(II) The two sources of sound in Fig. 16–15 face each other an...

Question

(II) The two sources of sound in Fig. 16–15 face each other and emit sounds of equal amplitude and equal frequency (305 Hz) but 180° out of phase. For what minimum separation of the two speakers will there be some point at which
(a) complete constructive interference occurs (Assume T = 20°C .) and

(a) complete constructive interference occurs (Assume T = 20°C .) and

Accepted Answer

Hey, everyone in this problem, two people are facing each other holding identical tuning forks in their hands. The tuning forks are struck simultaneously producing sound at a frequency of 250 Hertz. But due to a slight defect in one fork produces sound that is initially 180 degrees out of phase compared to the other. We are asked what should be the minimum separation between these two forks for complete constructive interference to occur at some point between the two people. We're told to assume the air temperature T is 25 °C. So we're given this diagram, we have our two tuning forks A and B and they're gonna have some separation. S and we're trying to figure out what that separation S should be so that we can get complete constructive interference at some point. We have four answer choices. Option A 0.69 m, option B 1.3 m, option C 2.7 m and option D, none of the above. So let's think about this. OK. So our tuning forks are gonna be some distance s apart and they are 180 degrees out of phase. OK. So if they are 100 and 80 degrees out of phase, what that means is when one is at its crust, yeah, there is at its trough. Hey, we can draw out what the sine waves would look like for each of them if we want to, they need to get a better idea. So if we draw kind of one cycle of this wave, we have the one in red and then we have the other in green, right? And they are 180 degrees at a phase. So when one is at its max, the other is at its minimum. And if we start to label the distances in this diagram, we know that the one complete cycle that will be the wavelength lambda. And so if we consider moving shifting this green curve over or shifting the red curve over, if we were to shift it half that distance LAMBDA divided by two, then the two would line up, our green curve would come over and line up with a red card. OK. So what we need to do is shift one of these by pi divided by two. And what that means is that our distant f must be equal two. And sorry, I said pi divided by two, I meant lambda divided by two, that wavelength lambda divided by two. So our distance s between our tuning forks needs to be the wavelength divided by two. And that's gonna give us that minimum separation distance where we can have complete constructive interference because the crest will line up. OK. So we have this expression for now, we need to figure out what is this wavelength lambda? Well, we're given information about the frequency and we're given some information about the temperature. So let's first recall that the wavelength and the frequency are related through the speed. OK. So we can write that the speed V is equal to the frequency F multiplied by the wavelength Lamar. Maybe we can rearrange to write the wavelength lambda is gonna be the speed divided by the frequency and we already have the frequency F. So what we need is the speed V and recall that the speed V is related to the temperature. OK? The speed of sound in this particular environment depends on the temperature. And so what we have is that V is equal to 331 was 0.6 multiplied by that temperature T and that is given in meters per second. Now, in this case, our temperature T is 25. So we get V is equal to 331 was 0.6 multiplied by 25 meters per second. And this gives us a speed V of 346 meters per second. OK. So we figured out this expression for that distance. S we know it depended on the wavel in order to calculate the wavelength, we need the speed and the frequency we had the frequency. Now we've calculated the speed. So we can come back to this calculation for LAMBDA, we can say that LAMBDA will be equal to this speed 346 m per second divided by the frequency of 250 Hertz, which gives a wavelength Lambda of about 1.384 m. And the final thing we need to do is to calculate this separation distance. S we said that it was gonna be half of the wavelength LAMBDA. So this is gonna be equal to 1.384 m divided by two, which is about 0.692 m. So in this case, in this problem, what we need is for these two tuning forks to be placed it at a minimum separation distance of 0.69 m in order to have complete constructive interference, this corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Constructive Interference

Phase Difference

Wavelength and Frequency Relationship

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