1
00:00:00,442 --> 00:00:01,786
- [Instructor] Alright, so we
saw that you could represent

2
00:00:01,786 --> 00:00:04,418
the motion of a simple harmonic oscillator

3
00:00:04,418 --> 00:00:07,228
on a horizontal position graph
and it looked kinda cool.

4
00:00:07,228 --> 00:00:08,455
It looks something like this.

5
00:00:08,455 --> 00:00:10,687
And the amplitude of that motion,

6
00:00:10,687 --> 00:00:13,375
the maximum displacement from equilibrium

7
00:00:13,375 --> 00:00:15,011
on this graph was just represented

8
00:00:15,011 --> 00:00:17,900
by the maximum displacement
from equilibrium,

9
00:00:17,900 --> 00:00:19,147
it looked like this.

10
00:00:19,147 --> 00:00:21,701
And the period, which was the time it took

11
00:00:21,701 --> 00:00:24,917
for this entire process
to reset, capital T,

12
00:00:24,917 --> 00:00:27,557
is the period, is the
time it takes to reset

13
00:00:27,557 --> 00:00:29,389
was the time it takes to reset,

14
00:00:29,389 --> 00:00:31,309
which would be from peak to peak

15
00:00:31,309 --> 00:00:33,108
or from trough to trough

16
00:00:33,108 --> 00:00:35,921
or from any point to any analogous point

17
00:00:35,921 --> 00:00:38,836
on that cycle, this was the period T.

18
00:00:38,836 --> 00:00:42,132
And so, with a graph
that's a sine or a cosine,

19
00:00:42,132 --> 00:00:44,340
you could represent any motion you want.

20
00:00:44,340 --> 00:00:47,876
So, if you had some oscillator
that had a larger amplitude,

21
00:00:47,876 --> 00:00:50,532
you can imagine just stretching
this thing vertically,

22
00:00:50,532 --> 00:00:51,710
the period would stay the same,

23
00:00:51,710 --> 00:00:53,540
but you could stretch out the amplitude.

24
00:00:53,540 --> 00:00:56,005
Or, if you had something
with a larger period,

25
00:00:56,005 --> 00:00:58,036
you can imagine stretching
it out horizontally

26
00:00:58,036 --> 00:00:59,761
and leaving the amplitude the same,

27
00:00:59,761 --> 00:01:03,060
or stretch it both ways to
represent any oscillator

28
00:01:03,060 --> 00:01:04,974
you want, which is kinda cool.

29
00:01:04,974 --> 00:01:07,588
However, a lot of times you also need

30
00:01:07,588 --> 00:01:09,247
the equation, in other words,

31
00:01:09,247 --> 00:01:11,996
you might wanna know what
equation would describe

32
00:01:11,996 --> 00:01:13,100
this graph right here.

33
00:01:13,100 --> 00:01:15,966
What equation would
represent this graph here?

34
00:01:15,966 --> 00:01:17,777
First of all, what do I even mean by like,

35
00:01:17,777 --> 00:01:19,777
the equation for this graph?

36
00:01:19,777 --> 00:01:21,897
What I mean is that this
graph's representing

37
00:01:21,897 --> 00:01:23,537
the horizontal position, X,

38
00:01:23,537 --> 00:01:25,777
which is how far the
mass has been displaced

39
00:01:25,777 --> 00:01:29,069
this way from equilibrium,
as a function of time.

40
00:01:29,069 --> 00:01:30,738
So, we want a function that will be,

41
00:01:30,738 --> 00:01:34,915
alright, what is the value
of the position of this mass

42
00:01:34,915 --> 00:01:36,346
as a function of time?

43
00:01:36,346 --> 00:01:38,230
So, what would this equation be?

44
00:01:38,230 --> 00:01:39,428
Gonna be a function, in other words,

45
00:01:39,428 --> 00:01:42,483
you're gonna feed this
function anytime you want,

46
00:01:42,483 --> 00:01:44,113
and the function's gonna give you,

47
00:01:44,113 --> 00:01:46,499
it's gonna spit out a
value for the position,

48
00:01:46,499 --> 00:01:48,562
and that should represent
whatever position

49
00:01:48,562 --> 00:01:50,977
this graph is representing,
where the mass is at,

50
00:01:50,977 --> 00:01:52,611
because the graph should agree

51
00:01:52,611 --> 00:01:54,314
with what this function's gonna tell us.

52
00:01:54,314 --> 00:01:56,147
And this function would
tell us where the mass is

53
00:01:56,147 --> 00:01:57,378
at any given moment.

54
00:01:57,378 --> 00:01:58,721
So, what would this look like?

55
00:01:58,721 --> 00:02:01,370
Now, we saw, like, this is
a sine or a cosine, right?

56
00:02:01,370 --> 00:02:02,906
So, this is either a sine or a cosine.

57
00:02:02,906 --> 00:02:04,190
That's the first choice.

58
00:02:04,190 --> 00:02:06,762
Do we wanna pick sine or cosine?

59
00:02:06,762 --> 00:02:08,739
And what I always do, is I
just look at the beginning

60
00:02:08,739 --> 00:02:10,925
and I say, alright, in a T equals zero,

61
00:02:10,925 --> 00:02:12,831
this one's starting at a maximum.

62
00:02:12,831 --> 00:02:13,999
So, I wanna use cosine

63
00:02:13,999 --> 00:02:16,423
because cosine starts at a maximum

64
00:02:16,423 --> 00:02:19,023
and by starting at a maximum,

65
00:02:19,023 --> 00:02:21,943
I mean, think about it,
for a cosine of zero,

66
00:02:21,943 --> 00:02:23,839
if you remember your trig functions,

67
00:02:23,839 --> 00:02:26,471
cosine of zero is equal to one.

68
00:02:26,471 --> 00:02:29,679
And so, because this is as
big as cosine ever gets.

69
00:02:29,679 --> 00:02:32,948
Sine and cosine can only
ever get as big as one.

70
00:02:32,948 --> 00:02:34,495
This thing's starting at a maximum.

71
00:02:34,495 --> 00:02:37,871
So, cosine starts at a
maximum at T equals zero.

72
00:02:37,871 --> 00:02:41,050
This function here starts at
a maximum at T equals zero.

73
00:02:41,050 --> 00:02:42,719
I'm gonna wanna use cosine

74
00:02:42,719 --> 00:02:44,759
but I'm gonna have to add
a few elements in here.

75
00:02:44,759 --> 00:02:46,935
Just cosine alone isn't
gonna do it for me,

76
00:02:46,935 --> 00:02:49,226
because cosine only gets as big as one.

77
00:02:49,226 --> 00:02:51,267
This thing has to get as big as A,

78
00:02:51,267 --> 00:02:54,138
whatever A is, this thing
has to get that big.

79
00:02:54,138 --> 00:02:56,711
So, in other words, my
simple harmonic oscillators

80
00:02:56,711 --> 00:02:59,439
aren't always gonna have
an amplitude of one,

81
00:02:59,439 --> 00:03:01,323
so I need some variable in here

82
00:03:01,323 --> 00:03:04,047
that will represent what the amplitude is

83
00:03:04,047 --> 00:03:06,455
for that given simple harmonic oscillator.

84
00:03:06,455 --> 00:03:07,863
Let me make this less abstract.

85
00:03:07,863 --> 00:03:09,620
Let me just say, let's say we happened

86
00:03:09,620 --> 00:03:13,575
to pull this thing back 20
centimeters for .2 meters.

87
00:03:13,575 --> 00:03:14,455
So, let's say our amplitude

88
00:03:14,455 --> 00:03:16,607
for a particular simple
harmonic oscillator

89
00:03:16,607 --> 00:03:18,439
happened to be .2 meters,

90
00:03:18,439 --> 00:03:19,615
that would mean that this here,

91
00:03:19,615 --> 00:03:22,031
I can represent this here with .2 meters,

92
00:03:22,031 --> 00:03:23,687
this doesn't even make it to one.

93
00:03:23,687 --> 00:03:25,547
So, if I just left this as cosine,

94
00:03:25,547 --> 00:03:27,549
that would say this thing's
gonna get as big as one

95
00:03:27,549 --> 00:03:29,347
at some point in time and that's a lie.

96
00:03:29,347 --> 00:03:32,549
This thing only gets as big
as .2, so it's easy though.

97
00:03:32,549 --> 00:03:33,860
You might realize, if you clever,

98
00:03:33,860 --> 00:03:36,034
we'll just multiply
the front of this thing

99
00:03:36,034 --> 00:03:38,471
by the amplitude, whatever
the amplitude is multiplied,

100
00:03:38,471 --> 00:03:40,752
'cause then one times amplitude

101
00:03:40,752 --> 00:03:43,958
means that this X only gets
as big as the amplitude,

102
00:03:43,958 --> 00:03:45,101
which is exactly what I want.

103
00:03:45,101 --> 00:03:46,691
I want this thing to be as big

104
00:03:46,691 --> 00:03:48,981
as whatever the amplitude
is of the motion.

105
00:03:48,981 --> 00:03:50,102
And then there's one more piece,

106
00:03:50,102 --> 00:03:51,285
you can't, you might be, like,

107
00:03:51,285 --> 00:03:53,677
alright, we're done, I'm
just gonna stick cosine of T

108
00:03:53,677 --> 00:03:56,042
in here, that's not gonna work.

109
00:03:56,042 --> 00:03:58,010
We do want this to be a
function of time, right?

110
00:03:58,010 --> 00:03:59,875
We wanna be able to plug in a time

111
00:03:59,875 --> 00:04:01,764
and have this function spit out

112
00:04:01,764 --> 00:04:04,308
what is the value of the
position of the graph,

113
00:04:04,308 --> 00:04:05,650
and that would represent where is it.

114
00:04:05,650 --> 00:04:09,317
So, is it at .2, is it
at .1, is it at .045,

115
00:04:10,728 --> 00:04:11,561
or something like that?

116
00:04:11,561 --> 00:04:13,307
That's what this
function's supposed to do.

117
00:04:13,307 --> 00:04:14,920
But just plugging in T here,

118
00:04:14,920 --> 00:04:17,160
just having T alone, isn't gonna be good

119
00:04:17,160 --> 00:04:18,505
because that would mean, look at,

120
00:04:18,505 --> 00:04:21,507
a cosine of zero, we know cosine's one.

121
00:04:21,507 --> 00:04:23,745
When does cosine get back to one?

122
00:04:23,745 --> 00:04:25,338
That's gonna be when the inside,

123
00:04:25,338 --> 00:04:27,409
the argument in here, is two pi.

124
00:04:27,409 --> 00:04:29,601
So, we're gonna be using radians.

125
00:04:29,601 --> 00:04:31,528
You could use degrees if you wanted to,

126
00:04:31,528 --> 00:04:33,659
but most physicists and
professors and teachers

127
00:04:33,659 --> 00:04:36,049
are gonna be using radians for this case.

128
00:04:36,049 --> 00:04:39,208
So, a cosine of two pi would again be one

129
00:04:39,208 --> 00:04:41,793
because that's when, if you
remember your unit circle,

130
00:04:41,793 --> 00:04:43,299
that's when this function for cosine

131
00:04:43,299 --> 00:04:44,744
has gone around one whole time

132
00:04:44,744 --> 00:04:45,937
and it gets back where it started, right?

133
00:04:45,937 --> 00:04:49,030
So, if something rotates
through an angle of two pi,

134
00:04:49,030 --> 00:04:52,206
you've reset the whole thing
and that process has reset.

135
00:04:52,206 --> 00:04:55,206
But that would mean this function resets

136
00:04:55,206 --> 00:04:57,758
every two pi seconds, right?

137
00:04:57,758 --> 00:05:00,776
'Cause at T equals zero,
the function was one,

138
00:05:00,776 --> 00:05:04,835
and then at T equals two pi,
the function's one again.

139
00:05:04,835 --> 00:05:07,921
That would mean the period for cosine of T

140
00:05:07,921 --> 00:05:12,482
is two pi but our period isn't
necessarily two pi, right?

141
00:05:12,482 --> 00:05:14,608
Unless you got a really special case,

142
00:05:14,608 --> 00:05:16,390
the period is whatever the period is.

143
00:05:16,390 --> 00:05:17,998
Let's say it happened to be,

144
00:05:17,998 --> 00:05:20,201
let's say our period happened
to be like six seconds

145
00:05:20,201 --> 00:05:22,017
for this particular case.

146
00:05:22,017 --> 00:05:23,347
So, if this was six seconds,

147
00:05:23,347 --> 00:05:25,168
we would not want a function

148
00:05:25,168 --> 00:05:27,448
that resets after two pi seconds,

149
00:05:27,448 --> 00:05:29,432
we need a function that resets after,

150
00:05:29,432 --> 00:05:30,961
for this case, six seconds.

151
00:05:30,961 --> 00:05:32,257
So, how do we do that?

152
00:05:32,257 --> 00:05:34,200
Well, we have to not just have T in here.

153
00:05:34,200 --> 00:05:35,813
We saw that if we just have T,

154
00:05:35,813 --> 00:05:37,613
the period is always two pi,

155
00:05:37,613 --> 00:05:40,302
'cause that's when cosine of T resets.

156
00:05:40,302 --> 00:05:41,178
How would we do this?

157
00:05:41,178 --> 00:05:42,261
Well, we're gonna be clever.

158
00:05:42,261 --> 00:05:43,588
And if you're really clever you realize,

159
00:05:43,588 --> 00:05:46,325
alright, I'm just gonna add
a little variable in here.

160
00:05:46,325 --> 00:05:48,733
I'm just gonna a little
variable, boom, omega

161
00:05:48,733 --> 00:05:50,922
and then multiply that by T,

162
00:05:50,922 --> 00:05:54,298
and then I can tune this
omega however I want, right?

163
00:05:54,298 --> 00:05:56,476
If I can make omega big or small,

164
00:05:56,476 --> 00:06:00,067
I can make the period of this
function whatever I want.

165
00:06:00,067 --> 00:06:01,426
And if you're curious, you might be like,

166
00:06:01,426 --> 00:06:04,474
wait a minute, omega,
we've used that before,

167
00:06:04,474 --> 00:06:05,510
and you'd be right.

168
00:06:05,510 --> 00:06:06,826
Omega we have used before.

169
00:06:06,826 --> 00:06:08,605
That was the angular velocity

170
00:06:08,605 --> 00:06:10,802
and remember, angular
velocity was delta theta

171
00:06:10,802 --> 00:06:13,183
over delta T, the amount
of change in angle

172
00:06:13,183 --> 00:06:14,402
over the amount of change in time,

173
00:06:14,402 --> 00:06:16,162
which you might think isn't relevant here

174
00:06:16,162 --> 00:06:18,202
'cause this mass is just
going back and forth.

175
00:06:18,202 --> 00:06:21,047
This mass isn't actually
rotating in a circle.

176
00:06:21,047 --> 00:06:25,050
However, you can represent
repeating processes,

177
00:06:25,050 --> 00:06:28,036
cyclic processes, processes
that go through a cycle

178
00:06:28,036 --> 00:06:29,629
on a unit circle.

179
00:06:29,629 --> 00:06:32,106
So, in other words, let's say
you start right here, right?

180
00:06:32,106 --> 00:06:34,210
So, at T equals zero, you start,

181
00:06:34,210 --> 00:06:36,060
we pulled this mass
back and then we let go.

182
00:06:36,060 --> 00:06:36,893
So, we start right there.

183
00:06:36,893 --> 00:06:39,587
That would be right
here on this unit circle

184
00:06:39,587 --> 00:06:42,800
and then it flies through
the equilibrium point,

185
00:06:42,800 --> 00:06:45,051
that would be through
a quarter of a cycle,

186
00:06:45,051 --> 00:06:48,211
that means it would have
made it to right here.

187
00:06:48,211 --> 00:06:50,202
And then it makes its
way over to this edge,

188
00:06:50,202 --> 00:06:53,176
fully compresses this thing
that would be over to here,

189
00:06:53,176 --> 00:06:54,786
that would be through half a cycle,

190
00:06:54,786 --> 00:06:56,324
and that would come back through,

191
00:06:56,324 --> 00:06:57,730
let me find another dark color,

192
00:06:57,730 --> 00:06:59,778
it would come back through
the equilibrium point

193
00:06:59,778 --> 00:07:01,194
and that would be down here.

194
00:07:01,194 --> 00:07:02,989
And then we would get
back to the initial point

195
00:07:02,989 --> 00:07:04,636
and that would be one whole cycle.

196
00:07:04,636 --> 00:07:07,776
So, you can see how we can
represent cyclic processes

197
00:07:07,776 --> 00:07:11,199
on a unit circle and that's
how this makes sense.

198
00:07:11,199 --> 00:07:13,590
That might seem abstract
but it's really useful

199
00:07:13,590 --> 00:07:14,990
'cause watch what we could do.

200
00:07:14,990 --> 00:07:16,419
Naively you might think, alright,

201
00:07:16,419 --> 00:07:18,410
how would we even define this?

202
00:07:18,410 --> 00:07:21,102
Well, one cycle on a unit circle

203
00:07:21,102 --> 00:07:22,667
is two pi radians, right?

204
00:07:22,667 --> 00:07:25,702
If we're using radians, then
one cycle would be two pi

205
00:07:25,702 --> 00:07:27,718
'cause two pi is once around the circle.

206
00:07:27,718 --> 00:07:29,017
And how long does that take?

207
00:07:29,017 --> 00:07:31,220
Well, I know for a simple
harmonic oscillator,

208
00:07:31,220 --> 00:07:34,798
we defined the period
to be the time it takes

209
00:07:34,798 --> 00:07:35,868
for one whole cycle.

210
00:07:35,868 --> 00:07:37,893
So, we'd have two pi over the period

211
00:07:37,893 --> 00:07:41,500
and this is what you
would plug in down here.

212
00:07:41,500 --> 00:07:43,500
So, it turns out this does work.

213
00:07:43,500 --> 00:07:47,452
So, even naively, just using
our ideas of angular velocity,

214
00:07:47,452 --> 00:07:49,052
plugging in two pi over the period,

215
00:07:49,052 --> 00:07:51,252
will give us a function that resets

216
00:07:51,252 --> 00:07:53,124
exactly when we want it to.

217
00:07:53,124 --> 00:07:54,532
And you might not be convinced.

218
00:07:54,532 --> 00:07:56,598
And if that doesn't make
sense, I don't blame ya.

219
00:07:56,598 --> 00:07:57,492
I might be confused too.

220
00:07:57,492 --> 00:07:58,548
So, let me show you what I mean.

221
00:07:58,548 --> 00:08:01,070
In other words, we take this function,

222
00:08:01,070 --> 00:08:03,989
instead of writing omega,
we can just do this.

223
00:08:03,989 --> 00:08:06,740
We can just be like, alright, forget this,

224
00:08:06,740 --> 00:08:09,612
taken this, omega is the angular velocity,

225
00:08:09,612 --> 00:08:11,732
sometimes it's called
the angular frequency,

226
00:08:11,732 --> 00:08:14,468
in this case, so people use
different terminologies.

227
00:08:14,468 --> 00:08:17,964
You'll hear it as angular
velocity or angular frequency.

228
00:08:17,964 --> 00:08:20,884
If you take this angular
velocity or angular frequency,

229
00:08:20,884 --> 00:08:22,540
we just smack that right in here.

230
00:08:22,540 --> 00:08:24,484
So, we just put that in there for omega,

231
00:08:24,484 --> 00:08:26,300
and then multiply by T.

232
00:08:26,300 --> 00:08:28,340
Watch what happens, this is beautiful.

233
00:08:28,340 --> 00:08:30,732
So, if we take this, now it's gonna work.

234
00:08:30,732 --> 00:08:31,580
So, we multiply by T.

235
00:08:31,580 --> 00:08:33,048
T is our variable.

236
00:08:33,048 --> 00:08:36,044
So, little t is our variable,
two pi's the constant,

237
00:08:36,044 --> 00:08:38,611
the period capital T is also a constant,

238
00:08:38,611 --> 00:08:41,891
it'll be different for
different harmonic oscillators.

239
00:08:41,892 --> 00:08:43,980
But for a given harmonic oscillator,

240
00:08:43,980 --> 00:08:46,380
capital T the period is a constant.

241
00:08:46,380 --> 00:08:47,381
So, watch what happens now.

242
00:08:47,381 --> 00:08:50,930
At T equals zero, this
whole inside becomes zero.

243
00:08:50,930 --> 00:08:52,996
So, let's say I plug in T equals zero.

244
00:08:52,996 --> 00:08:55,129
We get to plug in little
t whatever we want.

245
00:08:55,129 --> 00:08:57,428
That is our variable,
so if I plug in little t

246
00:08:57,428 --> 00:09:00,869
equals zero, cosine of zero gives me one.

247
00:09:00,869 --> 00:09:01,914
But now what happens?

248
00:09:01,914 --> 00:09:05,313
If I plug in t equals, alright,
after one whole process,

249
00:09:05,313 --> 00:09:06,437
right, after one whole cycle,

250
00:09:06,437 --> 00:09:07,935
it's gone through one whole period,

251
00:09:07,935 --> 00:09:11,386
so if I plug in little t
as capital T, the period.

252
00:09:11,386 --> 00:09:12,237
Look what happens.

253
00:09:12,237 --> 00:09:15,026
This capital T cancels with that capital T

254
00:09:15,026 --> 00:09:16,580
and you just get two pi in here

255
00:09:16,580 --> 00:09:19,439
and the cosine of two pi is also one.

256
00:09:19,439 --> 00:09:21,558
That means this thing goes through a cycle

257
00:09:21,558 --> 00:09:23,819
every capital T, period.

258
00:09:23,819 --> 00:09:24,652
That's what we wanted.

259
00:09:24,652 --> 00:09:26,692
We didn't want something
that always had to have

260
00:09:26,692 --> 00:09:28,227
two pi as the period.

261
00:09:28,227 --> 00:09:30,172
Now we've got a function
that we can plug in

262
00:09:30,172 --> 00:09:32,072
whatever our period is down here.

263
00:09:32,072 --> 00:09:35,692
That way, whenever this little
t makes it to the period,

264
00:09:35,692 --> 00:09:39,819
capital T, this whole argument
in here becomes two pi

265
00:09:39,819 --> 00:09:41,564
and the cosine resets itself

266
00:09:41,564 --> 00:09:44,787
and you get a graph or a function
that will give you a graph

267
00:09:44,787 --> 00:09:48,492
that resets every period, which
is exactly what we wanted.

268
00:09:48,492 --> 00:09:50,328
So, in other words, to
make this less abstract,

269
00:09:50,328 --> 00:09:51,187
let's take this thing here,

270
00:09:51,187 --> 00:09:53,466
for this particular function here,

271
00:09:53,466 --> 00:09:56,252
for this particular choice
of amplitude and period,

272
00:09:56,252 --> 00:09:59,404
we could say that the graph
that's representing this,

273
00:09:59,404 --> 00:10:01,482
so the function that
would represent this here,

274
00:10:01,482 --> 00:10:03,591
instead of amplitude, we'd plug in .2.

275
00:10:03,591 --> 00:10:06,564
So, 0.2, let me try to fit it in here,

276
00:10:06,564 --> 00:10:09,883
0.2, I don't wanna put
the units down here,

277
00:10:09,883 --> 00:10:13,875
meters times cosine,
remember, we wanted cosine

278
00:10:13,875 --> 00:10:15,707
'cause it starts at a maximum

279
00:10:15,707 --> 00:10:17,675
and this graph started at a maximum.

280
00:10:17,675 --> 00:10:21,291
If it started down here
and went up, I'd use sine

281
00:10:21,291 --> 00:10:22,771
because sine starts at zero.

282
00:10:22,771 --> 00:10:24,799
But this one started at a maximum.

283
00:10:24,799 --> 00:10:26,415
And I have two pi over the period,

284
00:10:26,415 --> 00:10:28,601
I can't just leave that as period T,

285
00:10:28,601 --> 00:10:31,305
that's a little bit vague,
I'd put in my actual period

286
00:10:31,305 --> 00:10:32,730
and we said that the actual period

287
00:10:32,730 --> 00:10:35,206
for this mass on a spring was six seconds.

288
00:10:35,206 --> 00:10:37,462
And then little t, a lot of
times people get confused,

289
00:10:37,462 --> 00:10:39,797
they're like, alright, what
do I plug in for little t?

290
00:10:39,797 --> 00:10:42,118
You don't, typically,
like, if you just want

291
00:10:42,118 --> 00:10:45,246
the function for the position
as a function of time,

292
00:10:45,246 --> 00:10:47,686
you leave little t as the variable.

293
00:10:47,686 --> 00:10:50,558
That's the variable that you
have sitting here, right?

294
00:10:50,558 --> 00:10:52,238
If I wanted to know what is the value

295
00:10:52,238 --> 00:10:55,637
of the position of this
mass at nine seconds,

296
00:10:55,637 --> 00:10:57,586
I would plug in nine seconds.

297
00:10:57,586 --> 00:10:59,150
I would calculate this function

298
00:10:59,150 --> 00:11:00,366
with the nine seconds in there,

299
00:11:00,366 --> 00:11:02,665
that would be the
position at nine seconds.

300
00:11:02,665 --> 00:11:06,498
Or, if I wanted the
position at 12.25 seconds,

301
00:11:07,374 --> 00:11:11,118
I'd plug in 12.25 seconds
for our little t time,

302
00:11:11,118 --> 00:11:13,349
calculate this function,
plug it into the calculator

303
00:11:13,349 --> 00:11:15,150
in other words and that would give me

304
00:11:15,150 --> 00:11:17,582
the position at 12.25 seconds.

305
00:11:17,582 --> 00:11:19,694
That's what this function can do for you.

306
00:11:19,694 --> 00:11:21,816
That's how it can represent the motion

307
00:11:21,816 --> 00:11:23,622
of a simple harmonic oscillator.

308
00:11:23,622 --> 00:11:26,110
And now you might be like,
dude, that took a long time.

309
00:11:26,110 --> 00:11:27,539
Do they all take that long?

310
00:11:27,539 --> 00:11:29,718
No, once you get good at
this, it's really easy.

311
00:11:29,718 --> 00:11:30,790
Watch, let me get rid of all that.

312
00:11:30,790 --> 00:11:32,175
Let's say you got this problem on a test

313
00:11:32,175 --> 00:11:34,058
or a quiz or whatever, on homework,

314
00:11:34,058 --> 00:11:36,293
and it was like, hey, make an equation

315
00:11:36,293 --> 00:11:38,754
that describes this simple
harmonic oscillator.

316
00:11:38,754 --> 00:11:39,587
It's easy.

317
00:11:39,587 --> 00:11:42,502
First thing you do, do I
want to use sine or cosine?

318
00:11:42,502 --> 00:11:44,277
So, you might be like, oh, crud,

319
00:11:44,277 --> 00:11:46,199
it doesn't start at a maximum

320
00:11:46,199 --> 00:11:48,335
and it doesn't even start at
zero, sine would start there.

321
00:11:48,335 --> 00:11:50,100
It starts down here, but that's okay.

322
00:11:50,100 --> 00:11:51,173
It starts at a minimum.

323
00:11:51,173 --> 00:11:52,718
So, we're still gonna use cosine.

324
00:11:52,718 --> 00:11:55,022
So, we're gonna say that
X as a function of time

325
00:11:55,022 --> 00:11:57,204
is gonna be, well, what's the amplitude?

326
00:11:57,204 --> 00:11:59,712
The amplitude here is three meters.

327
00:11:59,712 --> 00:12:01,712
So, three meters is our amplitude

328
00:12:01,712 --> 00:12:03,650
because that's the maximum displacement

329
00:12:03,650 --> 00:12:04,960
from equilibrium, so I'm gonna have

330
00:12:04,960 --> 00:12:07,197
three meters out front

331
00:12:07,197 --> 00:12:08,546
and then I'm gonna do cosine

332
00:12:08,546 --> 00:12:10,464
because it starts at an extreme value,

333
00:12:10,464 --> 00:12:13,224
like either a maximum or a minimum value.

334
00:12:13,224 --> 00:12:17,279
Cosine of, and then I need
two pi over the period.

335
00:12:17,279 --> 00:12:18,348
What is my period?

336
00:12:18,348 --> 00:12:19,727
I look at my graph and I ask,

337
00:12:19,727 --> 00:12:21,209
how long does it take to reset?

338
00:12:21,209 --> 00:12:22,653
So, started down here at a minimum,

339
00:12:22,653 --> 00:12:24,828
when does it get back to a minimum?

340
00:12:24,828 --> 00:12:26,274
That took four seconds.

341
00:12:26,274 --> 00:12:28,193
So, four seconds would be the period,

342
00:12:28,193 --> 00:12:30,484
so it'd be two pi over four seconds

343
00:12:30,484 --> 00:12:33,339
and then little t, what
do I plug in for little t?

344
00:12:33,339 --> 00:12:34,172
I don't.

345
00:12:34,172 --> 00:12:35,907
This is the variable that sits there

346
00:12:35,907 --> 00:12:37,835
and waits for me to
plug in whatever I want.

347
00:12:37,835 --> 00:12:41,735
So, that's my variable little
t that X is a function of.

348
00:12:41,735 --> 00:12:42,845
But I'm not done.

349
00:12:42,845 --> 00:12:45,187
This would be a graph that starts up here

350
00:12:45,187 --> 00:12:47,115
and goes down like that.

351
00:12:47,115 --> 00:12:49,195
This graph starts down
here but that's easy.

352
00:12:49,195 --> 00:12:51,939
You just multiply by a
negative sine out front

353
00:12:51,939 --> 00:12:54,747
and you've turned your
cosine into negative cosine

354
00:12:54,747 --> 00:12:57,334
and negative cosine starts down here.

355
00:12:57,334 --> 00:13:00,237
So, note our amplitude is still three.

356
00:13:00,237 --> 00:13:02,770
If the question asked,
what is the amplitude?

357
00:13:02,770 --> 00:13:06,033
The amplitude is the
magnitude of the displacement,

358
00:13:06,033 --> 00:13:09,515
maximum displacement, so that's
still positive three meters,

359
00:13:09,515 --> 00:13:11,452
even though it started down here,

360
00:13:11,452 --> 00:13:14,139
but you could just include
an extra negative out front

361
00:13:14,139 --> 00:13:16,347
that essentially goes
along with the cosine.

362
00:13:16,347 --> 00:13:18,273
That would give you negative cosine

363
00:13:18,273 --> 00:13:20,471
and there you have it, that
would be your function.

364
00:13:20,471 --> 00:13:22,099
So, keep in mind, it's good to remember,

365
00:13:22,099 --> 00:13:23,432
if you start up here,

366
00:13:23,432 --> 00:13:25,214
you're gonna wanna use cosine.

367
00:13:25,214 --> 00:13:26,652
If you start down here,

368
00:13:26,652 --> 00:13:28,870
you gonna wanna to use negative cosine.

369
00:13:28,870 --> 00:13:31,036
If you start right here,

370
00:13:31,036 --> 00:13:32,642
you're gonna wanna use sine.

371
00:13:32,642 --> 00:13:35,654
If you start here and go
up, that's gonna be sine.

372
00:13:35,654 --> 00:13:37,356
And if you start here and go down,

373
00:13:37,356 --> 00:13:39,029
that's gonna be negative sine.

374
00:13:39,029 --> 00:13:40,928
That's what those functions look like.

375
00:13:40,928 --> 00:13:42,724
So, recapping, you could use this equation

376
00:13:42,724 --> 00:13:45,239
to represent the motion of
a simple harmonic oscillator

377
00:13:45,239 --> 00:13:47,199
which is always gonna be plus or minus

378
00:13:47,199 --> 00:13:50,276
the amplitude, times either sine or cosine

379
00:13:50,276 --> 00:13:53,585
of two pi over the period times the time.

380
00:13:53,585 --> 00:13:55,253
This two pi over the period

381
00:13:55,253 --> 00:13:58,724
is representing the angular
frequency or angular velocity

382
00:13:58,724 --> 00:14:01,092
and you would choose positive cosine

383
00:14:01,092 --> 00:14:02,544
if you started at a max,

384
00:14:02,544 --> 00:14:05,284
negative cosine if you started at a min.

385
00:14:05,284 --> 00:14:09,188
Positive sine if you start at
zero or equilibrium and go up.

386
00:14:09,188 --> 00:00:00,000
Negative sine if you start
at equilibrium and go down.