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 pairs are not dimensionally identical, we need to analyze the dimensions of each pair given in the question. Let's go through the pairs step-by-step.

### Step 1: Analyze Heat Energy and Work
- **Heat Energy**: The dimension of heat energy is the same as that of work, which is energy.
- **Work**: Work is defined as force times displacement. The dimension of force is given by \( [F] = [M][L][T^{-2}] \) (mass times acceleration). Therefore, the dimension of work is:
\[ [W] = [F][L] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]
- **Conclusion**: Heat energy and work have the same dimension: \( [M][L^2][T^{-2}] \).

### Step 2: Analyze Impulse and Momentum
- **Impulse**: Impulse is defined as the change in momentum, which is the product of force and time. Therefore, the dimension of impulse is:
\[ [I] = [F][T] = [M][L][T^{-2}][T] = [M][L][T^{-1}] \]
- **Momentum**: Momentum is defined as mass times velocity. The dimension of momentum is:
\[ [P] = [M][V] = [M][L][T^{-1}] = [M][L][T^{-1}] \]
- **Conclusion**: Impulse and momentum have the same dimension: \( [M][L][T^{-1}] \).

### Step 3: Analyze Frequency and Angular Velocity
- **Frequency**: Frequency is defined as the number of cycles per unit time. Its dimension is:
\[ [f] = [T^{-1}] \]
- **Angular Velocity**: Angular velocity is defined as the rate of change of angular displacement with time. Its dimension is:
\[ [\omega] = [T^{-1}] \]
- **Conclusion**: Frequency and angular velocity have the same dimension: \( [T^{-1}] \).

### Step 4: Analyze Displacement and Angular Displacement
- **Displacement**: The dimension of linear displacement is simply:
\[ [d] = [L] \]
- **Angular Displacement**: Angular displacement is measured in radians, which is dimensionless. Therefore, its dimension is:
\[ [\theta] = [1] \text{ (dimensionless)} \]
- **Conclusion**: Displacement and angular displacement do not have the same dimension. Displacement has the dimension \( [L] \), while angular displacement is dimensionless.

### Final Answer
The pair that is not dimensionally identical is:
- **Displacement and Angular Displacement**