Which of the following physical quantities has neither units nor dimensions?

Options

A. 1. Relative velocity
B. 2. Relative density
C. 3. Angle
D. 4. Energy

Explanation

To determine which physical quantity has neither units nor dimensions, we will analyze each option provided in the question.

### Step-by-Step Solution:

1. **Understand the Options**: The question asks us to identify a physical quantity that has neither units nor dimensions. We will evaluate each option one by one.

2. **Evaluate Relative Velocity**:
- Relative velocity is defined as the velocity of one object with respect to another.
- It is calculated as \( V_{relative} = V_1 - V_2 \).
- The unit of velocity is meters per second (m/s), and its dimensional formula is \( [L T^{-1}] \).
- **Conclusion**: Relative velocity has both units and dimensions.

3. **Evaluate Relative Density**:
- Relative density (or specific gravity) is defined as the ratio of the density of a substance to the density of water.
- It is given by \( RD = \frac{\text{Density of substance}}{\text{Density of water}} \).
- Since both densities are expressed in kg/m³, the units cancel out, making relative density unitless.
- Additionally, it does not have any physical dimensions associated with it.
- **Conclusion**: Relative density has neither units nor dimensions.

4. **Evaluate Angle**:
- An angle is measured in radians.
- While radians are a unit of measurement, angles are considered dimensionless because they represent a ratio of lengths (arc length to radius).
- **Conclusion**: Angles have units (radians) but no dimensions.

5. **Evaluate Energy**:
- Energy is defined as the capacity to do work.
- The unit of energy is the joule (J), and its dimensional formula is \( [M L^2 T^{-2}] \).
- **Conclusion**: Energy has both units and dimensions.

6. **Final Conclusion**:
- From the analysis, we find that relative density is the only physical quantity that has neither units nor dimensions.

### Final Answer:
**Relative Density** is the physical quantity that has neither units nor dimensions.

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