Calculate the velocity of transverse wave in a copper wire 1 `mm^(2)` in cross-section, under the tension produced by 1 kg wt. The density of copper =`8.93 kg m^(-3)`

Answer

To calculate the velocity of a transverse wave in a copper wire, we can use the formula for the velocity of a wave on a string, which is given by:

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

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

### Step 1: Calculate the Tension (T)
The tension \( T \) is produced by a weight of 1 kg. The force due to this weight can be calculated using the formula:

\[ T = mg \]

where:
- \( m = 1 \, \text{kg} \) (mass),
- \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity).

Calculating \( T \):

\[ T = 1 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 9.81 \, \text{N} \]

### Step 2: Calculate the Cross-sectional Area (A)
The cross-sectional area \( A \) of the wire is given as \( 1 \, \text{mm}^2 \). We need to convert this to square meters:

\[ A = 1 \, \text{mm}^2 = 1 \times 10^{-6} \, \text{m}^2 \]

### Step 3: Calculate the Volume (V)
To find the mass per unit length, we need the volume of the wire per unit length. The volume \( V \) of the wire per unit length (1 meter) can be calculated as:

\[ V = A \times L \]

where \( L = 1 \, \text{m} \):

\[ V = 1 \times 10^{-6} \, \text{m}^2 \times 1 \, \text{m} = 1 \times 10^{-6} \, \text{m}^3 \]

### Step 4: Calculate the Mass (m)
Using the density \( \rho \) of copper, which is given as \( 8.93 \, \text{kg/m}^3 \), we can calculate the mass \( m \) of the wire per unit length:

\[ m = \rho \times V \]

\[ m = 8.93 \, \text{kg/m}^3 \times 1 \times 10^{-6} \, \text{m}^3 = 8.93 \times 10^{-6} \, \text{kg} \]

### Step 5: Calculate the Mass per Unit Length (\( \mu \))
The mass per unit length \( \mu \) is simply the mass we calculated above:

\[ \mu = 8.93 \times 10^{-6} \, \text{kg/m} \]

### Step 6: Calculate the Velocity (v)
Now we can substitute \( T \) and \( \mu \) into the wave velocity formula:

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

Substituting the values:

\[ v = \sqrt{\frac{9.81 \, \text{N}}{8.93 \times 10^{-6} \, \text{kg/m}}} \]

Calculating \( v \):

\[ v = \sqrt{\frac{9.81}{8.93 \times 10^{-6}}} \]

\[ v \approx \sqrt{1.099 \times 10^7} \]

\[ v \approx 3326.65 \, \text{m/s} \]

### Final Answer:
The velocity of the transverse wave in the copper wire is approximately \( 3326.65 \, \text{m/s} \).

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