The unit of `1//lsqrt((T//mu)`) is the same as that of (`l`-length, `T`-tension and `mu`-mass/unit length)

Options

A. wave-length
B. wave number
C. frequency
D. time period

Answer

To solve the problem, we need to find the unit of the expression \( \frac{1}{\sqrt{\frac{T}{\mu}}} \) where \( T \) is tension, \( \mu \) is mass per unit length, and \( l \) is length.

### Step-by-Step Solution:

1. **Identify the Units:**
- The unit of tension \( T \) is Newton (N).
- The unit of mass per unit length \( \mu \) is kilograms per meter (kg/m).
- The unit of length \( l \) is meters (m).

2. **Convert Tension to Base Units:**
- 1 Newton (N) is defined as \( 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \).
- Therefore, the unit of tension \( T \) can be expressed as:
\[ [T] = \text{kg} \cdot \text{m/s}^2 \]

3. **Express the Unit of \( \mu \):**
- The unit of mass per unit length \( \mu \) is:
\[ [\mu] = \text{kg/m} \]

4. **Calculate the Unit of \( \frac{T}{\mu} \):**
- Now, we can find the unit of \( \frac{T}{\mu} \):
\[ \frac{T}{\mu} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{kg/m}} = \frac{\text{kg} \cdot \text{m/s}^2 \cdot \text{m}}{\text{kg}} = \frac{\text{m}^2}{\text{s}^2} \]

5. **Find the Unit of \( \sqrt{\frac{T}{\mu}} \):**
- Taking the square root of \( \frac{T}{\mu} \):
\[ \sqrt{\frac{T}{\mu}} = \sqrt{\frac{\text{m}^2}{\text{s}^2}} = \frac{\text{m}}{\text{s}} \]

6. **Calculate the Unit of \( \frac{1}{\sqrt{\frac{T}{\mu}}} \):**
- Finally, we find the unit of \( \frac{1}{\sqrt{\frac{T}{\mu}}} \):
\[ \frac{1}{\sqrt{\frac{T}{\mu}}} = \frac{1}{\frac{\text{m}}{\text{s}}} = \frac{\text{s}}{\text{m}} \]

7. **Conclusion:**
- The unit of \( \frac{1}{\sqrt{\frac{T}{\mu}}} \) is \( \text{s/m} \), which is the same as the unit of wave number (inverse length).

### Final Answer:
The unit of \( \frac{1}{\sqrt{\frac{T}{\mu}}} \) is the same as that of \( \frac{1}{l} \) (where \( l \) is length), which is \( \text{s/m} \).

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