1 00:00:00,500 --> 00:00:02,502 - [Voiceover] The figure above shows a string 2 00:00:02,502 --> 00:00:05,038 with one end attached to an oscillator 3 00:00:05,038 --> 00:00:07,607 and the other end attached to a block there. 4 00:00:07,607 --> 00:00:08,708 There's our block. 5 00:00:08,708 --> 00:00:11,745 The string passes over a massless pulley that turns 6 00:00:11,745 --> 00:00:13,079 with negligible friction. 7 00:00:13,079 --> 00:00:15,081 There's our massless pulley that turns 8 00:00:15,081 --> 00:00:16,722 with negligible friction. 9 00:00:16,722 --> 00:00:20,887 Four such strings, A, B, C, and D are set up side-by-side 10 00:00:20,887 --> 00:00:23,757 as shown in the figure in the diagram below. 11 00:00:23,757 --> 00:00:26,559 So this is a top view, you can see oscillator. 12 00:00:26,559 --> 00:00:28,995 This is a top view of the oscillator string pulley 13 00:00:28,995 --> 00:00:31,798 mass system, and we have four of them. 14 00:00:31,798 --> 00:00:34,601 Each oscillator is adjusted to vibrate the string 15 00:00:34,601 --> 00:00:36,903 at its fundamental frequency f. 16 00:00:36,903 --> 00:00:38,905 So let's think about what it, I'll keep reading 17 00:00:38,905 --> 00:00:41,908 then we'll think about what fundamental frequency means. 18 00:00:41,908 --> 00:00:45,345 The distance between each oscillator and the pulley L 19 00:00:45,345 --> 00:00:46,346 is the same. 20 00:00:46,346 --> 00:00:48,681 So the length between the oscillator and the pulley 21 00:00:48,681 --> 00:00:52,519 is the same, and the mass of each block is the same. 22 00:00:52,519 --> 00:00:55,355 So the mass is what's providing the tension in the string. 23 00:00:55,355 --> 00:00:57,957 However, the fundamental frequency of each string 24 00:00:57,957 --> 00:00:59,092 is different. 25 00:00:59,092 --> 00:01:01,094 So let's just first of all think about 26 00:01:01,094 --> 00:01:03,096 what the fundamental frequency is, 27 00:01:03,096 --> 00:01:05,098 and then let's think about what makes them different. 28 00:01:05,098 --> 00:01:06,299 So the fundamental frequency, 29 00:01:06,299 --> 00:01:08,301 one way you could think about it is 30 00:01:08,301 --> 00:01:10,970 it's the lowest frequency that is going to produce 31 00:01:10,970 --> 00:01:13,592 a standing wave in your string. 32 00:01:13,592 --> 00:01:16,702 So it's the frequency that produces a standing wave 33 00:01:16,702 --> 00:01:18,522 that looks like this. 34 00:01:18,522 --> 00:01:23,522 It's a standing wave where the string is half a wavelength. 35 00:01:25,923 --> 00:01:27,487 I guess there's two ways to think about it. 36 00:01:27,487 --> 00:01:29,489 It's the lowest frequency we could produce 37 00:01:29,489 --> 00:01:31,491 the standing wave, or it's the frequency at which 38 00:01:31,491 --> 00:01:34,627 you're producing the wave, the standing wave, 39 00:01:34,627 --> 00:01:37,330 with the longest wavelength. 40 00:01:37,330 --> 00:01:39,332 So the string at the fundamental frequency 41 00:01:39,332 --> 00:01:41,334 is just going to go to, is gonna vibrate 42 00:01:41,334 --> 00:01:43,436 between those two positions. 43 00:01:43,436 --> 00:01:47,795 And you see here that the wavelength here 44 00:01:47,795 --> 00:01:49,404 is twice the length of the string. 45 00:01:49,404 --> 00:01:50,935 And if you want to see that a little bit clearer, 46 00:01:50,935 --> 00:01:53,105 if I were to continue with this wave, 47 00:01:53,105 --> 00:01:55,064 I would have to go another length of the string 48 00:01:55,064 --> 00:01:58,918 in order to complete one wavelength. 49 00:01:58,918 --> 00:01:59,957 Or another way to think about it, 50 00:01:59,957 --> 00:02:02,922 you're going up, down, and then you're going down back. 51 00:02:02,922 --> 00:02:05,925 It's reflecting back off of this end here. 52 00:02:05,925 --> 00:02:09,229 And as we mentioned, essentially what the mass is doing 53 00:02:09,229 --> 00:02:12,866 is providing the tension, the force of gravity on this mass 54 00:02:12,866 --> 00:02:15,869 is providing the tension in this string. 55 00:02:15,869 --> 00:02:18,638 So the oscillators is vibrating at the right frequency 56 00:02:18,638 --> 00:02:22,308 to produce this, the lowest frequency where you can produce 57 00:02:22,308 --> 00:02:24,664 this standing wave. 58 00:02:24,664 --> 00:02:28,114 So let's answer the questions now, and we have four 59 00:02:28,114 --> 00:02:31,551 of this set up and they all have different frequencies. 60 00:02:31,551 --> 00:02:35,154 The equation for the velocity v of a wave on a string 61 00:02:35,154 --> 00:02:38,491 is v is equal to the square root of the tension 62 00:02:38,491 --> 00:02:41,294 of the string divided by the mass of the string 63 00:02:41,294 --> 00:02:43,296 divided by the length of the string, 64 00:02:43,296 --> 00:02:45,298 where F sub T is tension of the string 65 00:02:45,298 --> 00:02:48,167 and m divided by L is the mass per unit length, 66 00:02:48,167 --> 00:02:50,169 linear mass density of the string. 67 00:02:50,169 --> 00:02:51,533 And hopefully this makes sense. 68 00:02:51,533 --> 00:02:54,340 It makes sense that it's going to be that if, 69 00:02:54,340 --> 00:02:57,043 if the tension increases, that your velocity will increase, 70 00:02:57,043 --> 00:02:59,712 the tension, you could think of the atoms of the string 71 00:02:59,712 --> 00:03:02,248 of how much they're pulling on each other. 72 00:03:02,248 --> 00:03:05,697 And so if there's higher tension, well, 73 00:03:05,697 --> 00:03:07,139 they're going to be able to move each other better, 74 00:03:07,139 --> 00:03:09,216 or I guess you could say accelerate each other better 75 00:03:09,216 --> 00:03:13,026 as the wave goes through the string. 76 00:03:13,026 --> 00:03:15,324 And also make sense that the larger the mass, 77 00:03:15,324 --> 00:03:17,033 if you hold everything else equal, 78 00:03:17,033 --> 00:03:18,586 that you're gonna have a slower velocity. 79 00:03:18,586 --> 00:03:22,004 Mass, you could view it 80 00:03:22,004 --> 00:03:24,054 as a measure of inertia, it's how hard it is 81 00:03:24,054 --> 00:03:25,538 to accelerate something. 82 00:03:25,538 --> 00:03:28,941 So if the string itself, especially the mass 83 00:03:28,941 --> 00:03:33,268 per unit length, if there's a lot of mass per unit length, 84 00:03:33,268 --> 00:03:34,910 actually, let me circle that because that's actually 85 00:03:34,910 --> 00:03:36,401 the more interesting thing. 86 00:03:36,401 --> 00:03:39,337 If there's a lot of mass per unit length, 87 00:03:39,337 --> 00:03:41,654 it makes sense that for a given amount of the string, 88 00:03:41,654 --> 00:03:44,757 it's gonna be harder to accelerate it back and forth 89 00:03:44,757 --> 00:03:45,992 as you vibrate it. 90 00:03:45,992 --> 00:03:50,129 And so this part right over here would be inversely related 91 00:03:50,129 --> 00:03:51,300 to the velocity. 92 00:03:51,300 --> 00:03:52,395 It's not proportional though. 93 00:03:52,395 --> 00:03:54,067 You have this square root here, 94 00:03:54,067 --> 00:03:55,708 but they're definitely, 95 00:03:55,708 --> 00:03:58,237 if the linear mass density increases, 96 00:03:58,237 --> 00:04:00,239 then you're gonna have a slower velocity. 97 00:04:00,239 --> 00:04:02,809 And if your tension increases, you're going to have 98 00:04:02,809 --> 00:04:03,810 a higher velocity. 99 00:04:03,810 --> 00:04:06,446 So hopefully this makes some intuitive sense. 100 00:04:06,446 --> 00:04:08,548 And they ask, what is the difference about the four string 101 00:04:08,548 --> 00:04:11,217 shown above that would result in having different 102 00:04:11,217 --> 00:04:13,119 fundamental frequencies? 103 00:04:13,119 --> 00:04:15,421 Explain how you arrive at your answer. 104 00:04:15,421 --> 00:04:17,423 And then part b is student graphs frequency 105 00:04:17,423 --> 00:04:20,994 is a function of the inverse of the linear mass density. 106 00:04:20,994 --> 00:04:22,729 Will the graph be linear? 107 00:04:22,729 --> 00:04:24,864 Explain how you arrived at your answer. 108 00:04:24,864 --> 00:04:27,266 All right, let's answer each of these. 109 00:04:27,266 --> 00:04:31,738 So a, part a, what is different about the four strings 110 00:04:31,738 --> 00:04:33,973 because they all have different fundamental frequencies? 111 00:04:33,973 --> 00:04:36,909 So the fundamental frequency, 112 00:04:36,909 --> 00:04:40,079 let's just go back to what we know about waves, 113 00:04:40,079 --> 00:04:44,283 that the velocity of a wave is equal to the frequency 114 00:04:44,283 --> 00:04:48,187 times the wavelength of the wave. 115 00:04:48,187 --> 00:04:51,257 Or you could say, you could divide both sides by lambda, 116 00:04:51,257 --> 00:04:53,259 you could say that the frequency of a wave 117 00:04:53,259 --> 00:04:57,764 is equal to the velocity over the velocity over 118 00:04:57,764 --> 00:04:58,998 the wavelength. 119 00:04:58,998 --> 00:05:02,114 So if we're talking about the fundamental frequency, 120 00:05:02,114 --> 00:05:04,937 if we're talking about the fundamental frequency, 121 00:05:04,937 --> 00:05:07,696 funda, let me just, 122 00:05:07,696 --> 00:05:09,809 let me write frequency and then let me write fundamental 123 00:05:09,809 --> 00:05:11,144 real small here. 124 00:05:11,144 --> 00:05:14,514 Fundamental, the fundamental frequency, 125 00:05:14,514 --> 00:05:17,917 is going to be the velocity of our waves 126 00:05:17,917 --> 00:05:21,054 divided by the wavelength is going to be twice the length 127 00:05:21,054 --> 00:05:24,524 of our string, divided by two L. 128 00:05:24,524 --> 00:05:27,527 And if you look at the expression that they gave us 129 00:05:27,527 --> 00:05:30,463 for the velocity of the wave on the string, 130 00:05:30,463 --> 00:05:32,765 well, this is going to be equal to 131 00:05:32,765 --> 00:05:36,836 the square root of the tension in the string 132 00:05:36,836 --> 00:05:41,836 divided by, divided by the linear mass density. 133 00:05:42,241 --> 00:05:44,243 Divided by the linear mass density. 134 00:05:44,243 --> 00:05:47,780 And all of that is going to be over two L. 135 00:05:47,780 --> 00:05:49,949 Now all of them have different fundamental frequencies, 136 00:05:49,949 --> 00:05:51,951 but let's think about what's different over here. 137 00:05:51,951 --> 00:05:54,587 They all have the same tension. 138 00:05:54,587 --> 00:05:56,689 They all have the same tension. 139 00:05:56,689 --> 00:05:57,824 How do I know that? 140 00:05:57,824 --> 00:05:59,826 Well, what's causing the tension is the masses 141 00:05:59,826 --> 00:06:01,797 hanging over the pulleys. 142 00:06:01,797 --> 00:06:03,616 So the weight of those masses. 143 00:06:03,616 --> 00:06:05,535 So that's all going to be the same for all of them, 144 00:06:05,535 --> 00:06:07,833 and they all have the same length. 145 00:06:07,833 --> 00:06:10,193 They all have the same length. 146 00:06:10,193 --> 00:06:12,004 So these are all the same. 147 00:06:12,004 --> 00:06:13,973 So the only way that you're going to have different 148 00:06:13,973 --> 00:06:18,010 fundamental frequencies is if you have different masses. 149 00:06:18,010 --> 00:06:19,679 So different masses. 150 00:06:19,679 --> 00:06:22,415 So that has to be different. 151 00:06:22,415 --> 00:06:24,650 Different. 152 00:06:24,650 --> 00:06:27,220 So to answer their question, 153 00:06:27,220 --> 00:06:29,455 the strings must have different mass, 154 00:06:29,455 --> 00:06:32,091 well, they would have different linear mass densities, 155 00:06:32,091 --> 00:06:33,181 but since they're all the same length, 156 00:06:33,181 --> 00:06:35,035 they would also have different masses. 157 00:06:35,035 --> 00:06:37,163 So let me write this down. 158 00:06:37,163 --> 00:06:41,467 Strings, strings must 159 00:06:42,667 --> 00:06:45,316 have different, 160 00:06:47,073 --> 00:06:49,610 different masses 161 00:06:51,277 --> 00:06:53,379 and mass densities, 162 00:06:53,379 --> 00:06:56,135 and mass densities, 163 00:06:57,383 --> 00:06:59,786 densities, since 164 00:07:02,288 --> 00:07:05,400 all other variables 165 00:07:07,293 --> 00:07:10,241 driving, driving 166 00:07:12,064 --> 00:07:14,400 fundamental frequency are the same, 167 00:07:14,400 --> 00:07:18,371 fundamental frequency 168 00:07:20,840 --> 00:07:23,342 are the same, 169 00:07:23,342 --> 00:07:27,280 are the same. 170 00:07:27,280 --> 00:07:29,745 All right, let's tackle part b now. 171 00:07:30,883 --> 00:07:33,820 A student graphs frequency as a function 172 00:07:33,820 --> 00:07:37,256 of the inverse of linear mass density. 173 00:07:37,256 --> 00:07:39,458 Will the graph be linear? 174 00:07:39,458 --> 00:07:41,928 Explain how you arrived at your answer. 175 00:07:41,928 --> 00:07:44,063 So student graphs frequency. 176 00:07:44,063 --> 00:07:45,965 Let me underline this. 177 00:07:45,965 --> 00:07:48,835 They're graphing frequency as a function 178 00:07:48,835 --> 00:07:51,871 of linear mass density. 179 00:07:51,871 --> 00:07:54,073 So we actually can write this down. 180 00:07:54,073 --> 00:07:56,242 If we wanted to write frequency 181 00:07:56,242 --> 00:07:58,611 as a function of linear mass densities, 182 00:07:58,611 --> 00:08:01,914 so we could write as a function of m over L, 183 00:08:01,914 --> 00:08:04,016 well, this is going to be equal to, 184 00:08:04,016 --> 00:08:05,451 is going to be equal to, 185 00:08:05,451 --> 00:08:07,653 we could rewrite this expression, 186 00:08:07,653 --> 00:08:09,655 actually, we could just leave it like this. 187 00:08:09,655 --> 00:08:12,225 This is the same thing as one over two L 188 00:08:12,225 --> 00:08:15,661 times the square root of the tension 189 00:08:15,661 --> 00:08:19,866 divided by the linear mass density. 190 00:08:19,866 --> 00:08:21,868 Or you can do this as being equal to, 191 00:08:21,868 --> 00:08:25,371 you could say this is a square root of 192 00:08:25,371 --> 00:08:28,007 the tension over two L, 193 00:08:28,007 --> 00:08:30,009 and I'm putting all of this separate because 194 00:08:30,009 --> 00:08:32,477 we're doing it as a function of our linear mass density. 195 00:08:32,477 --> 00:08:36,716 So we can assume that all of this is going to be a constant. 196 00:08:36,716 --> 00:08:39,751 And then times the square root 197 00:08:39,751 --> 00:08:43,990 of one over the linear mass density. 198 00:08:44,860 --> 00:08:48,928 So if you're plotting, if you're plotting frequency 199 00:08:48,928 --> 00:08:51,397 as a function of this, 200 00:08:51,397 --> 00:08:53,933 it's not going to be, the graph is definitely 201 00:08:53,933 --> 00:08:56,168 not going to be linear. 202 00:08:56,168 --> 00:08:58,404 You see here, one, I have the reciprocal of it, 203 00:08:58,404 --> 00:09:00,439 and then I take the square root of it. 204 00:09:00,439 --> 00:09:04,482 So let me write this down. 205 00:09:07,273 --> 00:09:12,273 This f or graph of f of two L, 206 00:09:12,396 --> 00:09:13,382 definitely not linear. 207 00:09:13,382 --> 00:09:15,012 Let me write it that way. 208 00:09:15,012 --> 00:09:19,428 F of m over L graph, 209 00:09:21,869 --> 00:09:25,833 definitely, definitely 210 00:09:26,784 --> 00:09:28,967 not linear. 211 00:09:30,355 --> 00:09:32,709 And you could point out that it has a reciprocal 212 00:09:32,709 --> 00:09:34,033 and it's a square root. 213 00:09:34,033 --> 00:09:38,678 Involves, involves square root 214 00:09:38,678 --> 00:09:42,048 and inverse, 215 00:09:42,048 --> 00:09:45,351 or I could say and reciprocal of the variable. 216 00:09:45,351 --> 00:09:49,165 And reciprocal, reciprocal 217 00:09:50,525 --> 00:09:52,858 of the linear mass density 218 00:09:52,858 --> 00:09:54,260 of m over L. 219 00:09:54,260 --> 00:09:55,561 So definitely not linear. 220 00:09:55,561 --> 00:09:58,364 All right, let's tackle part c. 221 00:09:58,364 --> 00:10:01,634 The frequency of the oscillator connected to string D 222 00:10:01,634 --> 00:10:04,236 is changed so that the string vibrates 223 00:10:04,236 --> 00:10:06,405 in its second harmonic. 224 00:10:06,405 --> 00:10:09,075 On the side view of string D below, 225 00:10:09,075 --> 00:10:11,677 mark and label the points on the string that have 226 00:10:11,677 --> 00:10:14,113 the greatest average vertical speed. 227 00:10:14,113 --> 00:10:16,115 So one way to think about the fundamental frequency, 228 00:10:16,115 --> 00:10:17,116 that's our first harmonic. 229 00:10:17,116 --> 00:10:19,118 So if they're talking about the first harmonic 230 00:10:19,118 --> 00:10:22,788 and the string is what I showed before, 231 00:10:22,788 --> 00:10:26,928 that's the lowest frequency that produces a standing wave 232 00:10:26,928 --> 00:10:29,228 or it's the frequency that produces the largest 233 00:10:29,228 --> 00:10:30,546 standing wave, I guess you could say the one 234 00:10:30,546 --> 00:10:31,774 with the largest wavelength. 235 00:10:31,774 --> 00:10:33,168 And if it was a first harmonic, 236 00:10:33,168 --> 00:10:35,091 the part of the string that moves the most 237 00:10:35,091 --> 00:10:36,737 is going to be that center. 238 00:10:36,737 --> 00:10:39,538 But our second harmonic is the next highest frequency 239 00:10:39,538 --> 00:10:41,807 that produces a standing wave, 240 00:10:41,807 --> 00:10:44,143 and that's going to be a situation in which 241 00:10:44,143 --> 00:10:46,746 the wave, instead of having a wavelength twice the length 242 00:10:46,746 --> 00:10:48,109 of the string, it's gonna have a wavelength 243 00:10:48,109 --> 00:10:49,988 equal to the length of the string. 244 00:10:49,988 --> 00:10:53,285 So now, it's going to look like this. 245 00:10:53,285 --> 00:10:56,657 So it's going to vibrate between this, 246 00:10:56,657 --> 00:10:59,125 actually, let me draw it a little bit neater than that. 247 00:10:59,125 --> 00:11:02,288 It's going to vibrate between, 248 00:11:03,429 --> 00:11:05,180 between this. 249 00:11:06,320 --> 00:11:09,335 And so 250 00:11:10,325 --> 00:11:13,636 it's gonna vibrate between that and this. 251 00:11:13,636 --> 00:11:15,508 And this right over here. 252 00:11:16,742 --> 00:11:19,578 So when you see this, this version of it, 253 00:11:19,578 --> 00:11:21,580 where now that we have half the wavelength, 254 00:11:21,580 --> 00:11:25,120 the wavelength is exactly the wavelength of, 255 00:11:25,120 --> 00:11:26,652 it's actually L this time, 256 00:11:26,652 --> 00:11:30,314 the part that moves the most is here. 257 00:11:30,314 --> 00:11:31,701 So that's going to move the most. 258 00:11:31,701 --> 00:11:33,259 And here. 259 00:11:33,259 --> 00:11:35,056 I didn't draw it perfectly. 260 00:11:35,056 --> 00:11:37,930 But one way to think about it, it's exactly 1/4 261 00:11:37,930 --> 00:11:39,732 and 3/4 of the way. 262 00:11:39,732 --> 00:11:42,134 Exactly halfway is not going to move as much, 263 00:11:42,134 --> 00:11:44,336 is not gonna move because it's a standing wave now 264 00:11:44,336 --> 00:11:46,338 or it's gonna move very imperceptibly. 265 00:11:46,338 --> 00:11:49,208 These two places are where you're gonna move the most 266 00:11:49,208 --> 00:11:53,145 at the first, or sorry, at the second harmonic. 267 00:11:53,145 --> 00:11:56,816 The second harmonic being the next highest frequency 268 00:11:56,816 --> 00:00:00,000 that produces a standing wave again.