Calculate the speed of a transverse wave in a wire of `1.0 mm^(2)` cross-section under a tension of `0.98 N`. Density of the material of wire is `9.8 xx 10^(3) kg//m^(3)`

Explanation

To calculate the speed of a transverse wave in a 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 (\( \mu \))

The mass per unit length (\( \mu \)) can be calculated using the formula:

\[ \mu = \text{density} \times \text{cross-sectional area} \]

Given:
- Density (\( \rho \)) = \( 9.8 \times 10^3 \, \text{kg/m}^3 \)
- Cross-sectional area (\( A \)) = \( 1.0 \, \text{mm}^2 = 1.0 \times 10^{-6} \, \text{m}^2 \)

Now, substituting the values:

\[ \mu = (9.8 \times 10^3 \, \text{kg/m}^3) \times (1.0 \times 10^{-6} \, \text{m}^2) \]

Calculating this gives:

\[ \mu = 9.8 \times 10^{-3} \, \text{kg/m} \]

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

Now that we have \( \mu \), we can substitute it into the wave speed formula along with the tension:

Given:
- Tension (\( T \)) = \( 0.98 \, \text{N} \)

Substituting into the formula:

\[ V = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{0.98 \, \text{N}}{9.8 \times 10^{-3} \, \text{kg/m}}} \]

### Step 3: Calculate the speed of the wave

Calculating the value inside the square root:

\[ V = \sqrt{\frac{0.98}{9.8 \times 10^{-3}}} \]

\[ V = \sqrt{100} \]

\[ V = 10 \, \text{m/s} \]

### Final Answer

The speed of the transverse wave in the wire is:

\[ V = 10 \, \text{m/s} \]

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