A 10 m long steel wire has mass 5 g. If the wire is under a tension of 80 N, the speed of transverse waves on the wire is

Options

A. `100ms^(-1)`
B. `200ms^(-1)`
C. `400ms^(-1)`
D. `500ms^(-1)`

Explanation

To find the speed of transverse waves on a steel wire, we can use the formula:

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

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

### Step 1: Calculate the mass per unit length (\( \mu \))

Given:
- Mass of the wire = 5 g = \( 5 \times 10^{-3} \) kg (since 1 g = \( 10^{-3} \) kg)
- Length of the wire = 10 m

The mass per unit length (\( \mu \)) is calculated as:

\[ \mu = \frac{\text{mass}}{\text{length}} = \frac{5 \times 10^{-3} \text{ kg}}{10 \text{ m}} = 0.5 \times 10^{-3} \text{ kg/m} = 5 \times 10^{-4} \text{ kg/m} \]

### Step 2: Substitute values into the wave speed formula

Now that we have \( \mu \), we can substitute the values into the wave speed formula.

Given:
- Tension \( T = 80 \) N
- Mass per unit length \( \mu = 5 \times 10^{-4} \) kg/m

Substituting these values into the formula:

\[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{80 \text{ N}}{5 \times 10^{-4} \text{ kg/m}}} \]

### Step 3: Calculate the speed \( v \)

Calculating the expression inside the square root:

\[ \frac{80}{5 \times 10^{-4}} = \frac{80}{0.0005} = 160000 \]

Now, taking the square root:

\[ v = \sqrt{160000} = 400 \text{ m/s} \]

### Final Answer

The speed of transverse waves on the wire is \( 400 \text{ m/s} \).

---