Which pair have not equal dimensions :

Options

A. Energy and torque
B. Force and impulse
C. Angular momentum and plank's constant
D. Elastic modulus and pressure

Answer

To solve the problem of identifying which pair of physical quantities does not have equal dimensions, we will analyze each option step by step.

### Step 1: Analyze Option 1 - Energy and Torque
- **Energy**: The dimension of energy is given by the formula \( \text{Energy} = \text{Force} \times \text{Distance} \).
- Force (F) has dimensions of \( [M][L][T^{-2}] \) (mass × acceleration).
- Distance (d) has dimensions of \( [L] \).
- Therefore, the dimensions of energy are:
\[ [E] = [F][d] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]

- **Torque**: Torque is defined as \( \text{Torque} = \text{Force} \times \text{Distance} \) (lever arm).
- Using the same dimensions for force and distance:
\[ [\tau] = [F][d] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]

- **Conclusion for Option 1**: Both energy and torque have the same dimensions of \( [M][L^2][T^{-2}] \).

### Step 2: Analyze Option 2 - Force and Impulse
- **Force**: As previously calculated, the dimensions of force are:
\[ [F] = [M][L][T^{-2}] \]

- **Impulse**: Impulse is defined as the product of force and time:
\[ \text{Impulse} = \text{Force} \times \text{Time} \]
- Time has dimensions of \( [T] \), thus:
\[ [I] = [F][T] = [M][L][T^{-2}][T] = [M][L][T^{-1}] \]

- **Conclusion for Option 2**: Force has dimensions \( [M][L][T^{-2}] \) and impulse has dimensions \( [M][L][T^{-1}] \). They are not equal.

### Step 3: Analyze Option 3 - Angular Momentum and Planck's Constant
- **Angular Momentum**: Defined as \( \text{Angular Momentum} = \text{Moment of Inertia} \times \text{Angular Velocity} \).
- Moment of Inertia has dimensions \( [M][L^2] \) and Angular Velocity has dimensions \( [T^{-1}] \):
\[ [L] = [M][L^2][T^{-1}] = [M][L^2][T^{-1}] \]

- **Planck's Constant**: It is defined as energy multiplied by time:
\[ [h] = [E][T] = [M][L^2][T^{-2}][T] = [M][L^2][T^{-1}] \]

- **Conclusion for Option 3**: Both angular momentum and Planck's constant have the same dimensions of \( [M][L^2][T^{-1}] \).

### Step 4: Analyze Option 4 - Elastic Modulus and Pressure
- **Elastic Modulus**: Defined as stress over strain. Stress has dimensions of pressure:
\[ [E] = \frac{[F]/[A]}{[L]} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]

- **Pressure**: Defined as force per unit area:
\[ [P] = \frac{[F]}{[A]} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]

- **Conclusion for Option 4**: Both elastic modulus and pressure have the same dimensions of \( [M][L^{-1}][T^{-2}] \).

### Final Conclusion
The only pair that does not have equal dimensions is **Option 2: Force and Impulse**.