Do the quantities having same dimensions always have same units?

Answer

To answer the question "Do the quantities having same dimensions always have same units?", we can follow these steps:

### Step 1: Understand the Concept of Dimensions and Units
- Dimensions refer to the physical nature of a quantity and are expressed in terms of fundamental quantities (mass, length, time, etc.).
- Units are the standard quantities used to measure these dimensions.

### Step 2: Identify the Question
- We need to determine whether having the same dimensions implies that two quantities must have the same units.

### Step 3: Provide Examples
- **Example 1: Angular Speed and Frequency**
- Angular speed (ω) is defined as \( \omega = \frac{2\pi}{T} \) where T is the time period.
- The unit of angular speed is radians per second (rad/s).
- The dimension of angular speed is \( [M^0 L^0 T^{-1}] \).

- Frequency (f) is defined as \( f = \frac{1}{T} \).
- The unit of frequency is Hertz (Hz), which is equivalent to \( s^{-1} \).
- The dimension of frequency is also \( [M^0 L^0 T^{-1}] \).

- **Comparison**: Both angular speed and frequency have the same dimension \( [M^0 L^0 T^{-1}] \), but their units are different (radians per second vs. Hertz).

### Step 4: Provide Another Example
- **Example 2: Torque and Energy**
- Torque (τ) is defined as \( \tau = \text{Force} \times \text{Distance} \).
- The unit of torque is Newton-meter (N·m).
- The dimension of torque is \( [M^1 L^2 T^{-2}] \).

- Energy (E) is defined as \( E = \text{Force} \times \text{Distance} \).
- The unit of energy is Joule (J), which is also equivalent to \( N·m \).
- The dimension of energy is also \( [M^1 L^2 T^{-2}] \).

- **Comparison**: Here, torque and energy have the same dimension \( [M^1 L^2 T^{-2}] \), but they have different units (Newton-meter for torque and Joule for energy).

### Step 5: Conclusion
- From the examples provided, we can conclude that quantities with the same dimensions do not necessarily have the same units. Therefore, 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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