A steel cable of cross-sectional area `2.83 xx 10^(-3) m^2` is kept under a tension of `1.00 xx 10^4 N`. The density of steel is 7860 `"kg/m"^(3)` (this is not the linear density). At what speed does a transverse wave move along the cable?

Options

A. 22.7m/s
B. 21.2m/s
C. 19.9m/s
D. 17.6m/s

Answer

To find the speed of a transverse wave moving along a steel cable, we will follow these steps:

### Step 1: Identify the given values
- Cross-sectional area of the cable, \( A = 2.83 \times 10^{-3} \, \text{m}^2 \)
- Tension in the cable, \( T = 1.00 \times 10^4 \, \text{N} \)
- Density of steel, \( \rho = 7860 \, \text{kg/m}^3 \)

### Step 2: Calculate the mass per unit length (\( \mu \))
The mass per unit length (\( \mu \)) can be calculated using the formula:
\[ \mu = \rho \times A \]
Substituting the values:
\[ \mu = 7860 \, \text{kg/m}^3 \times 2.83 \times 10^{-3} \, \text{m}^2 \]
Calculating this gives:
\[ \mu = 22.24 \, \text{kg/m} \]

### Step 3: Use the wave speed formula
The speed of a transverse wave (\( v \)) in a medium under tension is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
Substituting the values of tension and mass per unit length:
\[ v = \sqrt{\frac{1.00 \times 10^4 \, \text{N}}{22.24 \, \text{kg/m}}} \]

### Step 4: Calculate the speed
Calculating the value inside the square root:
\[ v = \sqrt{449.64} \approx 21.2 \, \text{m/s} \]

### Final Answer
The speed of the transverse wave moving along the cable is approximately \( 21.2 \, \text{m/s} \).

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