What is the wave speed in a brass wire with a radius of 0.500 mm stretched with a tension of 125
N? The density of brass is $8.60 \times 10 ^ { 3 } \mathrm {~kg} / \mathrm { m } ^ { 3 }$
A) 68.8 m/s
B) 100 m/s
C) 136 m/s
D) 0.121 m/s
E) 500 m/s

Question

What is the wave speed in a brass wire with a radius of  0.500 mm  stretched with a tension of 125
N? The density of brass is $8.60 \times 10 ^ { 3 } \mathrm {~kg} / \mathrm { m } ^ { 3 }$
A)  68.8 m/s
B)  100 m/s
C)  136 m/s
D)  0.121 m/s
E)  500 m/s

Quizplus · Accepted Answer

The wave speed $v$ in a stretched string or wire can be calculated using the formula $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension in the wire and $\mu$ is the linear mass density (mass per unit length) of the wire. The linear mass density $\mu$ can be found from the volume density $\rho$ (given as $8.60 \times 10^3 \, \text{kg/m}^3$) and the cross-sectional area $A$ of the wire. The cross-sectional area of a wire with a radius $r$ is $A = \pi r^2$. Given $r = 0.500 \, \text{mm} = 0.500 \times 10^{-3} \, \text{m}$, we find $A = \pi (0.500 \times 10^{-3})^2 \, \text{m}^2$. The linear mass density $\mu = \rho A = 8.60 \times 10^3 \times \pi (0.500 \times 10^{-3})^2 \, \text{kg/m}$. Substituting $\mu$ and $T = 125 \, \text{N}$ into the wave speed formula gives $v = \sqrt{\frac{125}{8.60 \times 10^3 \times \pi (0.500 \times 10^{-3})^2}}$, which calculates to approximately $136 \, \text{m/s}$.

What Is the Wave Speed in a Brass Wire with a Radius