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. lmpulse 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 each pair provided in the options.

### Step-by-Step Solution:

1. **Identify the pairs of physical quantities**:
- Work and Torque
- Angular Momentum and Planck's Constant
- Tension and Surface Tension
- Impulse and Linear Momentum

2. **Calculate the dimensional formula for each pair**:

- **Work (W)**:
- Formula: Work = Force × Displacement
- Force (F) = Mass × Acceleration = MLT⁻²
- Displacement (s) = L
- Therefore, Work = F × s = (MLT⁻²) × L = ML²T⁻²

- **Torque (τ)**:
- Formula: Torque = Force × Distance
- Using the same Force as above, Torque = F × r = (MLT⁻²) × L = ML²T⁻²

- **Conclusion for Work and Torque**: Both have the same dimensional formula: **ML²T⁻²**.

3. **Angular Momentum (L)**:
- Formula: Angular Momentum = Mass × Velocity × Radius = MVR
- Velocity (V) = L/T
- Therefore, Angular Momentum = M × (L/T) × L = ML²T⁻¹

- **Planck's Constant (h)**:
- Formula: Planck's constant relates energy (E) and frequency (ν): h = E/ν
- Energy (E) = Work = ML²T⁻²
- Frequency (ν) = 1/T
- Therefore, h = (ML²T⁻²) / (1/T) = ML²T⁻¹

- **Conclusion for Angular Momentum and Planck's Constant**: Both have the same dimensional formula: **ML²T⁻¹**.

4. **Tension (T)**:
- Formula: Tension = Force = MLT⁻²

- **Surface Tension (σ)**:
- Formula: Surface Tension = Force/Length = (MLT⁻²) / L = MT⁻²

- **Conclusion for Tension and Surface Tension**: Tension has dimensional formula **MLT⁻²** and Surface Tension has **MT⁻²**. They are different.

5. **Impulse (I)**:
- Formula: Impulse = Force × Time = (MLT⁻²) × T = MLT⁻¹

- **Linear Momentum (p)**:
- Formula: Linear Momentum = Mass × Velocity = M × (L/T) = MLT⁻¹

- **Conclusion for Impulse and Linear Momentum**: Both have the same dimensional formula: **MLT⁻¹**.

6. **Final Conclusion**: The pair of physical quantities that do not have the same dimensional formula is **Tension and Surface Tension**.

### Final Answer:
**Tension and Surface Tension do not have the same dimensional formula.**