1
00:00:00,000 --> 00:00:00,780
 

2
00:00:00,780 --> 00:00:04,310
So where I left off in the last
video, I'd just rewritten

3
00:00:04,310 --> 00:00:05,440
the spring equation.

4
00:00:05,440 --> 00:00:08,900
And I just wrote force is
mass times acceleration.

5
00:00:08,900 --> 00:00:11,980
And I was in the process of
saying, well if x is a

6
00:00:11,980 --> 00:00:14,000
function of t, what's
acceleration?

7
00:00:14,000 --> 00:00:17,170
Well, velocity is this
derivative of x with respect

8
00:00:17,170 --> 00:00:17,790
to time, right?

9
00:00:17,790 --> 00:00:20,060
Your change in position
over change of time.

10
00:00:20,060 --> 00:00:23,350
And acceleration is the
derivative of velocity, or the

11
00:00:23,350 --> 00:00:25,635
second derivative of position.

12
00:00:25,635 --> 00:00:29,350
So you take the derivative
twice of x of t, right?

13
00:00:29,350 --> 00:00:35,930
So let's rewrite this equation
in those terms. Let me erase

14
00:00:35,930 --> 00:00:37,570
all this--I actually want to
keep all of this, just so we

15
00:00:37,570 --> 00:00:40,710
remember what we're talking
about this whole time.

16
00:00:40,710 --> 00:00:43,800
Let me see if I can
erase it cleanly.

17
00:00:43,800 --> 00:00:45,050
That's pretty good.

18
00:00:45,050 --> 00:00:48,260
 

19
00:00:48,260 --> 00:00:50,270
Let me erase all of this.

20
00:00:50,270 --> 00:00:56,600
 

21
00:00:56,600 --> 00:00:57,850
All of this.

22
00:00:57,850 --> 00:00:59,100
I'll even erase this.

23
00:00:59,100 --> 00:01:02,430
 

24
00:01:02,430 --> 00:01:05,280
That's pretty good, all right.

25
00:01:05,280 --> 00:01:08,350
Now back to work.

26
00:01:08,350 --> 00:01:11,070
So, we know that-- or hopefully
we know-- that

27
00:01:11,070 --> 00:01:13,390
acceleration is the second
derivative of x as

28
00:01:13,390 --> 00:01:14,060
a function of t.

29
00:01:14,060 --> 00:01:18,910
So we can rewrite this as
mass times the second

30
00:01:18,910 --> 00:01:21,290
derivative of x.

31
00:01:21,290 --> 00:01:23,870
So I'll write that as-- well,
I think the easiest notation

32
00:01:23,870 --> 00:01:26,940
would just be x prime prime.

33
00:01:26,940 --> 00:01:29,640
That's just the second
derivative of x as

34
00:01:29,640 --> 00:01:31,180
a function of t.

35
00:01:31,180 --> 00:01:33,530
I'll write the function
notation, just so you remember

36
00:01:33,530 --> 00:01:36,190
this is a function of time.

37
00:01:36,190 --> 00:01:43,520
Is equal to minus
k times x of t.

38
00:01:43,520 --> 00:01:46,080
And what you see here, what
I've just written, this is

39
00:01:46,080 --> 00:01:50,270
actually a differential
equation.

40
00:01:50,270 --> 00:01:51,650
And so what is a differential
equation?

41
00:01:51,650 --> 00:01:55,030
Well, it's an equation where,
in one expression, or in one

42
00:01:55,030 --> 00:01:57,560
equation, on both sides of
this, you not only have a

43
00:01:57,560 --> 00:02:01,100
function, but you have
derivatives of that function.

44
00:02:01,100 --> 00:02:05,670
And the solution to a
differential equation isn't

45
00:02:05,670 --> 00:02:06,880
just a number, right?

46
00:02:06,880 --> 00:02:10,250
A solution to equations that
we've done in the past are

47
00:02:10,250 --> 00:02:14,420
numbers, essentially, or a set
of numbers, or maybe a line.

48
00:02:14,420 --> 00:02:17,660
But the solution to differential
equations is

49
00:02:17,660 --> 00:02:20,250
actually going to be a function,
or a class of

50
00:02:20,250 --> 00:02:22,470
functions, or a set
of functions.

51
00:02:22,470 --> 00:02:25,700
So it'll take a little time to
get your head around it, but

52
00:02:25,700 --> 00:02:28,100
this is as good an example as
ever to be exposed to it.

53
00:02:28,100 --> 00:02:31,390
And we're not going to solve
this differential equation

54
00:02:31,390 --> 00:02:32,130
analytically.

55
00:02:32,130 --> 00:02:35,550
We're going to use our intuition
behind what we did

56
00:02:35,550 --> 00:02:37,710
earlier in the previous video.

57
00:02:37,710 --> 00:02:41,040
We're going to use that to guess
at what a solution to

58
00:02:41,040 --> 00:02:42,710
this differential equation is.

59
00:02:42,710 --> 00:02:46,070
And then, if it works out, then
we'll have a little bit

60
00:02:46,070 --> 00:02:46,640
more intuition.

61
00:02:46,640 --> 00:02:49,140
And then we'll actually know
what the position is, at any

62
00:02:49,140 --> 00:02:52,870
given time, of this mass
attached to the spring.

63
00:02:52,870 --> 00:02:53,700
So this is exciting.

64
00:02:53,700 --> 00:02:56,140
This is a differential
equation.

65
00:02:56,140 --> 00:02:58,580
When we drew the position-- our
intuition for the position

66
00:02:58,580 --> 00:03:01,070
over time-- our intuition tells
us that it's a cosine

67
00:03:01,070 --> 00:03:02,790
function, with amplitude A.

68
00:03:02,790 --> 00:03:08,130
So we said it's A cosine omega
t, where this is the angular

69
00:03:08,130 --> 00:03:11,400
velocity of-- well, I don't want
to go into that just yet,

70
00:03:11,400 --> 00:03:13,080
we'll get a little bit more
intuition in a second.

71
00:03:13,080 --> 00:03:17,490
And now, what we can do is,
let's test this expression--

72
00:03:17,490 --> 00:03:23,980
this function-- to see if it
satisfies this equation.

73
00:03:23,980 --> 00:03:25,230
Right?

74
00:03:25,230 --> 00:03:28,260
 

75
00:03:28,260 --> 00:03:39,880
If we say that x of t is equal
to A cosine of wt, what is the

76
00:03:39,880 --> 00:03:42,530
derivative of this?
x prime of t.

77
00:03:42,530 --> 00:03:45,450
And you could review
the derivative

78
00:03:45,450 --> 00:03:47,560
videos to remember this.

79
00:03:47,560 --> 00:03:50,100
Well, it's the derivative of the
inside, so it'll be that

80
00:03:50,100 --> 00:03:53,680
omega, times the
outside scalar.

81
00:03:53,680 --> 00:03:56,650
A omega.

82
00:03:56,650 --> 00:03:58,915
And then the derivative-- I'm
just doing the chain rule--

83
00:03:58,915 --> 00:04:01,700
the derivative of cosine of t is
minus sine of whatever's in

84
00:04:01,700 --> 00:04:03,070
the inside.

85
00:04:03,070 --> 00:04:04,780
I'll put the minus outside.

86
00:04:04,780 --> 00:04:11,040
So it's minus sine of wt.

87
00:04:11,040 --> 00:04:15,510
And then, if we want the second
derivative-- so that's

88
00:04:15,510 --> 00:04:17,120
x prime prime of t.

89
00:04:17,120 --> 00:04:20,899
 

90
00:04:20,899 --> 00:04:22,490
Let me do this in a different
color, just so it doesn't get

91
00:04:22,490 --> 00:04:23,430
monotonous.

92
00:04:23,430 --> 00:04:25,540
That's the derivative
of this, right?

93
00:04:25,540 --> 00:04:28,270
So what's the derivative
of-- these are just

94
00:04:28,270 --> 00:04:29,260
scalar values, right?

95
00:04:29,260 --> 00:04:30,570
These are just constants.

96
00:04:30,570 --> 00:04:32,870
So the derivative of the
inside is an omega.

97
00:04:32,870 --> 00:04:35,270
I multiply the omega times
the scalar constant.

98
00:04:35,270 --> 00:04:43,190
I get minus A omega squared.

99
00:04:43,190 --> 00:04:45,350
And then the derivative of
sine is just cosine.

100
00:04:45,350 --> 00:04:46,910
But the minus is still there,
because I had the minus to

101
00:04:46,910 --> 00:04:47,950
begin with.

102
00:04:47,950 --> 00:04:54,422
Minus cosine of omega t.

103
00:04:54,422 --> 00:04:56,560
Now let's see if this is true.

104
00:04:56,560 --> 00:05:07,840
So if this is true, I should
be able to say that m times

105
00:05:07,840 --> 00:05:10,740
the second derivative of x of
t, which is in this case is

106
00:05:10,740 --> 00:05:21,750
this, times minus Aw
squared cosine wt.

107
00:05:21,750 --> 00:05:31,430
That should be equal to minus k
times my original function--

108
00:05:31,430 --> 00:05:32,380
times x of t.

109
00:05:32,380 --> 00:05:34,020
And x of t is a cosine wt.

110
00:05:34,020 --> 00:05:37,210
 

111
00:05:37,210 --> 00:05:39,530
I'm running out of space.

112
00:05:39,530 --> 00:05:41,590
Hopefully you understand
what I'm saying.

113
00:05:41,590 --> 00:05:44,870
I just substituted x prime
prime, the second derivative,

114
00:05:44,870 --> 00:05:51,310
into this, and I just
substituted x of t, which I

115
00:05:51,310 --> 00:05:53,800
guess that's that, in here.

116
00:05:53,800 --> 00:05:55,070
And now I got this.

117
00:05:55,070 --> 00:05:56,570
And let me see if
I can rewrite.

118
00:05:56,570 --> 00:05:58,670
Maybe I can get rid of
the spring up here.

119
00:05:58,670 --> 00:05:59,510
I'm trying to look for space.

120
00:05:59,510 --> 00:06:00,930
I don't want to get rid of this,
because I think this

121
00:06:00,930 --> 00:06:04,200
gives us some intuition
of what we're doing.

122
00:06:04,200 --> 00:06:06,250
One of those days that I wish
I had a larger blackboard.

123
00:06:06,250 --> 00:06:10,680
 

124
00:06:10,680 --> 00:06:13,160
Erase the spring.

125
00:06:13,160 --> 00:06:16,490
Hopefully you can remember
that image in your mind.

126
00:06:16,490 --> 00:06:19,190
And actually, I can
erase that.

127
00:06:19,190 --> 00:06:21,945
I can erase that.

128
00:06:21,945 --> 00:06:24,600
I can erase all of this, just so
I have some space, without

129
00:06:24,600 --> 00:06:27,150
getting rid of that nice curve I
took the time to draw in the

130
00:06:27,150 --> 00:06:28,650
last video.

131
00:06:28,650 --> 00:06:29,750
Almost there.

132
00:06:29,750 --> 00:06:31,860
OK.

133
00:06:31,860 --> 00:06:33,110
Back to work.

134
00:06:33,110 --> 00:06:35,470
 

135
00:06:35,470 --> 00:06:37,420
Make sure my pen feels
right, OK.

136
00:06:37,420 --> 00:06:42,960
So all I did is I took-- we
said that by the spring

137
00:06:42,960 --> 00:06:46,900
constant, if you rewrite force
as mass times acceleration,

138
00:06:46,900 --> 00:06:47,580
you get this.

139
00:06:47,580 --> 00:06:49,340
Which is essentially a
differential equation, I just

140
00:06:49,340 --> 00:06:52,220
rewrote acceleration as
the second derivative.

141
00:06:52,220 --> 00:06:56,450
Then I took a guess, that this
is x of t, just based on our

142
00:06:56,450 --> 00:06:58,500
intuition of the drawing.

143
00:06:58,500 --> 00:06:59,680
I took a guess.

144
00:06:59,680 --> 00:07:01,430
And then I took the second
derivative of it.

145
00:07:01,430 --> 00:07:01,490
Right?

146
00:07:01,490 --> 00:07:04,160
This is the first derivative,
this is the second derivative.

147
00:07:04,160 --> 00:07:06,300
And then I substituted the
second derivative here, and I

148
00:07:06,300 --> 00:07:07,450
substituted the function here.

149
00:07:07,450 --> 00:07:08,770
And this is what I got.

150
00:07:08,770 --> 00:07:12,500
And so let me see if I can
simplify that a little bit.

151
00:07:12,500 --> 00:07:28,550
So if I rewrite there, I get
minus mAw squared cosine of wt

152
00:07:28,550 --> 00:07:37,720
is equal to minus
kA cosine of wt.

153
00:07:37,720 --> 00:07:38,900
Well it looks good so far.

154
00:07:38,900 --> 00:07:41,685
Let's see, we can get rid of the
minus signs on both sides.

155
00:07:41,685 --> 00:07:46,450
 

156
00:07:46,450 --> 00:07:47,680
Get rid of the A's
on both sides.

157
00:07:47,680 --> 00:07:47,750
Right?

158
00:07:47,750 --> 00:07:50,335
We can divide both sides by A.

159
00:07:50,335 --> 00:07:54,090
Let me do this in black, just
so it really erases it.

160
00:07:54,090 --> 00:07:57,780
So if we get rid of A on both
sides, we're left with that.

161
00:07:57,780 --> 00:08:01,210
 

162
00:08:01,210 --> 00:08:04,690
And then-- so let's see, we
have mw squared cosine of

163
00:08:04,690 --> 00:08:08,620
omega t is equal to k
cosine of omega t.

164
00:08:08,620 --> 00:08:12,660
So this equation holds
true if what is true?

165
00:08:12,660 --> 00:08:21,120
This equation holds true if mw
squared-- or omega squared, I

166
00:08:21,120 --> 00:08:21,520
think that's omega.

167
00:08:21,520 --> 00:08:24,796
I always forget my--
is equal to k.

168
00:08:24,796 --> 00:08:29,110
Or another way of saying
it, if omega squared is

169
00:08:29,110 --> 00:08:32,909
equal to k over m.

170
00:08:32,909 --> 00:08:40,308
Or, omega is equal to the
square root of k over m.

171
00:08:40,308 --> 00:08:41,219
So there we have it.

172
00:08:41,220 --> 00:08:44,080
We have figured out what
x of t has to be.

173
00:08:44,080 --> 00:08:47,650
We said that this differential
equation is true, if this is x

174
00:08:47,650 --> 00:08:49,840
of t, and omega is
equal to this.

175
00:08:49,840 --> 00:08:55,520
So now we've figured out the
actual function that describes

176
00:08:55,520 --> 00:08:58,350
the position of that spring
as a function of time.

177
00:08:58,350 --> 00:09:04,886
x of t is going to be equal to--
we were right about the

178
00:09:04,886 --> 00:09:07,590
A, and that's just intuition,
right, because the amplitude

179
00:09:07,590 --> 00:09:13,820
of this cosine function is A--
A cosine-- and instead of

180
00:09:13,820 --> 00:09:18,880
writing w, we can now write the
square root of k over m.

181
00:09:18,880 --> 00:09:26,060
The square root of k over m t.

182
00:09:26,060 --> 00:09:27,490
That to me is amazing.

183
00:09:27,490 --> 00:09:34,140
We have now, using not too
sophisticated calculus, solved

184
00:09:34,140 --> 00:09:34,980
a differential equation.

185
00:09:34,980 --> 00:09:38,860
And now can-- if you tell me at
5.8 seconds, where is x, I

186
00:09:38,860 --> 00:09:39,920
can tell you.

187
00:09:39,920 --> 00:09:42,180
And I just realized that I am
now running out of time, so I

188
00:09:42,180 --> 00:09:44,210
will see you in the
next video.

189
00:09:44,210 --> 00:00:00,000