Which one of the following pairs are not dimensionally identical?

Options

A. Heat energy and work
B. Impulse and momentum
C. Frequency and angular velocity
D. Displacement and angular displacement

Answer

To determine which pair of quantities is not dimensionally identical, we will analyze each option step by step.

### Step 1: Analyze Option A - Heat and Work
- **Heat** is a form of energy, which can be expressed as work done. The formula for both heat and work is given by:
\[ \text{Work} = \text{Force} \times \text{Displacement} = \text{Mass} \times \text{Acceleration} \times \text{Displacement} = MLT^{-2} \times L = ML^2T^{-2} \]
- Therefore, the dimension of both heat and work is:
\[ [\text{Heat}] = [\text{Work}] = ML^2T^{-2} \]
- **Conclusion**: Heat and work are dimensionally identical.

### Step 2: Analyze Option B - Impulse and Momentum
- **Impulse** is defined as the change in momentum. The formula for momentum is:
\[ \text{Momentum} = \text{Mass} \times \text{Velocity} = M \times (LT^{-1}) = MLT^{-1} \]
- Impulse can also be expressed as:
\[ \text{Impulse} = \text{Force} \times \text{Time} = (MLT^{-2}) \times T = MLT^{-1} \]
- **Conclusion**: Impulse and momentum are dimensionally identical.

### Step 3: Analyze Option C - Frequency and Angular Velocity
- **Frequency** is defined as the number of cycles per unit time:
\[ [\text{Frequency}] = T^{-1} \]
- **Angular Velocity** is defined as the rate of change of angular displacement:
\[ \text{Angular Velocity} = \frac{\text{Angular Displacement}}{\text{Time}} = \frac{\text{(Dimensionless)}}{T} = T^{-1} \]
- **Conclusion**: Frequency and angular velocity are dimensionally identical.

### Step 4: Analyze Option D - Displacement and Angular Displacement
- **Displacement** has the dimension:
\[ [\text{Displacement}] = L \]
- **Angular Displacement** (θ) is defined as the ratio of arc length to radius:
\[ \text{Angular Displacement} = \frac{\text{Arc Length}}{\text{Radius}} = \frac{L}{L} = \text{Dimensionless} \]
- **Conclusion**: Displacement and angular displacement are not dimensionally identical.

### Final Conclusion
The pair that is not dimensionally identical is **Option D: Displacement and Angular Displacement**.

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