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 determine which pair of physical quantities does not have the same dimensional formula, we will analyze the dimensional formulas of each pair provided in the options.

### Step-by-Step Solution:

1. **Identify the pairs of physical quantities**: The question provides several pairs of physical quantities. We will analyze each pair to find their dimensional formulas.

2. **Analyze Work and Torque**:
- **Work (W)**:
- Formula: \( W = F \cdot S \)
- Dimension of Force (F): \( [F] = MLT^{-2} \)
- Dimension of Displacement (S): \( [S] = L \)
- Therefore, \( [W] = [F] \cdot [S] = (MLT^{-2}) \cdot (L) = ML^2T^{-2} \)

- **Torque (τ)**:
- Formula: \( τ = F \cdot R \)
- Dimension of Perpendicular Distance (R): \( [R] = L \)
- Therefore, \( [τ] = [F] \cdot [R] = (MLT^{-2}) \cdot (L) = ML^2T^{-2} \)

- **Conclusion**: Both Work and Torque have the same dimensional formula: \( ML^2T^{-2} \).

3. **Analyze Angular Momentum and Planck's Constant**:
- **Angular Momentum (L)**:
- Formula: \( L = M \cdot V \cdot R \)
- Dimension of Linear Momentum (P): \( [P] = MLT^{-1} \)
- Therefore, \( [L] = [P] \cdot [R] = (MLT^{-1}) \cdot (L) = ML^2T^{-1} \)

- **Planck's Constant (h)**:
- Formula: \( h = E \cdot T \)
- Dimension of Energy (E): \( [E] = ML^2T^{-2} \)
- Therefore, \( [h] = [E] \cdot [T] = (ML^2T^{-2}) \cdot (T) = ML^2T^{-1} \)

- **Conclusion**: Both Angular Momentum and Planck's Constant have the same dimensional formula: \( ML^2T^{-1} \).

4. **Analyze Tension and Surface Tension**:
- **Tension (T)**:
- Tension is a force, so its dimension is the same as force: \( [T] = MLT^{-2} \)

- **Surface Tension (σ)**:
- Formula: \( σ = \frac{F}{L} \)
- Therefore, \( [σ] = \frac{[F]}{[L]} = \frac{MLT^{-2}}{L} = MT^{-2} \)

- **Conclusion**: Tension has the dimensional formula \( MLT^{-2} \) while Surface Tension has \( MT^{-2} \). Thus, they do not have the same dimensional formula.

5. **Analyze Impulse and Linear Momentum**:
- **Impulse (J)**:
- Formula: \( J = ΔP \)
- Dimension of Change in Momentum (ΔP): \( [ΔP] = [P] = MLT^{-1} \)

- **Linear Momentum (P)**:
- Dimension: \( [P] = MLT^{-1} \)

- **Conclusion**: Both Impulse and Linear Momentum have the same dimensional formula: \( MLT^{-1} \).

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

---