Which of the following pairs does not have similar dimensions ?

Options

A. Tension and surface tension
B. Stress and pressure
C. Plank's constant and angular momentum
D. Angle and strain

Answer

To solve the problem of identifying which pair of quantities does not have similar dimensions, we will analyze the dimensions of each pair provided in the question.

### Step-by-Step Solution:

1. **Identify the quantities in the first pair: Tension and Surface Tension.**
- **Surface Tension (T)**:
- Surface tension is defined as force per unit length.
- Force can be expressed as mass times acceleration (F = ma).
- Acceleration can be expressed in terms of velocity (v) and time (t): a = v/t.
- Therefore, the dimension of force is:
\[ [F] = [M][L][T^{-2}] = MLT^{-2} \]
- Now, surface tension is:
\[ [\text{Surface Tension}] = \frac{[F]}{[\text{Length}]} = \frac{MLT^{-2}}{L} = MT^{-2} \]

- **Tension**:
- Tension is defined as the magnitude of force, hence:
\[ [\text{Tension}] = [F] = MLT^{-2} \]

- **Conclusion for Pair 1**:
- Dimensions of Surface Tension: \( MT^{-2} \)
- Dimensions of Tension: \( MLT^{-2} \)
- Since \( MT^{-2} \) is not equal to \( MLT^{-2} \), these two do not have similar dimensions.

2. **Identify the quantities in the second pair: Stress and Pressure.**
- **Stress**:
- Stress is defined as force per unit area.
- Therefore, the dimension of stress is:
\[ [\text{Stress}] = \frac{[F]}{[\text{Area}]} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]

- **Pressure**:
- Pressure is also defined as force per unit area, hence:
\[ [\text{Pressure}] = \frac{[F]}{[\text{Area}]} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]

- **Conclusion for Pair 2**:
- Both Stress and Pressure have the same dimensions: \( ML^{-1}T^{-2} \).

3. **Identify the quantities in the third pair: Planck's Constant and Angular Momentum.**
- **Planck's Constant (h)**:
- It relates energy (E) and frequency (ν): \( E = hν \).
- The dimension of energy is \( [E] = ML^2T^{-2} \) and frequency is \( [ν] = T^{-1} \).
- Thus, the dimension of Planck's constant is:
\[ [h] = \frac{[E]}{[ν]} = \frac{ML^2T^{-2}}{T^{-1}} = ML^2T^{-1} \]

- **Angular Momentum (L)**:
- Angular momentum is defined as \( L = mvr \) where \( m \) is mass, \( v \) is velocity, and \( r \) is radius.
- The dimension of angular momentum is:
\[ [L] = [M][L][T^{-1}] = ML^2T^{-1} \]

- **Conclusion for Pair 3**:
- Both Planck's Constant and Angular Momentum have the same dimensions: \( ML^2T^{-1} \).

4. **Identify the quantities in the fourth pair: Angle and Strain.**
- **Angle**:
- Angles are dimensionless quantities, often measured in radians.
- Therefore, the dimension of angle is:
\[ [\text{Angle}] = 1 \]

- **Strain**:
- Strain is defined as the ratio of change in length to original length, which is also dimensionless.
- Therefore, the dimension of strain is:
\[ [\text{Strain}] = 1 \]

- **Conclusion for Pair 4**:
- Both Angle and Strain are dimensionless quantities.

### Final Conclusion:
The only pair that does not have similar dimensions is **Tension and Surface Tension**.