Do the quantities having same dimensions always have same units?

Answer

To answer the question "Do the quantities having the same dimensions always have the same units?" we will analyze the concepts of dimensions and units in physics.

### Step-by-Step Solution:

1. **Understanding Dimensions**:
- Dimensions of a physical quantity represent the nature of the quantity in terms of the fundamental quantities (mass, length, time, etc.). For example, the dimension of velocity is [L T^-1], which indicates that it involves length and time.

2. **Understanding Units**:
- Units are the standard measures used to quantify physical quantities. For instance, velocity can be measured in meters per second (m/s), kilometers per hour (km/h), etc.

3. **Same Dimensions, Different Quantities**:
- Two different physical quantities can have the same dimensions but can be measured in different units. For example, angular velocity and frequency both have the dimension of T^-1 (time inverse).

4. **Example of Angular Velocity and Frequency**:
- Angular velocity is measured in radians per second (rad/s), while frequency is measured in hertz (Hz), which is equivalent to seconds inverse (s^-1).
- Both have the same dimension of T^-1, but their units are different due to the nature of the quantities they represent.

5. **Conclusion**:
- Therefore, it is not necessary for quantities with the same dimensions to have the same units. The answer to the question is "No".

### Final Answer:
No, quantities having the same dimensions do not always have the same units.

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