A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x, t)=2.30mm cos[(16.98 rad/m^)x+(742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.

Video transcript

Hey everyone, welcome back in this problem. We have a student who's using a frequency generator and an oscillator key and they're producing a wave along the strength. Were told of information about the length of the string, the mass of the string and we're told that the produced wave can be modeled by the function Y x t. 12 millimeters time signed by over two radiance per meter, x minus 100 pi radian per second. T. Okay. And what we're asked to do is to determine the wave's amplitude, frequency and weight blank. Okay. Alright, so let's first start off with, what does a general equation of a wave look like? Okay, so we have wine of X of T. Is equal to a sign of K X minus. Oh my God. T. Okay and then we have our equation Y. Is equal to X of T. 12. Okay. And I'm just gonna leave out the units just right here where we're writing this just to make it clear how we can compare them, but the units do matter and we will refer back to them so pi over two x minus 100 I. T. Okay, so that's our equation. Now let's start with the first thing, the first thing we're asked to do is find the amplitude. Well the amplitude is given just by this constant out in front and so that's easy in this case, in this case the amplitude A that we're looking for, It's just 12. Okay and again we care about the unit. So in this case it's going to be 12 mm. So the amplitude a is equal to 12 mm. So that's it for part one. Okay now let's go to part two. So for part two we're looking for the frequency. Now let's think about how we can relate frequency to the wave equation that we have in the wave function. Okay, so frequency isn't a direct term. It's not a this A or K. Or omega. Okay. But we know that frequency is related to omega. Okay, so we can write omega is equal to two pi times the frequency F. Alright, so let's figure out what that omega is then. Well, omega here in this equation is 100 pie in our equation. Okay, coefficient with the t The time variable. Okay, so it's 100 pies. So we can substitute that in for omega here. Okay, we're gonna have 100 oops. 100 pie on the left hand side and two pi times the frequency F on the right hand side. Okay, dividing by two pi we're gonna get that the frequency is equal to 50 and our unit here is going to be our unit 100 pi radiance for second to pi radiance case. We're gonna have 50 Um one over second And this is just equal to 50 hertz. Okay. Alright, so we found the frequency. So that answers part two and the last thing we want to do is to find the wavelength. Okay, so let's do the wavelength over here. All right, so same thing with the frequency, the wave length isn't the K or the omega or the aid directly in this equation. But we know we can relate it to this equation. Now let's recall how can we do that? We can do that through K. Okay, So we are looking for the wavelength and we can relate that to our equation through. K is equal to two pi over lambda. Ok. And lambda is what we're looking for. That's the wavelength. Alright, well, what is K? Well, let's look at our equation again. Okay, we have K here with the extra. So in our equation K is pi over two. Ok. So our value of K is going to be pi over two. On the right hand side. We have two pi over λ. Okay, multiplying. We get lambda times pi is equal to four pi And dividing by pi we get lambda is equal to four. And when we're looking at units. Okay, let's go back. And in our equation we have radiance per minute. Okay, or sorry, per meter? So our unit of wavelength is going to be meters. Alright, so that's the wavelength lambda. Now we've found everything. We were looking for the amplitude is going to be 12. The frequency is going to be 50 hertz and the wavelength is four m. That corresponds with answer be that's it for this one. Thanks everyone for watching