Sound Waves Legacy Problem #18 Guided Solution

Here's a difficult problem, and like any difficult problem, one needs to practice the habits of an effective problem solver. Reading it carefully, identifying what is known in terms of variables of physics equations, and then plotting out a strategy as to how to get from the known information to the unknown information. We have a steel piano wire, 72.9 centimeters long. That's not a wavelength, that's a length. So I list L equals 72.9 centimeters. And then I'm going to just do a quick conversion to meters, because I know I'm going to have to do that eventually anyway. So I say it's also equal to 0.729 meters. It has a mass, an m, of 4.54 times 10 to the negative third kilograms. I need to pay attention to units here. The fundamental frequency of this wire is 262 hertz. That's f, f subscripted 1, the fundamental frequency, or first harmonic frequency. Determine the tension to which the wire is pulled in order to vibrate with this frequency. I wish to find T, or tension. Now, as I think about the relationships here and the equations that I know, I'm thinking this tension of the wire is related to one thing, and that's the speed of waves within the wire. So eventually, I'm going to have to find the speed of waves within this wire. Now, once I do, I can find the tension using the mass and the length in the equation, speed equal tension, square root of tension divided by mass per length. Now, how do I get the speed? Well, I can get the speed if I know the frequency, check, I got it, and the wavelength, which I don't have right now. So how can I get the wavelength so I can get the speed? Well, I can take the 72.9 centimeters, which is the length of this piano wire, and I can sketch out within it the first harmonic wave pattern and recognize that within that 72.9 centimeters, there is a half of a wavelength. I can solve for a wavelength by doubling the 72.9 centimeters. OK, we're going to execute now. If I take that 72.9 centimeters and I double it, I get a wavelength of the first harmonic of 1.458 meters. Now, I have a frequency of the first harmonic, 262 hertz. So if I go wavelength times frequency using the wave equation, I get the wave speed of 381.9960, and that's in units of meters per second. Now, I need to get the mass per length. That's part of the equation, speed equals square root of tension divided by mass per length. So the mass is given, and so is the length. And what's important about mass per length calculations is it needs to be done in kilograms per meter in order to get the tension of that equation to be in newtons. So I'm going to take the 4.54 times 10 to the negative 3rd kilograms. I'm going to divide it by the 0.729 meters. That's the length of the string. I end up getting 6.2277 times 10 to the negative 3rd kilograms per meter. OK, that's the mass per length. Now, I'm going to take the equation, speed equals square root of tension over mass per length. I'm going to solve for tension. I know two of the three quantities there. So to do that, I'm going to square both sides of the equation. It becomes now v squared equals tension divided by mass per length. And then I'm going to multiply both sides of the equation by m per hour, mass per length. I end up with the equation, tension equals speed squared times mass per length. Substituting in the values of mass per length of 6.2277 times 10 to the negative 3rd in speed, 381.9960, into this equation, making sure I square the speed, I will get 908.75. And that's in units of newtons. And I can round that to three significant figures, 909 newtons.