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) 36.2 m/s
B) 27.2 m/s
C) 100 m/s
D) 24.9 m/s
E) 81.0 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 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. Given that the speed in the first wire is 54.0 m/s, the speed in the second wire will be $54.0 \times \sqrt{\frac{0.610^2}{0.910^2}} = 36.2$ m/s.

Two Steel Wires Are Stretched with the Same Tension