Which pair do not have equal dimensions?

Options

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

Answer

To determine which pair of physical quantities do not have equal dimensions, we will analyze each option provided in the question step by step.

### Step 1: Analyze Energy and Torque
- **Energy** is defined as work done, which is given by the formula:
\[ \text{Energy} = \text{Work} = \text{Force} \times \text{Distance} \]
- The dimension of force is given by:
\[ \text{Force} = \text{Mass} \times \text{Acceleration} = [M][L][T^{-2}] \]
- Therefore, the dimension of energy (or work) is:
\[ [E] = [F][L] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]
- **Torque** is defined as:
\[ \text{Torque} = \text{Force} \times \text{Distance} \]
- Thus, the dimension of torque is also:
\[ [\tau] = [F][L] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]
- **Conclusion**: The dimensions of energy and torque are equal.

### Step 2: Analyze Force and Impulse
- **Force** has the dimension:
\[ [F] = [M][L][T^{-2}] \]
- **Impulse** is defined as the change in momentum, which can be expressed as:
\[ \text{Impulse} = \Delta P = \text{Force} \times \Delta T \]
- The dimension of impulse is:
\[ [I] = [M][L][T^{-2}][T] = [M][L][T^{-1}] \]
- **Conclusion**: The dimensions of force and impulse are not equal.

### Step 3: Analyze Angular Momentum and Planck's Constant
- **Angular Momentum** is given by:
\[ L = M \times V \times R = [M][L][T^{-1}][L] = [M][L^2][T^{-1}] \]
- **Planck's Constant** has the dimension:
\[ [H] = [E][T] = [M][L^2][T^{-2}][T] = [M][L^2][T^{-1}] \]
- **Conclusion**: The dimensions of angular momentum and Planck's constant are equal.

### Step 4: Analyze Elastic Modulus, Young's Modulus, and Pressure
- **Elastic Modulus (Young's Modulus)** is defined as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{\text{Force}/\text{Area}}{\text{Length/Length}} = \frac{F/A}{1} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]
- **Pressure** is defined as:
\[ P = \frac{F}{A} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]
- **Conclusion**: The dimensions of elastic modulus (Young's modulus) and pressure are equal.

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