1 T^{-2}] \]

### Step 3: Determine the dimensional formula of Surface Tension
Surface tension is defined as the force per unit length. Therefore, its dimensional formula can be derived as follows:
\[ \text{Surface Tension} = \frac{\text{Force}}{\text{Length}} = \frac{[M^1 L^1 T^{-2}]}{[L^1]} = [M^1 L^0 T^{-2}] \]

### Step 4: Compare the dimensional formulas
Now we compare the dimensional formulas:
- Dimensional formula of Tension: \([M^1 L^1 T^{-2}]\)
- Dimensional formula of Surface Tension: \([M^1 L^0 T^{-2}]\)

### Step 5: Conclusion
Since the dimensional formula of tension includes [L^1] while that of surface tension includes [L^0], we can conclude that:
- Tension and Surface Tension do not have the same dimensional formula.

### Final Answer
The pair of physical quantities that does not have the same dimensional formula is **Tension and Surface Tension**.

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