Average Power of Waves on Strings - Video Tutorials & Practice Problems

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concept

Energy & Power of Waves on Strings

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Hey everyone. So in some problems throughout this chapter, you have to calculate something called the average power, the average power that you have to supply to the string in order to continuously produce waves, I'm gonna show you how to do that in this video. And it really comes down to just one simple equation that you need to know for the average power of waves on strings. Now, before we go any further, I want to point out something really important that you need to know a conceptual point which is that waves actually carry energy throughout space. Not matter when you whip a string up and down, nothing is actually moving left or to the right the particles on the string. Remember are actually just going up and down like this. What is happening though is that the overall wave pattern moves to the right. And that's because you whip the string up and down to do that you had to supply, you had to do some work onto the string. So waves actually carry energy as they go from left to right, not matter. Now, in order to continuously produce a wave, you have to continuously flick the string up and down, you have to supply some energy over time. And remember that energy over time is the definition of power. Now, in this, in these kind of problems here, you're going to be asked for a specific kind of power, which is the average power and I'm just going to give you the equation here. Your textbooks are going to do some really lengthy derivation using a bunch of math that you don't need to know. I'm just gonna give you the equation because you're most likely gonna be given it on a formula sheet. It's one half of omega squared A squared V times mu. So unfortunately, there's four variables in here. But the way I always like to remember it is that this equation kind of looks like the word wave except the mu replaces the E and also the first two variables are squared. All right. So it kind of looks like the word wave. All right. So that's basically it. That's the equation. Let's go ahead and take a look at our example. So we are, are continuously producing a wave on a taut string. We're told what the mass density is. That's mu 0.05. We're told that the tension is 100 that's FT is 100. We wanna create waves with a frequency of 60. So that's gonna be F and the amplitude of the waves is going to be six centimeters. So we have to convert that, that's just gonna be A and that's going to be 0.06. So given these values here, what is the average power? We're just going to use the equation that we just saw up here and we're going to write it out again, one half omega squared A squared. So both of the first two variables are squared and then we're gonna do V times mu. So it kind of looks like the word wave again. All right. So which variables do we know? We don't know the angular frequency. We are given the amplitude, we don't know the wave speed, but we are given the mass density. We have two out of four. I need to find out the other two before I plug anything in. So I want to figure out the omega first. So how do we figure out omega well, omega unfortunately, pops up in a lot of different equations, but the one that you should always try out first is this equation here two pi root period or two pi times frequency. Now, hopefully realize guys that we're just gonna use omega equals two pi F because we're given here, the F is equal to 60. So we just go ahead and do that real quick. This is gonna be two pi times 60 you're gonna get omega here of 372nd, 77 and this is gonna be radiance per second. All right. So that's, that's done. So now all we have to do is figure out the wave speed. So we have to figure out V now remember that we're a wave, we're producing a wave on a string. So we actually have a couple of different equations that we can use for strings. Only we can use this equation here. This V equals screw root of tension over mu. We can also use lambda F remember because that applies to all kinds of waves. Now, if we look through our variables here, we actually have what the tension is and we have what the mass density is. So we're just going to go ahead and stop there and we're just going to use this one right here. So we're going to use uh V equals this is gonna be square roots of 100 over mu which is going to be 0.05. If you go ahead and work this out, what you're gonna get is a wave speed of 44.7 m per second and it's not squared, right? So now all we have to do is just plug our values for V and Omega back into our equation here. So P average is just gonna be one half. Now, you just plug in everything carefully. We got 377 squared, then we've got our amplitude which is gonna be 0.06 squared. And then we've got our V which is 44.7 and then we've got our mass density, which is 0.05. If you go ahead and plug in everything carefully, you should get an average power of 572 watts. So that's how much power it takes to continuously produce waves of this, of these characteristics here. All right. So that's it for this one guys. And let me know if you have any questions.

2

Problem

Problem

A horizontal string is stretched with a tension of 90 N, and the speed of transverse waves for the wire is 400 m/s. What must the amplitude of a 70.0 Hz traveling wave be for the average power carried by the wave to be 0.365 W?

A

2.9 mm

B

4.1 mm

C

0.2 mm

D

0.017 mm

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