A rope with length 2.50 m and mass 0.10 kg is stretched and pulled to create transverse waves of frequency 40.0 Hz and wavelength of 0.750 m. How much tension is exerted on the rope?

Hey, guys. So hopefully you've got this one on your own, but just in case you didn't, here's a little bit of help. So we have this rope. We're told some characteristics about the mass and length, but we're creating some transverse waves by pulling and stretching it. Right? So I'm gonna go ahead and sort of draw this out real quick. So I've got a sketch of my wave like this. I'm told that the length of this wave or sorry, the length of the string is 2.5 meters, and the mass is 0.1. Now I'm also told that the frequency, remember that's the number of cycles that pass through per second, is 40 hertz. And I'm told that the wavelength, which remember the wavelength is the horizontal distance between crest to crest or trough to trough or whatever, this lambda is equal to 0.75. So those are my variables. Right? Just draw everything out. And then we wanna figure out what the tension that's exerted on the rope is. So what variable is that? So, hopefully, you guys realize that's actually feet. Right? So we wanna figure out what that is. So how do we calculate that? Well, I'm gonna need an equation. And And remember the equation for the velocity on strings that includes this tension variable, this v string, is equal to the square root of tension over mu. Remember, this mu here is the mass density and this mu here is equal to the mass per length. So I've got these 4 variables if you look at it. I've got v string. I've got f of t. Sorry, the tension, I've got the mass, and the length. So if I wanna figure out this f t here, I need to know all the other variables. So do I have the velocity of the string? I don't. I don't have that variable here. Do I have the mass? Yes. Do I have the length? Yes. So I have 2 out of 4, and so all I need to do is figure out what the velocity of this string is and then I can find out what this tension is. So how do we do that? Well, remember, we have one other equation when it comes to velocity, and this works for all kinds of waves, no matter what it is. And this is just equal to lambda times frequency. So, basically, what happens is if v string is equal to this and vstring is also equal to this, then these two things are equal to each other and you can just set them equal to each other. So we can kind of just drop this for a second. We're only just gonna focus on these last two sort of terms here. And so, basically, I do know the wavelength and I also do know the frequency. Right? I have both of those numbers up here. So now I actually can go ahead and solve for the tension. So that's kind of the game plan. Alright. So, basically, what I'm gonna have to do here is I'm gonna have to square this first side. I have to get rid of the square roots. So this is gonna be square root of tension over, this is gonna be mass per length, and this is gonna be squared, and this is gonna equal lambda squared actually, lambda frequency and then whatever I have to do one side, I have to do the other. So if I square the left side, I have square the right side. When you square the left side, basically the square and the square root cancel. And I'm actually just going to go ahead and start plugging in some numbers here. So I've got Feet on the bottom. This is equal to, sorry, divided by mass per length. The mass is 0.1. The length is 2.5. That's over here. And then this is gonna equal So I've got lambda times frequency squared. Remember what happens is you distribute the square inside to both of those terms in there when they're multiplied. So this just ends up being 0.75 squared times, and this is gonna be 40 squared. Right? So that's the wavelength and frequency. Alright. So now when you actually go ahead and plug this into your calculator, what you should get is, you should get 900. So the last thing I need to do here is just figure out what this is over here, this little term. And what I can do here is I can actually just move this whole entire term over to the other side. This whole entire fraction here, it's sort of like a fraction inside of a fraction. But I can take this thing and I can just move it, to the other side, and I don't have to, you know, do the reciprocal or anything like that. So, basically, what I end up with here is that Feet is equal to 900 times, and this is gonna be 0.1 divided by 2.5, and you should get 36 Newtons. And that is the answer, so that's how much tension is on the rope. So you look through your answer choices, and that's answer choice a. So let me know if you guys have any questions with this one, and I'll see you in the next video.

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A rope with length 2.50 m and mass 0.10 kg is stretched and pulled to create transverse waves of frequency 40.0 Hz and wavelength of 0.750 m. How much tension is exerted on the rope?

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