Sound Waves Legacy Problem #17 Guided Solution

An effective problem solver reads a problem carefully, begins to develop a mental picture of what's going on, records the known quantities and the unknown quantity, and then plots out a strategy as to how to get from the known to the unknown. Here the question is divided up into sections, which make the whole process of solving the problem easier. What we know is we have a pin and a wire, and it's pulled to a tension of 448 Newtons. I write down tension equal to 448 Newtons. Here's a mass density or linear density of 0.00621 kilograms per meter. So I write down m per l, that's the mass density, equal 0.00621 kilograms per meter. Now the string is 61.8 centimeters long. That's not the wavelength, that's the length of the string. So I write down l equals 61.8 centimeters. And finally it's vibrating with a fundamental frequency, in other words, with a first harmonic wave pattern. Now in part A I'm to determine the speed at which waves travel through the wire. I have an equation, you'll find it if you click the link to the overview page for this set of problems. The equation relates the speed to the tension and the linear density. It goes something like this. The speed, v, equals square root of tension divided by mass per length. So what I need to do is take the 448 Newtons and divide by the 0.00621 kilograms per meter, and then take the square root of the result. What I do, I get a speed of 268.5921 meters per second, and I can round that to three significant digits, 269 meters per second. In part B I'm to find the wavelength of these waves. Well what I know is it's vibrating in the first harmonic. So I draw the wave pattern for the first harmonic. There's a node on each end and it has an antinode right in the middle, and that antinode's vibrating up and down. Within the length of the string, there's a half of a wavelength. So I can say 61.8 centimeters equals one half times a lambda, where lambda's the wavelength. Multiplying both sides of the equation by two gets me the wavelength value. And that comes out to be 123.6 centimeters, and I can change that to meters, in which case it becomes 1.236 meters. And then I can round it to three significant digits, 1.24 meters. In part C of this problem, I have to calculate the frequency that causes this first harmonic wave pattern. That's simply a matter of using the wave equation. I now have calculated v and lambda, and I can use the wave equation which states that the v, or speed, is equal to the f frequency times lambda wavelength. Now I can rearrange that to solve for frequency, f equal v divided by lambda. I can substitute in my values of 268.5921 meters per second and 1.236 meters, and I'll end up calculating a frequency of 217.3075 hertz, and I can round that also to three significant digits.