A transverse wave is propagating on a stretched string of mass per unit length`32 gm^(-1)`. The tension on the string is 80 N. The speed of the wave over the string is

Options

A. `(5)/(2)ms^(-1)`
B. `sqrt((5)/(2))ms^(-1)`
C. `sqrt((2)/(5))ms^(-1)`
D. `50 ms^(-1)`

Answer

To find the speed of a transverse wave propagating on a stretched string, we can use the formula:

\[ v = \sqrt{\frac{T}{\mu}} \]

where:
- \( v \) is the speed of the wave,
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length of the string.

### Step-by-step Solution:

1. **Identify the Given Values:**
- Tension, \( T = 80 \, \text{N} \)
- Mass per unit length, \( \mu = 32 \, \text{g/m} \)

2. **Convert Mass per Unit Length to Kilograms:**
- Since \( 1 \, \text{g} = 10^{-3} \, \text{kg} \),
- Convert \( \mu \):
\[ \mu = 32 \, \text{g/m} = 32 \times 10^{-3} \, \text{kg/m} = 0.032 \, \text{kg/m} \]

3. **Substitute the Values into the Wave Speed Formula:**
- Using the formula \( v = \sqrt{\frac{T}{\mu}} \):
\[ v = \sqrt{\frac{80 \, \text{N}}{0.032 \, \text{kg/m}}} \]

4. **Calculate the Value Inside the Square Root:**
- First, compute \( \frac{80}{0.032} \):
\[ \frac{80}{0.032} = 2500 \]

5. **Calculate the Square Root:**
- Now take the square root:
\[ v = \sqrt{2500} = 50 \, \text{m/s} \]

6. **Final Result:**
- The speed of the wave on the string is:
\[ v = 50 \, \text{m/s} \]

### Conclusion:
The speed of the transverse wave on the stretched string is \( 50 \, \text{m/s} \).