1 00:00:00,281 --> 00:00:03,049 - [Instructor] So, as far as simple harmonic oscillators go, 2 00:00:03,049 --> 00:00:06,293 masses on springs are the most common example, 3 00:00:06,293 --> 00:00:09,397 but the next most common example is the pendulum. 4 00:00:09,397 --> 00:00:10,861 So, that's what I wanna talk to you about 5 00:00:10,861 --> 00:00:12,150 in this video. 6 00:00:12,150 --> 00:00:14,410 And a pendulum is just a mass, m, 7 00:00:14,410 --> 00:00:17,356 connected to a string of some length, L, 8 00:00:17,356 --> 00:00:19,798 that you can then pull back a certain amount 9 00:00:19,798 --> 00:00:21,964 and then you let it swing back and forth. 10 00:00:21,964 --> 00:00:25,120 So, this is gonna swing forward and then backward, 11 00:00:25,120 --> 00:00:26,434 and then forward and backward. 12 00:00:26,434 --> 00:00:29,835 It oscillates just like a simple harmonic oscillator 13 00:00:29,835 --> 00:00:31,090 and so that's why we study it 14 00:00:31,090 --> 00:00:33,300 when we study simple harmonic oscillators. 15 00:00:33,300 --> 00:00:34,958 And technically speaking, I should say 16 00:00:34,958 --> 00:00:37,588 that this is actually a simple pendulum 17 00:00:37,588 --> 00:00:41,732 because this is simply a mass connected to a string. 18 00:00:41,732 --> 00:00:42,788 It's not complicated. 19 00:00:42,788 --> 00:00:44,524 You could have more complicated examples. 20 00:00:44,524 --> 00:00:46,374 Let's say you connect another string, 21 00:00:46,374 --> 00:00:48,655 with another mass down here. 22 00:00:48,655 --> 00:00:50,364 This gets really complicated. 23 00:00:50,364 --> 00:00:53,523 In fact, it gets, what physicists call chaotic, 24 00:00:53,523 --> 00:00:54,396 which is kind of cool. 25 00:00:54,396 --> 00:00:55,321 If you've never seen it, 26 00:00:55,321 --> 00:00:57,572 look up double pendulum, it's pretty sweet. 27 00:00:57,572 --> 00:01:00,309 But really complicated to describe mathematically. 28 00:01:00,309 --> 00:01:01,616 So, we're not gonna bother with that. 29 00:01:01,616 --> 00:01:03,068 We've got enough things to study 30 00:01:03,068 --> 00:01:04,708 by just studying simple pendulums. 31 00:01:04,708 --> 00:01:06,620 We can learn a lot about the motion 32 00:01:06,620 --> 00:01:08,316 just by looking at this case. 33 00:01:08,316 --> 00:01:09,782 So, what do we mean that the pendulum 34 00:01:09,782 --> 00:01:11,775 is a simple harmonic oscillator? 35 00:01:11,775 --> 00:01:13,666 Well, we mean that there's a restoring force 36 00:01:13,666 --> 00:01:15,536 proportional to the displacement 37 00:01:15,536 --> 00:01:16,895 and we mean that its motion 38 00:01:16,895 --> 00:01:20,604 can be described by the simple harmonic oscillator equation. 39 00:01:20,604 --> 00:01:23,772 So, if you remember that was described by an equation 40 00:01:23,772 --> 00:01:25,417 that looked like this, X, 41 00:01:25,417 --> 00:01:27,815 some variable X is a function of time 42 00:01:27,815 --> 00:01:32,662 was equal to some amplitude times cosine or sine, 43 00:01:32,662 --> 00:01:36,199 I'm just gonna write cosine, of two pi 44 00:01:36,199 --> 00:01:39,272 divided by the period, times the time 45 00:01:39,272 --> 00:01:42,269 and you can if you want add a phase constant. 46 00:01:42,269 --> 00:01:44,100 I'm not gonna write it 'cause usually you can 47 00:01:44,100 --> 00:01:45,657 get away with not using that one. 48 00:01:45,657 --> 00:01:48,729 So, this is the simple harmonic oscillator equation. 49 00:01:48,729 --> 00:01:50,536 So, how would I apply this equation 50 00:01:50,536 --> 00:01:52,338 to this case of a pendulum? 51 00:01:52,338 --> 00:01:53,849 Well, I wouldn't use X. 52 00:01:53,849 --> 00:01:56,888 The far more useful and common example 53 00:01:56,888 --> 00:01:58,888 of using a variable to describe a pendulum 54 00:01:58,888 --> 00:02:02,065 is the angle that the pendulum is at. 55 00:02:02,065 --> 00:02:03,708 So, consider the fact that this mass 56 00:02:03,708 --> 00:02:05,026 is gonna be at different angles 57 00:02:05,026 --> 00:02:06,577 at different moments in time. 58 00:02:06,577 --> 00:02:09,503 So, it starts over here, maybe it's at like, 30 degrees 59 00:02:09,503 --> 00:02:12,360 and it swings it's only at like 20 and then 10 60 00:02:12,360 --> 00:02:14,143 and then zero 'cause we're measuring angles 61 00:02:14,143 --> 00:02:15,879 from the center line. 62 00:02:15,879 --> 00:02:18,161 And then it swings through, maybe it's at negative 10, 63 00:02:18,161 --> 00:02:19,633 negative 20, negative 30 64 00:02:19,633 --> 00:02:21,321 and then this whole process repeats. 65 00:02:21,321 --> 00:02:23,913 So, instead of using X, we're gonna use theta. 66 00:02:23,913 --> 00:02:26,306 So, this is gonna be an angle as a function of time. 67 00:02:26,306 --> 00:02:29,724 So, I'll write theta as a function of time 68 00:02:29,724 --> 00:02:33,013 is gonna equal some amplitude, but again, 69 00:02:33,013 --> 00:02:34,173 since I'm measuring theta, 70 00:02:34,173 --> 00:02:37,224 my amplitude is not going to be a distance in X, 71 00:02:37,224 --> 00:02:38,723 or a displacement in X, 72 00:02:38,723 --> 00:02:42,236 this is gonna be not the maximum regular displacement, 73 00:02:42,236 --> 00:02:45,075 it's gonna be the maximum angular displacement 74 00:02:45,075 --> 00:02:46,754 from equilibrium right here. 75 00:02:46,754 --> 00:02:48,714 This line here would be equilibrium 76 00:02:48,714 --> 00:02:49,844 'cause if you put the mass there 77 00:02:49,844 --> 00:02:52,041 and let it sit it would just continue to sit there, 78 00:02:52,041 --> 00:02:53,714 there'd be no net force on it. 79 00:02:53,714 --> 00:02:55,530 Only when you displace the mass 80 00:02:55,530 --> 00:02:57,534 from this equilibrium position 81 00:02:57,534 --> 00:02:59,419 does it have a restoring force. 82 00:02:59,419 --> 00:03:01,225 So, this would be the maximum, 83 00:03:01,225 --> 00:03:03,308 I'll just call it theta maximum, 84 00:03:03,308 --> 00:03:06,449 'cause this is the maximum angular displacement 85 00:03:06,449 --> 00:03:07,764 when you pull this back, 86 00:03:07,764 --> 00:03:10,043 the maximum angle you pull it to, whatever that it. 87 00:03:10,043 --> 00:03:12,192 Maybe it's 30 degrees, maybe it's 20, 88 00:03:12,192 --> 00:03:14,376 that would be the angle that I plug in here. 89 00:03:14,376 --> 00:03:16,304 And then we'll multiply by cosine 90 00:03:16,304 --> 00:03:18,681 and it will have the same argument in here. 91 00:03:18,681 --> 00:03:21,378 Two pi over whatever the period is, 92 00:03:21,378 --> 00:03:23,240 and the period is the time it takes 93 00:03:23,240 --> 00:03:26,824 for this pendulum to reset or to complete a whole cycle 94 00:03:26,824 --> 00:03:28,672 and we always have to multiply by T, 95 00:03:28,672 --> 00:03:29,888 that's our variable, 96 00:03:29,888 --> 00:03:31,880 that's what makes this a function, 97 00:03:31,880 --> 00:03:33,960 it's a function of time. 98 00:03:33,960 --> 00:03:36,368 Alright, so I gotta come clean about something now. 99 00:03:36,368 --> 00:03:38,760 Technically speaking, the simple pendulum 100 00:03:38,760 --> 00:03:42,361 is not a perfect simple harmonic oscillator, 101 00:03:42,361 --> 00:03:45,023 it's only extremely close to being 102 00:03:45,023 --> 00:03:46,808 a simple harmonic oscillator. 103 00:03:46,808 --> 00:03:50,128 In fact, for small angles, this will only be off 104 00:03:50,128 --> 00:03:53,349 by very small amounts, like less than a per cent. 105 00:03:53,349 --> 00:03:54,682 So, because of that, we often treat 106 00:03:54,682 --> 00:03:57,537 a simple pendulum as a simple harmonic oscillator, 107 00:03:57,537 --> 00:03:59,553 but technically speaking it only works 108 00:03:59,553 --> 00:04:01,383 really well if you're less 109 00:04:01,383 --> 00:04:04,389 than say a certain amount, say 20 degrees. 110 00:04:04,389 --> 00:04:07,081 As you get to larger maximum amplitudes, 111 00:04:07,081 --> 00:04:09,329 this is gonna deviate more and more. 112 00:04:09,329 --> 00:04:11,146 It'll still be reasonably close, 113 00:04:11,146 --> 00:04:13,106 maybe within like 20 per cent, 114 00:04:13,106 --> 00:04:16,124 but only for small angles is it extremely close. 115 00:04:16,124 --> 00:04:17,454 But if you are at small angles. 116 00:04:17,454 --> 00:04:20,326 So, if you're considering a pendulum that has small angles. 117 00:04:20,326 --> 00:04:22,679 Like, maybe this is only 20 degrees or less, 118 00:04:22,679 --> 00:04:25,158 that pendulum would be described really well 119 00:04:25,158 --> 00:04:27,286 by this equation because it would be extremely close 120 00:04:27,286 --> 00:04:29,870 to being a simple harmonic oscillator. 121 00:04:29,870 --> 00:04:31,313 Alright, so let's assume we're in 122 00:04:31,313 --> 00:04:33,443 that small angle approximation 123 00:04:33,443 --> 00:04:35,374 where this amplitude is small. 124 00:04:35,374 --> 00:04:36,499 What can we say? 125 00:04:36,499 --> 00:04:37,646 Well, one question we can ask 126 00:04:37,646 --> 00:04:41,368 is what's the period of this pendulum gonna depend on? 127 00:04:41,368 --> 00:04:42,722 Right, this period here, 128 00:04:42,722 --> 00:04:45,352 what could we change that would change this period here? 129 00:04:45,352 --> 00:04:47,064 So, what might this depend on? 130 00:04:47,064 --> 00:04:50,180 My first guess might be, well, maybe it's the mass. 131 00:04:50,180 --> 00:04:51,865 So, let's think about this. 132 00:04:51,865 --> 00:04:55,760 If we increased the mass on this pendulum, 133 00:04:55,760 --> 00:04:57,656 do you think that would increase the period 134 00:04:57,656 --> 00:04:59,875 or decrease the period or leave it the same? 135 00:04:59,875 --> 00:05:01,468 Some people might say, well, 136 00:05:01,468 --> 00:05:04,243 I think an increase in mass would increase 137 00:05:04,243 --> 00:05:06,400 the inertia of this system. 138 00:05:06,400 --> 00:05:08,216 Right, it's gonna be harder to move. 139 00:05:08,216 --> 00:05:10,184 When the mass of something goes up, 140 00:05:10,184 --> 00:05:12,599 it's more sluggish to accelerations, 141 00:05:12,599 --> 00:05:14,296 it's more difficult to move around 142 00:05:14,296 --> 00:05:16,072 and change its direction. 143 00:05:16,072 --> 00:05:19,437 That means it should take longer to complete a cycle. 144 00:05:19,437 --> 00:05:22,040 Maybe that means that the period should increase 145 00:05:22,040 --> 00:05:24,001 because the time would increase. 146 00:05:24,001 --> 00:05:26,024 But other people might say, wait a minute, 147 00:05:26,024 --> 00:05:28,014 if we increase the mass, 148 00:05:28,014 --> 00:05:31,144 that would increase the gravitational force. 149 00:05:31,144 --> 00:05:32,744 Right, gravity's going to be pulling down 150 00:05:32,744 --> 00:05:34,864 harder now on this mass, 151 00:05:34,864 --> 00:05:37,801 and gravity is the force that's gonna be restoring 152 00:05:37,801 --> 00:05:39,896 this mass back to equilibrium. 153 00:05:39,896 --> 00:05:41,426 Gravity's gonna be pulling down 154 00:05:41,426 --> 00:05:44,008 and if it pulls down with a greater force, 155 00:05:44,008 --> 00:05:45,744 you might think this mass is gonna swing 156 00:05:45,744 --> 00:05:48,785 with a greater speed and if it's got a greater speed, 157 00:05:48,785 --> 00:05:52,314 it'll complete this cycle in less time 158 00:05:52,314 --> 00:05:53,992 because it's moving faster, 159 00:05:53,992 --> 00:05:55,426 and since it takes less time, 160 00:05:55,426 --> 00:05:57,943 you might think that the period goes down, 161 00:05:57,943 --> 00:06:01,328 but these two effects exactly cancel. 162 00:06:01,328 --> 00:06:04,401 So, the fact that the mass is gonna have more inertia, 163 00:06:04,401 --> 00:06:06,800 with greater mass, that means it's harder to move, 164 00:06:06,800 --> 00:06:09,119 and the force is gonna increase 165 00:06:09,119 --> 00:06:11,171 due to the force of gravity getting larger. 166 00:06:11,171 --> 00:06:14,048 Those offset perfectly and this mass 167 00:06:14,048 --> 00:06:16,186 will not affect the period. 168 00:06:16,186 --> 00:06:18,024 So, it turns out, it's kinda weird, 169 00:06:18,024 --> 00:06:21,441 changing the mass on here does not affect the period 170 00:06:21,441 --> 00:06:23,946 at which this swings back and forth. 171 00:06:23,946 --> 00:06:24,779 So, imagine this. 172 00:06:24,779 --> 00:06:26,752 So, if you go get on a swing at the park, 173 00:06:26,752 --> 00:06:28,271 and you swing back and forth, 174 00:06:28,271 --> 00:06:30,152 and then a little kid, tiny kid, 175 00:06:30,152 --> 00:06:32,590 five year old comes on and swings back and forth, 176 00:06:32,590 --> 00:06:35,338 they should have the same period of motion as you do 177 00:06:35,338 --> 00:06:36,646 because the mass at the end here 178 00:06:36,646 --> 00:06:39,141 does not affect the period. 179 00:06:39,141 --> 00:06:40,659 So, that's a little weird but it's true 180 00:06:40,659 --> 00:06:41,617 and you should keep that in mind. 181 00:06:41,617 --> 00:06:44,550 Mass does not affect the period. 182 00:06:44,550 --> 00:06:46,416 So, what does affect the period? 183 00:06:46,416 --> 00:06:48,592 Well, I'm just gonna write the formula down for you. 184 00:06:48,592 --> 00:06:50,072 I'm not gonna derive this. 185 00:06:50,072 --> 00:06:52,432 The derivation requires calculus. 186 00:06:52,432 --> 00:06:53,611 It's an awesome derivation. 187 00:06:53,611 --> 00:06:55,832 If you know calculus, you should go check it out. 188 00:06:55,832 --> 00:06:57,712 But just in case you haven't seen calculus, 189 00:06:57,712 --> 00:06:59,081 I'm just gonna write this down, 190 00:06:59,081 --> 00:07:01,744 give you a little tour of this equation. 191 00:07:01,744 --> 00:07:03,729 Show you why it should make sense 192 00:07:03,729 --> 00:07:05,608 and hopefully give you a little intuition 193 00:07:05,608 --> 00:07:07,905 about why the variables are in here that they are. 194 00:07:07,905 --> 00:07:09,721 So, the first variable is L. 195 00:07:09,721 --> 00:07:12,072 L goes on top, the length of the string, 196 00:07:12,072 --> 00:07:13,962 and then the acceleration due to gravity, 197 00:07:13,962 --> 00:07:15,434 little g goes on the bottom. 198 00:07:15,434 --> 00:07:17,308 So, why is this the formula? 199 00:07:17,308 --> 00:07:20,152 Well, the two pi is just a constant, you get a square root. 200 00:07:20,152 --> 00:07:22,804 L is on top, that means if you increase the length 201 00:07:22,804 --> 00:07:25,710 of the string, you're gonna get a greater period. 202 00:07:25,710 --> 00:07:28,813 So, increasing the length should increase the period. 203 00:07:28,813 --> 00:07:30,134 Why is that? 204 00:07:30,134 --> 00:07:31,215 Well, think about this. 205 00:07:31,215 --> 00:07:34,862 A mass on a string rotating back and forth, 206 00:07:34,862 --> 00:07:37,392 if there's rotation, a quantity that's useful 207 00:07:37,392 --> 00:07:39,929 to think about is the moment of inertia. 208 00:07:39,929 --> 00:07:42,527 So, the moment of inertia of this mass on a string 209 00:07:42,527 --> 00:07:45,209 would be equal to, this is a point mass, 210 00:07:45,209 --> 00:07:47,573 rotating about an axis, 211 00:07:47,573 --> 00:07:50,540 so the axis of rotation is this point right here. 212 00:07:50,540 --> 00:07:52,710 And a point mass rotating around an axis 213 00:07:52,710 --> 00:07:54,710 is just given by mr squared. 214 00:07:54,710 --> 00:07:56,674 That would be the moment of inertia. 215 00:07:56,674 --> 00:07:59,970 But this r is the distance from the axis to the mass, 216 00:07:59,970 --> 00:08:02,765 so this is just mL squared. 217 00:08:02,765 --> 00:08:04,554 This is the moment of inertia. 218 00:08:04,554 --> 00:08:06,695 And look it, if we increase the length, 219 00:08:06,695 --> 00:08:09,254 we increase the moment of inertia. 220 00:08:09,254 --> 00:08:11,866 So, bigger L gives us bigger moment of inertia. 221 00:08:11,866 --> 00:08:13,130 What does that mean? 222 00:08:13,130 --> 00:08:16,048 Moment of inertia is a measure of how difficult it is 223 00:08:16,048 --> 00:08:18,570 to angularly accelerate something. 224 00:08:18,570 --> 00:08:20,674 So, it's a measure of how sluggish 225 00:08:20,674 --> 00:08:24,558 this mass is gonna be to changes in its angular velocity. 226 00:08:24,558 --> 00:08:26,147 So, bigger moment of inertia 227 00:08:26,147 --> 00:08:28,762 means it's gonna be harder to take this mass 228 00:08:28,762 --> 00:08:30,418 and whip it around back and forth 229 00:08:30,418 --> 00:08:32,058 and change its direction. 230 00:08:32,058 --> 00:08:34,785 So, since it's harder to move this mass around, 231 00:08:34,785 --> 00:08:37,977 it's gonna take longer to move it back and forth, 232 00:08:37,977 --> 00:08:41,197 that's why bigger length means bigger moment of inertia 233 00:08:41,198 --> 00:08:43,241 and bigger moment of inertia means it takes 234 00:08:43,241 --> 00:08:45,542 longer to move this thing back and forth, 235 00:08:45,542 --> 00:08:47,550 that's why the period gets bigger. 236 00:08:47,550 --> 00:08:48,922 Now, some people out there might object. 237 00:08:48,922 --> 00:08:51,010 If you're really clever, you might say, 238 00:08:51,010 --> 00:08:53,948 wait a minute, if this length increases, 239 00:08:53,948 --> 00:08:56,794 the thing causing this to angularly accelerate 240 00:08:56,794 --> 00:08:59,706 is the torque, and I know the formula for torque. 241 00:08:59,706 --> 00:09:02,010 The formula for torque looks like this. 242 00:09:02,010 --> 00:09:04,010 Torque is rf sine theta. 243 00:09:05,490 --> 00:09:07,898 And r is the distance from the axis 244 00:09:07,898 --> 00:09:09,692 to the point where the force is applied. 245 00:09:09,692 --> 00:09:12,146 So since gravity's supplying the torque, 246 00:09:12,146 --> 00:09:14,554 that r would also be this L. 247 00:09:14,554 --> 00:09:17,194 It'd go from the axis to the point where gravity's applied, 248 00:09:17,194 --> 00:09:21,974 so I'd have L times the force of gravity times sine theta. 249 00:09:21,974 --> 00:09:25,132 So, you might say, look, if the length increases, 250 00:09:25,132 --> 00:09:27,018 so would the amount of torque. 251 00:09:27,018 --> 00:09:29,093 So, I've got more torque trying to make 252 00:09:29,093 --> 00:09:31,986 this thing move around, I've also got more inertia, 253 00:09:31,986 --> 00:09:33,983 so it's harder to move around. 254 00:09:33,983 --> 00:09:37,554 Do those offset like so many of these other things offset? 255 00:09:37,554 --> 00:09:38,458 They don't. 256 00:09:38,458 --> 00:09:40,274 Look it, this torque will increase 257 00:09:40,274 --> 00:09:42,203 but it only increases with L, 258 00:09:42,203 --> 00:09:44,034 it's only proportional to L. 259 00:09:44,034 --> 00:09:46,930 This moment of inertia's proportional to L squared. 260 00:09:46,930 --> 00:09:48,883 So, if you double the length, 261 00:09:48,883 --> 00:09:51,355 you've quadrupled how difficult it is to move 262 00:09:51,355 --> 00:09:53,968 this mass around but you've only doubled 263 00:09:53,968 --> 00:09:56,885 the ability of this torque to move it around. 264 00:09:56,885 --> 00:09:58,618 That's means it's gonna take longer 265 00:09:58,618 --> 00:10:00,139 to go through a whole cycle 266 00:10:00,139 --> 00:10:02,306 and that period is gonna increase. 267 00:10:02,306 --> 00:10:04,631 This larger torque is not gonna compensate 268 00:10:04,631 --> 00:10:07,042 for the fact that this mass is harder to move 269 00:10:07,042 --> 00:10:10,706 as there's more inertia to the rotation of this mass. 270 00:10:10,706 --> 00:10:13,035 Alright, so that's why increasing the length, 271 00:10:13,035 --> 00:10:14,426 increases the period. 272 00:10:14,426 --> 00:10:17,938 But why does increasing g, the gravitational acceleration, 273 00:10:17,938 --> 00:10:19,282 decrease the period? 274 00:10:19,282 --> 00:10:20,273 Well, think about it, 275 00:10:20,273 --> 00:10:22,843 if I increase the gravitational acceleration, 276 00:10:22,843 --> 00:10:25,096 so I take this pendulum to some planet 277 00:10:25,096 --> 00:10:27,322 that's extremely dense or massive 278 00:10:27,322 --> 00:10:30,007 and it's pulling down with a huge force of gravity. 279 00:10:30,007 --> 00:10:32,844 So, bigger g means a bigger force of gravity, 280 00:10:32,844 --> 00:10:35,174 pulling downward on this mass, 281 00:10:35,174 --> 00:10:37,746 that gives me a larger restoring force. 282 00:10:37,746 --> 00:10:39,826 So, a larger force means it's gonna pull 283 00:10:39,826 --> 00:10:41,490 this mass more quickly, 284 00:10:41,490 --> 00:10:43,580 it's gonna have larger acceleration, 285 00:10:43,580 --> 00:10:45,018 that means it's gonna have a larger speed, 286 00:10:45,018 --> 00:10:46,939 it's gonna move back and forth faster, 287 00:10:46,939 --> 00:10:48,194 and if it moves faster, 288 00:10:48,194 --> 00:10:50,599 it takes less time to complete a cycle. 289 00:10:50,599 --> 00:10:53,667 That's why increasing the gravitational acceleration 290 00:10:53,667 --> 00:10:57,474 increases the force and it decreases the period. 291 00:10:57,474 --> 00:10:59,354 Essentially, if you're cool with torque, 292 00:10:59,354 --> 00:11:00,794 if you know about torque, 293 00:11:00,794 --> 00:11:04,642 you increased the force that increases the torque 294 00:11:04,642 --> 00:11:07,162 which would increase the angular acceleration 295 00:11:07,162 --> 00:11:08,730 and it would take less time for this thing 296 00:11:08,730 --> 00:11:09,883 to go back and forth, 297 00:11:09,883 --> 00:11:11,878 that's why the period goes down 298 00:11:11,878 --> 00:11:14,916 if you increase the gravitational acceleration. 299 00:11:14,916 --> 00:11:15,786 Now, if you're really clever, 300 00:11:15,786 --> 00:11:17,851 you'll be like, wait a minute. 301 00:11:17,851 --> 00:11:21,258 This is just like the formula for the mass on a spring. 302 00:11:21,258 --> 00:11:22,946 If you take the period from a mass on a spring, 303 00:11:22,946 --> 00:11:27,834 it was two pi, square root, something over something, 304 00:11:27,834 --> 00:11:30,581 and the term on top for the mass on a spring 305 00:11:30,581 --> 00:11:33,567 was the mass that was connected to the spring, 306 00:11:33,567 --> 00:11:36,146 and the term on the bottom, was the spring constant. 307 00:11:36,146 --> 00:11:39,626 And so you might say, wait, this is the same idea. 308 00:11:39,626 --> 00:11:42,082 Increasing the mass is just increasing 309 00:11:42,082 --> 00:11:44,442 the inertia of that system. 310 00:11:44,442 --> 00:11:47,647 That's why it's taking longer to go through a cycle. 311 00:11:47,647 --> 00:11:48,655 Just like over here. 312 00:11:48,655 --> 00:11:52,116 Increasing the length is increasing the inertia, 313 00:11:52,116 --> 00:11:53,515 at least the rotational inertia, 314 00:11:53,515 --> 00:11:55,535 the moment of inertia of that system, 315 00:11:55,535 --> 00:11:57,957 so it takes longer to go through a cycle. 316 00:11:57,957 --> 00:12:00,539 And you might say, increasing the k value, 317 00:12:00,539 --> 00:12:03,068 that's increasing the force on the system, 318 00:12:03,068 --> 00:12:05,259 and if you increase the force on the system, 319 00:12:05,259 --> 00:12:07,667 you make that system have a larger acceleration, 320 00:12:07,667 --> 00:12:11,532 greater speeds takes less time to go through a period, 321 00:12:11,532 --> 00:12:13,998 that's why this force constant k for the spring 322 00:12:13,998 --> 00:12:16,419 appears on the bottom, same as this g. 323 00:12:16,419 --> 00:12:20,445 Increasing the g, increases the force on the system 324 00:12:20,445 --> 00:12:22,477 which gives you a larger acceleration, 325 00:12:22,477 --> 00:12:26,302 greater speeds takes less time to go through a period. 326 00:12:26,302 --> 00:12:28,158 So, these formulas are very similar 327 00:12:28,158 --> 00:12:29,646 and they're completely analogous. 328 00:12:29,646 --> 00:12:31,439 There's an inertia term on top, 329 00:12:31,439 --> 00:12:32,890 a force term on the bottom, 330 00:12:32,890 --> 00:12:35,635 and they both affect the period in the same way. 331 00:12:35,635 --> 00:12:37,130 One more thing you should notice, 332 00:12:37,130 --> 00:12:40,720 amplitude does not affect the period of a mass on a spring, 333 00:12:40,720 --> 00:12:43,144 and the amplitude, this theta maximum 334 00:12:43,144 --> 00:12:46,159 will not affect the period of a pendulum either, 335 00:12:46,159 --> 00:12:48,966 as long as your amplitudes are small. 336 00:12:48,966 --> 00:12:52,700 So, we've gotta assume we're in this small amplitude region 337 00:12:52,700 --> 00:12:54,733 where this mass on a string 338 00:12:54,733 --> 00:12:57,316 is acting like a simple harmonic oscillator. 339 00:12:57,316 --> 00:13:00,324 And if that's true for small angles, 340 00:13:00,324 --> 00:13:03,004 the amplitude does not affect the period 341 00:13:03,004 --> 00:13:05,320 of a pendulum just like amplitude doesn't affect 342 00:13:05,320 --> 00:13:07,356 the period of a mass on a spring. 343 00:13:07,356 --> 00:13:09,244 Let me tell you about one last thing here. 344 00:13:09,244 --> 00:13:10,978 This simple pendulum only acts 345 00:13:10,978 --> 00:13:14,124 like a simple harmonic oscillator for small angles. 346 00:13:14,124 --> 00:13:16,788 And that means this period formula for the pendulum 347 00:13:16,788 --> 00:13:19,365 is only true for small angles. 348 00:13:19,365 --> 00:13:21,804 But how small does the angle have to be? 349 00:13:21,804 --> 00:13:22,916 So, to give you an idea, 350 00:13:22,916 --> 00:13:24,366 let's say your theta maximum, 351 00:13:24,366 --> 00:13:28,644 this amplitude for how far back you pull this pendulum 352 00:13:28,644 --> 00:13:32,100 to start it, is, let's say, less than 20 degrees. 353 00:13:32,100 --> 00:13:34,988 If you pull it back less than 20 degrees, 354 00:13:34,988 --> 00:13:38,186 the amount that this formula is gonna be off by 355 00:13:38,186 --> 00:13:41,324 compared to the true period of the pendulum, 356 00:13:41,324 --> 00:13:43,137 is gonna be less than one per cent. 357 00:13:43,137 --> 00:13:45,300 So, this formula gets you really close 358 00:13:45,300 --> 00:13:47,988 to the true actual value of the pendulum. 359 00:13:47,988 --> 00:13:49,507 I mean, it's really close to being 360 00:13:49,507 --> 00:13:51,264 a simple harmonic oscillator here. 361 00:13:51,264 --> 00:13:54,395 And let's say the theta maximum was less than 40 degrees, 362 00:13:54,395 --> 00:13:57,494 you're still only gonna be off by less than three per cent. 363 00:13:57,494 --> 00:13:59,184 So, the value you get from this equation 364 00:13:59,184 --> 00:14:01,998 is only off from the true value by three per cent. 365 00:14:01,998 --> 00:14:04,638 And let's say your theta maximum was less than 70 degrees, 366 00:14:04,638 --> 00:14:06,607 you get all the way up to 70 degrees, 367 00:14:06,607 --> 00:14:08,687 the error that this formula's gonna be 368 00:14:08,687 --> 00:14:10,894 is still less than 10 per cent. 369 00:14:10,894 --> 00:14:14,068 So, not nearly as good but still not bad. 370 00:14:14,068 --> 00:14:16,169 So, this formula gives you the period of the pendulum. 371 00:14:16,169 --> 00:14:18,960 It works really well for small angles. 372 00:14:18,960 --> 00:14:20,358 As that angle gets bigger, 373 00:14:20,358 --> 00:14:22,107 the value you get from this formula 374 00:14:22,107 --> 00:14:25,447 will deviate from the true value by more and more. 375 00:14:25,447 --> 00:14:28,953 So, recapping, for small angles, i.e. small amplitudes, 376 00:14:28,953 --> 00:14:31,601 you could treat a pendulum as a simple harmonic oscillator, 377 00:14:31,601 --> 00:14:33,125 and if the amplitude is small, 378 00:14:33,125 --> 00:14:34,982 you can find the period of a pendulum 379 00:14:34,982 --> 00:14:37,159 using two pi root, L over g, 380 00:14:37,159 --> 00:14:39,077 where L is the length of the string, 381 00:14:39,077 --> 00:14:41,693 and g is the acceleration due to gravity 382 00:14:41,693 --> 00:00:00,000 at the location where the pendulum is swinging.