Two steel wires are stretched with the same tension.The first wire has a diameter of 0.610 mm and the second wire has a diameter of 0.910 mm.If the speed of waves traveling along the first wire is 54.0 m/s,what is the speed of waves traveling along the second wire?
A)24.9 m/s
B)27.2 m/s
C)36.2 m/s
D)81.0 m/s
E)100 m/s

Question

Quizplus · Accepted Answer

The speed of waves in a stretched string or wire is given by the formula $v = \sqrt{\frac{T}{\mu}}$, where $v$ is the speed of the wave, $T$ is the tension in the wire, and $\mu$ is the linear mass density of the wire. The linear mass density $\mu$ is inversely proportional to the cross-sectional area of the wire, which in turn is proportional to the square of the diameter of the wire. Therefore, if the diameter of the second wire is $\frac{0.910}{0.610}$ times the diameter of the first wire, the speed of waves in the second wire will be $\sqrt{\frac{0.610^2}{0.910^2}}$ times the speed in the first wire, which is $\sqrt{\frac{0.610^2}{0.910^2}} \times 54.0 \, \text{m/s} = \sqrt{\frac{0.3721}{0.8281}} \times 54.0 \, \text{m/s} \approx 0.67 \times 54.0 \, \text{m/s} = 36.2 \, \text{m/s}$.

Two Steel Wires Are Stretched with the Same Tension