A copper wire `2.4 mm` in diameter is `3 m` long and is used to suspend a `2 kg` mass from a beam. If a trasverse disturbance is sent along the wire by striking it lightly with a pencil, how fast will the disturbance travel? The density of copper is `8920 kg//m^(3)`.

Explanation

To find the speed of the transverse disturbance traveling along the copper 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 linear mass density of the wire.

### Step 1: Calculate the tension \( T \)
The tension in the wire is equal to the weight of the mass suspended from it. The weight \( W \) can be calculated using the formula:

\[ T = W = m \cdot g \]

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

So,

\[ T = 2 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 19.62 \, \text{N} \]

### Step 2: Calculate the cross-sectional area \( A \) of the wire
The diameter of the wire is given as \( 2.4 \, \text{mm} \). First, convert this to meters:

\[ d = 2.4 \, \text{mm} = 2.4 \times 10^{-3} \, \text{m} \]

The area \( A \) of the wire can be calculated using the formula for the area of a circle:

\[ A = \frac{\pi d^2}{4} \]

Substituting the diameter:

\[ A = \frac{\pi (2.4 \times 10^{-3})^2}{4} \]

Calculating this gives:

\[ A \approx \frac{\pi (5.76 \times 10^{-6})}{4} \approx 4.52 \times 10^{-6} \, \text{m}^2 \]

### Step 3: Calculate the linear mass density \( \mu \)
The linear mass density \( \mu \) can be calculated using the formula:

\[ \mu = \rho \cdot A \]

where \( \rho \) is the density of copper, given as \( 8920 \, \text{kg/m}^3 \).

Substituting the values:

\[ \mu = 8920 \, \text{kg/m}^3 \cdot 4.52 \times 10^{-6} \, \text{m}^2 \]

Calculating this gives:

\[ \mu \approx 0.0403 \, \text{kg/m} \]

### Step 4: Calculate the speed \( v \)
Now we can substitute the values of \( T \) and \( \mu \) into the wave speed formula:

\[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{19.62 \, \text{N}}{0.0403 \, \text{kg/m}}} \]

Calculating this gives:

\[ v \approx \sqrt{486.1} \approx 22.0 \, \text{m/s} \]

### Final Answer
The speed of the transverse disturbance traveling along the copper wire is approximately \( 22.0 \, \text{m/s} \).

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