A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire, so that the speed of transverse wave on the wire equals the speed of sound in dry air at `20^@C ?` `(i.e, 343ms^(-1))?`

Answer

To find the tension in the steel wire so that the speed of transverse waves equals the speed of sound in dry air at 20°C (343 m/s), we can use the formula for the speed of a wave on a string:

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

where:
- \( V \) is the speed of the wave,
- \( T \) is the tension in the wire,
- \( \mu \) is the linear mass density of the wire, given by \( \mu = \frac{m}{L} \),
- \( m \) is the mass of the wire,
- \( L \) is the length of the wire.

### Step 1: Calculate the linear mass density (\( \mu \))
Given:
- Mass of the wire, \( m = 2.10 \, \text{kg} \)
- Length of the wire, \( L = 12.0 \, \text{m} \)

We can calculate \( \mu \) as follows:

\[ \mu = \frac{m}{L} = \frac{2.10 \, \text{kg}}{12.0 \, \text{m}} = 0.175 \, \text{kg/m} \]

### Step 2: Rearrange the wave speed formula to find tension (\( T \))
From the wave speed formula, we can rearrange it to solve for tension:

\[ V = \sqrt{\frac{T}{\mu}} \implies T = V^2 \cdot \mu \]

### Step 3: Substitute the values into the tension formula
Now, substitute \( V = 343 \, \text{m/s} \) and \( \mu = 0.175 \, \text{kg/m} \) into the tension formula:

\[ T = (343 \, \text{m/s})^2 \cdot 0.175 \, \text{kg/m} \]

### Step 4: Calculate the tension
Calculating \( (343)^2 \):

\[ (343)^2 = 117649 \, \text{m}^2/\text{s}^2 \]

Now, substituting this value into the tension equation:

\[ T = 117649 \cdot 0.175 = 20584.575 \, \text{N} \]

### Step 5: Round the answer
Rounding this to two significant figures gives:

\[ T \approx 2.06 \times 10^4 \, \text{N} \]

### Final Answer
The tension in the wire should be approximately \( 2.06 \times 10^4 \, \text{N} \).

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