A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed v, frequency f, amplitude A, and wavelength λ. (b) What is the amplitude of the motion at the points located at (i) x = λ/2, (ii) x = λ/4, and (iii) x = λ/8, from the left-hand end of the string?

Video transcript

Hey everyone, welcome back in this problem. We have a wire K. Of length L. And it's gonna be stretched between two clamps. Okay. The tension is adjusted until we have standing waves, We have the amplitude and wavelength of the wave or four cm and 1.6 m. And we're asked to find the amplitude of the wire particle at particular distances away from the left clamp. Okay. Alright. So we are looking for amplitude at particular distances. So what we wanna do is we want to find a wave equation or a function that represents this wave. Okay, we have standing waves. So let's recall the general form of the equation for a standing wave. So we have Y of X. T. Is equal to to a sine K. X. Sign omega T. Alright, the information we're given, we have the amplitude A. Is four centimeters But the wavelength λ is 1.6 m, we have the value of K for our equation. Okay, we aren't given K directly but let's recall that. We can write K as two pi over lambda. Okay, so this is two pi divided by 1.6 m. There's gonna be 1.25 pi. That's all the information we're given. So we can write our equation as Y. Of X. T. Is equal to eight. Sign Over 1.25 pi x. I'm sign omega. Okay. Alright, so let's go ahead and think about the maximum amplitude. Okay, if we're at a particular point we're going to have an X. Value. Okay we want to find the amplitude at that location. And we're talking amplitude here, we're talking about what's the biggest displacement that we're going to have? Well that's going to occur when this sine omega T. Is equal to one. Okay, This is gonna give us our max displacement, which is gonna be our amplitude. So we don't necessarily know the T. Value of the mega, but we don't need to we know that sine omega T will be one. That's the maximum value we can have. Okay, so we know that our amplitude at a given location will be given by Y of X. And it's just gonna be Y of X. Because we've eliminated that variable of T. Is going to be eight, sign 1.255 X. Okay, so we're looking for again that max displacement. The maximum of this function occurs when sine omega T. Is one. So we can set that to one and we don't need to worry about the particular time that it's going to occur at. Okay, Alright, so for part one We are asked to find the max or the amplitude story at 0.8 m. So we have y 0.8 m is equal to eight sign of 1.25 pi times 0.8. Okay, and just be sure that your units match. Okay, so our unit of wavelength was in meters and so inside of these brackets were working in meters. Okay, so we can have our 0.8 m. And then on the outside, this amplitude on the outside wasn't centimeters. Okay. So you just need to make sure that the units inside of that sign match. Okay, well this is going to be eight sign 1.25 times 0.8 is gonna give us one. So we're just gonna have sine of pi which we know is equal to zero. Okay. All right, let's give ourselves some more room. Right? So we know that the amplitude at .8 m is just going to be zero. Okay? And now let's do part two same thing. But this time at 0.4 m from the left clamp, We're gonna have eight sign 1. Pi times 0.4. This time we're going to get eight Sine of pi over two. Okay, we know that sine of pi over two is one, so we're gonna get eight centimeters. Okay, So this is going to be our amplitude for that max displacement At 0.4 m from the left clamp. And finally, for part three, We have y at 0.2 m is equal to eight sign 1.25 pi times 0.2. Who's gonna give us eight? Sine of pi over four. Okay. Which is going to be eight times one over the square root of two Or 5.7 cm. Alright, so that's it. Okay, We found now the amplitude at each of the points we were interested in. So when we're 0.8 m from the clamp, our amplitude is zero. When we're 00.4 m from the clamp, our amplitude is eight centimeters. And when were 80.2 m from the clamp, our amplitude is 5.7 centimeters. That's going to correspond with Answer C. Thanks everyone for watching. See you in the next video.