A steel wire of length 64 cm weighs 5 g. If it is stretched by a force of 8 N, what would be the speed of a transverse wave passing on it ?

Explanation

To find the speed of a transverse wave passing through a steel wire, we can use the formula for wave speed on a string:

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

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

### Step-by-Step Solution:

**Step 1: Convert the given length from centimeters to meters.**
- The length of the wire is given as 64 cm.
- To convert to meters:
\[ L = 64 \, \text{cm} = \frac{64}{100} = 0.64 \, \text{m} \]

**Step 2: Convert the mass from grams to kilograms.**
- The mass of the wire is given as 5 g.
- To convert to kilograms:
\[ M = 5 \, \text{g} = \frac{5}{1000} = 0.005 \, \text{kg} \]

**Step 3: Calculate the mass per unit length (\( \mu \)).**
- The mass per unit length \( \mu \) is calculated as:
\[ \mu = \frac{M}{L} = \frac{0.005 \, \text{kg}}{0.64 \, \text{m}} \approx 0.0078125 \, \text{kg/m} \]

**Step 4: Use the tension in the wire.**
- The tension \( T \) is given as 8 N.

**Step 5: Substitute the values into the wave speed formula.**
- Now we can calculate the speed \( v \):
\[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{8 \, \text{N}}{0.0078125 \, \text{kg/m}}} \]

**Step 6: Calculate the value inside the square root.**
- First, calculate \( \frac{8}{0.0078125} \):
\[ \frac{8}{0.0078125} = 1024 \]

**Step 7: Take the square root to find the speed.**
- Finally, calculate the speed:
\[ v = \sqrt{1024} = 32 \, \text{m/s} \]

### Final Answer:
The speed of the transverse wave passing through the steel wire is \( 32 \, \text{m/s} \).

---