The wave speed in a brass wire with a radius of 0.750 mm is 340 m/s.What is the tension in the wire? The density of brass is 8.60 * 10³ kg/m³. A)2320 N B)250 N C)1900 N D)2900 N E)1760 N

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Quizplus · Accepted Answer

The wave speed $v$ in a string or wire is given by the formula $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension in the wire and $\mu$ is the linear mass density of the wire. The linear mass density $\mu$ can be calculated using the formula $\mu = \frac{m}{L} = \frac{\rho \cdot V}{L} = \rho \cdot A$, where $\rho$ is the density of the material, $V$ is the volume, $L$ is the length, and $A$ is the cross-sectional area of the wire.Given:- Wave speed, $v = 340$ m/s- Radius of the wire, $r = 0.750$ mm = $0.750 \times 10^{-3}$ m- Density of brass, $\rho = 8.60 \times 10^3$ kg/m$^3$First, calculate the cross-sectional area $A$ of the wire:$$A = \pi r^2 = \pi (0.750 \times 10^{-3})^2$$Then, calculate the linear mass density $\mu$:$$\mu = \rho \cdot A = 8.60 \times 10^3 \cdot \pi (0.750 \times 10^{-3})^2$$Finally, solve for $T$ using the wave speed formula:$$v = \sqrt{\frac{T}{\mu}} \Rightarrow T = v^2 \cdot \mu$$Substituting the values:$$T = (340)^2 \cdot 8.60 \times 10^3 \cdot \pi (0.750 \times 10^{-3})^2$$$$T \approx 1760 \, \text{N}$$Therefore, the tension in the wire is approximately 1760 N.

The Wave Speed in a Brass Wire with a Radius