Your friend needs advice on her newest "acoustic sculpture." She attaches one end of a steel wire, of diameter \(4.00 \mathrm{mm}\) and density $7860 \mathrm{kg} / \mathrm{m}^{3},$ to a wall. After passing over a pulley, located \(1.00 \mathrm{m}\) from the wall, the other end of the wire is attached to a hanging weight. Below the horizontal length of wire she places a 1.50 -m-long hollow tube, open at one end and closed at the other. Once the sculpture is in place, air will blow through the tube, creating a sound. Your friend wants this sound to cause the steel wire to oscillate at the same resonant frequency as the tube. What weight (in newtons) should she hang from the wire if the temperature is \(18.0^{\circ} \mathrm{C} ?\)

The speed of sound in air (v) depends on its temperature and can be calculated using the formula: v = \(331.5 \sqrt{1 + \frac{T}{273}}\), where T is the temperature in Celsius. Let's plug the given temperature (18°C) into the formula and solve for the speed of sound: v = \(331.5 \sqrt{1+\frac{18}{273}} \approx 343.46 \;\mathrm{m/s}\)

An open-closed tube has its fundamental (first harmonic) frequency given by the formula: \(f = \frac{v}{4L}\), where L is the length of the column. Given that L = 1.50 m and v = 343.46 m/s, we can find the fundamental frequency: \(f = \frac{343.46}{4 \cdot 1.50} \approx 57.244 \;\mathrm{Hz}\) So, the fundamental frequency of the air column is approximately 57.244 Hz.

We're given the diameter (D) and density (ρ) of the steel wire. We can determine the linear mass density (μ) using the formula: \(\mu = \frac{\pi D^2 \rho}{4}\) Plugging the given values (D = 0.004 m, ρ = 7860 kg/m³), we can calculate the linear mass density of the wire: \(\mu = \frac{\pi (0.004)^2 \cdot 7860}{4} \approx 0.0993 \;\mathrm{kg/m}\)

The fundamental frequency of oscillation of the wire (f') is given by the formula: \(f' = \frac{1}{2L'}\sqrt{\frac{T}{\mu}}\), where L' is the length of the wire and T is the tension in the wire. Tension is the force acting on the wire due to the hanging weight (W) and can be expressed as T = W. Since we want the wire's frequency f' to match the air column's fundamental frequency f, we can set f' = f and solve for the hanging weight W in terms of L', μ, and f: \(W = T = 4L'^2 \mu f^2\) Substituting the known parameters (L' = 1.00 m, μ = 0.0993 kg/m, and f = 57.244 Hz) into the equation, we can find the hanging weight: \(W = 4(1.00)^2 \cdot 0.0993 \cdot (57.244)^2 \approx 1286.33 \;\mathrm{N}\) So, we conclude that the hanging weight required to make the steel wire oscillate at the same resonant frequency as the air column in the tube is approximately 1286.33 N.