1
00:00:03,650 --> 00:00:05,970
So when we analyzed Newton's
Second Law-- applied

2
00:00:05,970 --> 00:00:08,650
to this compound system--
we had two equations

3
00:00:08,650 --> 00:00:13,369
for Object 1 and Object 2.

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00:00:13,369 --> 00:00:15,410
And what we found is that
we had three unknowns--

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00:00:15,410 --> 00:00:18,070
the tension in the string and
the accelerations of the two

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00:00:18,070 --> 00:00:18,830
objects.

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00:00:18,830 --> 00:00:21,590
Now, how do we
solve this system?

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00:00:21,590 --> 00:00:23,540
Well, we're missing
one condition, which

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00:00:23,540 --> 00:00:24,960
is a constraint condition.

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00:00:24,960 --> 00:00:28,310
Which is, as Object 2
moves and Object 1 moves,

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00:00:28,310 --> 00:00:31,060
they have to move in some
relationship together.

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00:00:31,060 --> 00:00:32,560
And now what we'd
like to do is show

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00:00:32,560 --> 00:00:36,470
an analytic approach for finding
that constraint condition.

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00:00:36,470 --> 00:00:39,270
And the way we think
about it is that,

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00:00:39,270 --> 00:00:43,870
we'll call l equal to
the length of the string,

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00:00:43,870 --> 00:00:48,540
and this quantity is a constant.

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00:00:48,540 --> 00:00:50,510
And what we'd like
to do is introduce

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00:00:50,510 --> 00:00:53,520
coordinate functions for
our Object 1 and our Object

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00:00:53,520 --> 00:00:57,170
2 and express l in terms of
those coordinate functions.

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00:00:57,170 --> 00:01:01,190
And then, take two derivatives
of l, set that equal to 0

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00:01:01,190 --> 00:01:03,020
because it's a
constant, and that

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00:01:03,020 --> 00:01:05,710
will give us a relationship
between the accelerations

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00:01:05,710 --> 00:01:07,420
of Objects 1 and 2.

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00:01:07,420 --> 00:01:08,900
Now, this is a
little bit tricky.

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00:01:08,900 --> 00:01:11,220
And so, what we want to
do is, very carefully,

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00:01:11,220 --> 00:01:13,800
show how we introduce
coordinate functions.

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00:01:13,800 --> 00:01:20,390
Recall that we had j hat 1
down and j hat 2 downwards.

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00:01:20,390 --> 00:01:24,110
What that implies is that
we're choosing some origin

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00:01:24,110 --> 00:01:27,650
and we're-- let's choose
an origin up here--

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00:01:27,650 --> 00:01:30,240
and for a coordinate
function for Object 1,

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00:01:30,240 --> 00:01:34,771
it has to be consistent with our
choice of what we mean by j hat

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00:01:34,771 --> 00:01:35,380
1.

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00:01:35,380 --> 00:01:39,610
So, in this sense, y1
is a positive quantity

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00:01:39,610 --> 00:01:42,940
when we're going downward.

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00:01:42,940 --> 00:01:45,110
Now, what about
coordinate functions

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00:01:45,110 --> 00:01:47,229
for the other objects
in the system?

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00:01:47,229 --> 00:01:51,020
Well, let's look at
a few things first.

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00:01:51,020 --> 00:01:53,020
This is a fixed
distance-- we'll call

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00:01:53,020 --> 00:02:01,760
it s1-- between the ceiling
and the center of the pulley.

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00:02:01,760 --> 00:02:06,160
And let's make each of
the pulleys a radius, r.

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00:02:06,160 --> 00:02:12,540
And let's call this
a function y b.

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00:02:12,540 --> 00:02:17,070
And let's make this y2 of t.

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00:02:17,070 --> 00:02:19,470
So now we have coordinate
functions for 2

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00:02:19,470 --> 00:02:21,120
and coordinate functions for 1.

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00:02:21,120 --> 00:02:26,100
And again, recall that
this distance here-- s2--

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00:02:26,100 --> 00:02:28,870
is a fixed distance.

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00:02:28,870 --> 00:02:31,600
And when we define these
coordinate functions

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00:02:31,600 --> 00:02:35,650
in this fashion, we know that
the second derivative-- d

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00:02:35,650 --> 00:02:41,100
squared y1 dt squared--
this is precisely what we

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00:02:41,100 --> 00:02:44,510
mean by the acceleration
of Object 1.

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00:02:44,510 --> 00:02:46,430
And in the similar
fashion, d squared

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00:02:46,430 --> 00:02:53,130
y2 dt squared is what we mean
by the acceleration of Object 2.

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00:02:53,130 --> 00:02:55,040
So we've introduced
a coordinate system,

54
00:02:55,040 --> 00:02:58,540
we've made it very clear what we
mean by the accelerations of a1

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00:02:58,540 --> 00:03:02,460
and a2, and now let's look
at our constraint condition

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00:03:02,460 --> 00:03:05,090
that the length of the
string is constant.

57
00:03:05,090 --> 00:03:07,510
So what we're going
to do is try to see

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00:03:07,510 --> 00:03:12,190
if we can express the length
of this string in terms of all

59
00:03:12,190 --> 00:03:14,060
the coordinate
functions and some

60
00:03:14,060 --> 00:03:16,660
of these ancillary quantities.

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00:03:16,660 --> 00:03:22,160
So what we have here is that
the length of the string

62
00:03:22,160 --> 00:03:26,820
is y b going down here.

63
00:03:26,820 --> 00:03:30,980
So the length of the
string has a factor y b.

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00:03:30,980 --> 00:03:36,750
It wraps around Pulley
b-- so that's pi R--

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00:03:36,750 --> 00:03:40,720
and it goes up to
this length here.

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00:03:40,720 --> 00:03:46,520
Now, this length
is y b minus s1.

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00:03:46,520 --> 00:03:51,290
So that's y b minus s1.

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00:03:51,290 --> 00:03:56,420
We wrap around the pulley
again-- that's pi R.

69
00:03:56,420 --> 00:04:05,880
And now, we have this length
here, which is y1 minus s1.

70
00:04:05,880 --> 00:04:07,910
s1?

71
00:04:07,910 --> 00:04:09,140
Yeah.

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00:04:09,140 --> 00:04:13,120
And let's just make sure we
have all of our quantities here.

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00:04:13,120 --> 00:04:17,230
Now, we also have another
constraint condition,

74
00:04:17,230 --> 00:04:20,790
that this length of
the string-- we're

75
00:04:20,790 --> 00:04:27,760
going to call this l1-- we
have a second string here, l2.

76
00:04:27,760 --> 00:04:38,770
And l2 is given by y2-- this
length is y2 minus y b--

77
00:04:38,770 --> 00:04:42,710
and that was what we
called this constant, s2.

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00:04:42,710 --> 00:04:48,972
So we now have two string
lengths-- l2 equals y2 minus y

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00:04:48,972 --> 00:04:54,070
b-- and both of these
string lengths are constant.

80
00:04:54,070 --> 00:04:55,990
And we have the following facts.

81
00:04:55,990 --> 00:04:59,460
Let's start with this one first,
that the second derivative

82
00:04:59,460 --> 00:05:02,830
of l2 dt squared-- because
the length of the string

83
00:05:02,830 --> 00:05:05,620
is a constant, that's 0.

84
00:05:05,620 --> 00:05:08,810
And that tells us, two
derivatives of this

85
00:05:08,810 --> 00:05:13,540
is a2 and two derivatives
of that is a b.

86
00:05:13,540 --> 00:05:18,270
And this is something that we
saw before, that block b and 2

87
00:05:18,270 --> 00:05:20,290
are moving together.

88
00:05:20,290 --> 00:05:23,620
So when we treated the system
as just b and 2 together,

89
00:05:23,620 --> 00:05:25,920
we see that the
acceleration of 2

90
00:05:25,920 --> 00:05:28,520
and the acceleration of
Pulley b are the same.

91
00:05:28,520 --> 00:05:32,200
So we could have just
said that before we began.

92
00:05:32,200 --> 00:05:38,530
Now, let's put these equations
aside for the moment.

93
00:05:38,530 --> 00:05:47,900
And now let's consider taking
two derivatives of String 1.

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00:05:47,900 --> 00:05:51,960
Recall, String 1 is
this object here.

95
00:05:51,960 --> 00:05:58,970
We'll call this String
1 and this String 2.

96
00:05:58,970 --> 00:06:01,600
Now, again, if we take
two derivatives-- let's

97
00:06:01,600 --> 00:06:04,590
look at our
expression first-- we

98
00:06:04,590 --> 00:06:06,940
see that we have
two factors of y b

99
00:06:06,940 --> 00:06:09,580
and we have a bunch of
constants whose derivative is 0.

100
00:06:09,580 --> 00:06:12,850
So we don't have to worry about
the wrap around distances--

101
00:06:12,850 --> 00:06:17,150
the pi R's-- we don't have to
worry about the constants-- s1.

102
00:06:17,150 --> 00:06:19,140
All we have to think
about is which quantities

103
00:06:19,140 --> 00:06:20,540
are changing in time.

104
00:06:20,540 --> 00:06:25,130
So we have 2 a b when
we take two derivatives

105
00:06:25,130 --> 00:06:29,650
and we have one factor
of a1-- and that's 0.

106
00:06:29,650 --> 00:06:35,450
And now, because the
block and 2 and Pulley b

107
00:06:35,450 --> 00:06:38,820
are accelerating together,
we have our condition,

108
00:06:38,820 --> 00:06:48,590
which is 2 a2 plus a1 is 0-- or
that a1 is equal to minus 2 a2.

109
00:06:48,590 --> 00:06:53,180
And that is the extra
constraint condition

110
00:06:53,180 --> 00:06:57,770
that will enable us to solve
the system of equations.