1
00:00:00,313 --> 00:00:03,663
- So just to show you how
powerful this approach is of

2
00:00:03,663 --> 00:00:06,754
treating multiple objects as
if they were a single mass

3
00:00:06,754 --> 00:00:09,317
let's look at this one,
this would be a hard one.

4
00:00:09,317 --> 00:00:11,854
We've got a 9kg mass hanging from a rope

5
00:00:11,854 --> 00:00:14,429
that rope passes over a
pulley then it's connected

6
00:00:14,429 --> 00:00:17,708
to a 4kg mass sitting on an incline.

7
00:00:17,708 --> 00:00:19,514
And this incline is at 30 degrees,

8
00:00:19,514 --> 00:00:21,670
and let's step it up let's make it hard,

9
00:00:21,670 --> 00:00:24,242
let's say the coefficient
of kinetic friction

10
00:00:24,242 --> 00:00:27,825
between the incline
and the 4kg mass is 0.2

11
00:00:29,265 --> 00:00:30,623
And that's the coefficient.

12
00:00:30,623 --> 00:00:32,517
so there's going to be friction as well.

13
00:00:32,517 --> 00:00:34,644
If you tried to solve this the hard way it

14
00:00:34,644 --> 00:00:36,615
would be challenging, it's do-able

15
00:00:36,615 --> 00:00:38,453
but you're going to
have multiple equations

16
00:00:38,453 --> 00:00:41,625
with multiple unknowns,
if you try to analyze each

17
00:00:41,625 --> 00:00:45,149
box separately using Newton's second law.

18
00:00:45,149 --> 00:00:49,250
But because these boxes have
to accelerate at the same rate

19
00:00:49,250 --> 00:00:52,126
well at least the same
magnitude of acceleration,

20
00:00:52,126 --> 00:00:54,942
then we're just going to be
able to find the system's

21
00:00:54,942 --> 00:00:59,530
acceleration, at least the
magnitude of it, the size of it.

22
00:00:59,530 --> 00:01:02,230
This 4 kg mass is going to
have acceleration in this way

23
00:01:02,230 --> 00:01:04,909
of a certain magnitude, and this 9 kg mass

24
00:01:04,909 --> 00:01:06,213
is going to have acceleration this way

25
00:01:06,213 --> 00:01:08,810
and because our rope is not
going to break or stretch,

26
00:01:08,810 --> 00:01:10,906
these accelerations are
going to have to be the same.

27
00:01:10,906 --> 00:01:12,369
So we get to use this
trick where we treat these

28
00:01:12,369 --> 00:01:16,084
multiple objects as if
they are a single mass.

29
00:01:16,084 --> 00:01:18,464
And the acceleration of the single mass

30
00:01:18,464 --> 00:01:22,426
only depends on the external
forces on that mass.

31
00:01:22,426 --> 00:01:24,458
So we're only looking
at the external forces,

32
00:01:24,458 --> 00:01:27,009
and we're gonna divide by the total mass.

33
00:01:27,009 --> 00:01:28,460
So what would that be?

34
00:01:28,460 --> 00:01:31,975
If we wanted to find
the acceleration of this

35
00:01:31,975 --> 00:01:36,142
4 kg mass, let's say what the
magnitude of this acceleration

36
00:01:37,321 --> 00:01:40,185
This 9 kg mass is much more massive than

37
00:01:40,185 --> 00:01:43,021
the 4 kg mass and so this whole system

38
00:01:43,021 --> 00:01:46,312
is going to accelerate in that direction,

39
00:01:46,312 --> 00:01:48,913
let's just call that direction positive.

40
00:01:48,913 --> 00:01:50,503
So that's one weird part about treating

41
00:01:50,503 --> 00:01:53,185
multiple objects as if
they're a single mass

42
00:01:53,185 --> 00:01:55,911
is defining the direction
which is positive

43
00:01:55,911 --> 00:01:59,161
is a little bit sketchy to some people.

44
00:02:00,657 --> 00:02:02,359
We're just saying the direction of motion

45
00:02:02,359 --> 00:02:05,180
this way is what we're calling positive.

46
00:02:05,180 --> 00:02:07,989
And that works just
fine, so when I plug in

47
00:02:07,989 --> 00:02:11,917
and go to solve for
what is the acceleration

48
00:02:11,917 --> 00:02:14,323
I'm gonna plug in forces which go this way

49
00:02:14,323 --> 00:02:18,490
as positive and forces which
go the other way as negative.

50
00:02:19,332 --> 00:02:22,125
What do I plug in up top?
What forces make this go?

51
00:02:22,125 --> 00:02:25,692
The force of gravity on
this 9 kg mass is driving

52
00:02:25,692 --> 00:02:29,065
this system, this is the
force which makes the

53
00:02:29,065 --> 00:02:31,787
whole system move if I were
to just let go of these masses

54
00:02:31,787 --> 00:02:35,160
it would start accelerating
this way because of

55
00:02:35,160 --> 00:02:37,404
this force of gravity right here.

56
00:02:37,404 --> 00:02:40,614
So that's going to be 9 kg times

57
00:02:40,614 --> 00:02:45,197
9.8 meters per second
squared and that's going

58
00:02:45,197 --> 00:02:48,982
to be positive because
it's making the system go.

59
00:02:48,982 --> 00:02:51,992
There's no other forces
that make this system go.

60
00:02:51,992 --> 00:02:54,125
So now I'm only going to
subtract forces that resist

61
00:02:54,125 --> 00:02:58,066
the acceleration, what forces
resist the acceleration?

62
00:02:58,066 --> 00:03:02,089
The gravity of this 4 kg
mass resists acceleration,

63
00:03:02,089 --> 00:03:04,162
but not all of the gravity.

64
00:03:04,162 --> 00:03:06,405
The gravity of this 4
kg mass points straight

65
00:03:06,405 --> 00:03:10,155
down, but it's only
this component this way

66
00:03:10,155 --> 00:03:14,666
which resists the motion of
this system in this direction.

67
00:03:14,666 --> 00:03:15,824
What is this component?

68
00:03:15,824 --> 00:03:19,226
This is "m" "g" "sin(theta)"
so if that doesn't

69
00:03:19,226 --> 00:03:21,646
make any sense go back and look
at the videos about inclines

70
00:03:21,646 --> 00:03:25,434
or the article on inclines
and you'll see the component

71
00:03:25,434 --> 00:03:27,642
of gravity that points down an incline

72
00:03:27,642 --> 00:03:29,500
parallel to the surface is equal

73
00:03:29,500 --> 00:03:31,143
to "m" "g" "sin(theta)" so I'm gonna

74
00:03:31,143 --> 00:03:34,114
have to subtract 4 kg times

75
00:03:34,114 --> 00:03:36,364
4 kg times 9.8 which is "g"

76
00:03:40,505 --> 00:03:44,172
times sin of the angle,
which is 30 degrees.

77
00:03:47,537 --> 00:03:50,014
We need more room up here
because there are more

78
00:03:50,014 --> 00:03:52,301
forces that try to prevent the system

79
00:03:52,301 --> 00:03:55,207
from moving, there's one more
force, the force of friction

80
00:03:55,207 --> 00:03:57,024
is going to try to prevent this system

81
00:03:57,024 --> 00:03:58,402
from moving and that force of friction

82
00:03:58,402 --> 00:04:01,778
is gonna also point in this direction.

83
00:04:01,778 --> 00:04:04,492
It's not equal to "m" "g" "sin(theta)"

84
00:04:04,492 --> 00:04:08,075
it's equal to the force
of kinetic friction

85
00:04:09,269 --> 00:04:12,269
"mu" "k" times "Fn" and the "mu" "k"

86
00:04:13,409 --> 00:04:17,766
is going to be 0.2, you
have to be careful because

87
00:04:17,766 --> 00:04:20,418
the "Fn" is not just equal to "m" "g"

88
00:04:20,418 --> 00:04:23,446
the reason is that on an
incline the normal force

89
00:04:23,446 --> 00:04:26,157
points this way so the
normal force doesn't

90
00:04:26,157 --> 00:04:28,931
have to counteract all
of gravity on an incline

91
00:04:28,931 --> 00:04:31,906
it just has to counteract
that component of gravity

92
00:04:31,906 --> 00:04:34,790
that's directed
perpendicular to the incline

93
00:04:34,790 --> 00:04:38,588
and that happens to be
"m" "g" "cos(theta)"

94
00:04:38,588 --> 00:04:41,476
for an object on an
incline and if that makes

95
00:04:41,476 --> 00:04:43,894
no sense go back and look
at the video on inclines

96
00:04:43,894 --> 00:04:46,039
or look at the article on inclines and

97
00:04:46,039 --> 00:04:48,668
you'll see that this component of gravity

98
00:04:48,668 --> 00:04:51,635
pointing into the
surface is "m" "g" cosine

99
00:04:51,635 --> 00:04:55,007
that means that normal
force is "m" "g" cosine.

100
00:04:55,007 --> 00:04:57,527
Because there's no acceleration in this

101
00:04:57,527 --> 00:05:00,043
perpendicular direction
and I have to multiply

102
00:05:00,043 --> 00:05:02,566
by 0.2 because I'm not
really plugging in the

103
00:05:02,566 --> 00:05:04,995
normal force up here
or the force of gravity

104
00:05:04,995 --> 00:05:07,349
in this perpendicular direction.

105
00:05:07,349 --> 00:05:10,246
I'm plugging in the
kinetic frictional force

106
00:05:10,246 --> 00:05:14,646
this 0.2 turns this
perpendicular force into

107
00:05:14,646 --> 00:05:16,962
this parallel force,
so I'm plugging in the

108
00:05:16,962 --> 00:05:18,817
force of kinetic friction and it just

109
00:05:18,817 --> 00:05:22,059
so happens that it depends
on the normal force.

110
00:05:22,059 --> 00:05:23,611
That's why I'm plugging that in,

111
00:05:23,611 --> 00:05:26,944
I'm gonna need a negative 0.2 times 4 kg

112
00:05:30,148 --> 00:05:33,148
times 9.8 meters per second squared.

113
00:05:36,793 --> 00:05:40,090
And then I need to multiply
by cosine of the angle

114
00:05:40,090 --> 00:05:42,935
in this case the angle is 30 degrees.

115
00:05:42,935 --> 00:05:45,927
Alright, now finally I divide by my

116
00:05:45,927 --> 00:05:48,481
total mass because I have no other forces

117
00:05:48,481 --> 00:05:50,525
trying to propel this system or to make it

118
00:05:50,525 --> 00:05:54,069
stop and my total mass
is going to be 13 kg.

119
00:05:54,069 --> 00:05:56,457
You might object and think wait a minute,

120
00:05:56,457 --> 00:05:58,660
there's other forces
here like this tension

121
00:05:58,660 --> 00:06:01,403
going this way, why don't we include that?

122
00:06:01,403 --> 00:06:03,720
Well that's internal force and the whole

123
00:06:03,720 --> 00:06:05,844
benefit and appeal of treating this

124
00:06:05,844 --> 00:06:08,871
two-mass system as if
it were a single mass

125
00:06:08,871 --> 00:06:10,944
is that we don't have to worry about these

126
00:06:10,944 --> 00:06:15,094
internal forces, it's
there but that tension

127
00:06:15,094 --> 00:06:17,665
is also over here and on this side it's

128
00:06:17,665 --> 00:06:19,595
resisting the motion because it's pointing

129
00:06:19,595 --> 00:06:21,688
opposite the directional motion.

130
00:06:21,688 --> 00:06:23,891
On this side it's helping the motion,

131
00:06:23,891 --> 00:06:25,546
it's an internal force the internal force

132
00:06:25,546 --> 00:06:27,313
is canceled that's why we don't care about

133
00:06:27,313 --> 00:06:29,734
them, that's what this
trick allows us to do

134
00:06:29,734 --> 00:06:31,885
by treating this two-mass
system as a single

135
00:06:31,885 --> 00:06:34,900
object we get to neglect
any internal forces

136
00:06:34,900 --> 00:06:38,592
because internal forces
always cancel on that object.

137
00:06:38,592 --> 00:06:42,331
So if we just solve
this now and calculate,

138
00:06:42,331 --> 00:06:45,414
we get 4.75 meters per second squared

139
00:06:48,037 --> 00:06:50,153
is the acceleration of this system.

140
00:06:50,153 --> 00:06:53,270
So this 4 kg mass will
accelerate up the incline

141
00:06:53,270 --> 00:06:55,578
parallel to it with an acceleration

142
00:06:55,578 --> 00:06:58,309
of 4.75 meters per second squared.

143
00:06:58,309 --> 00:07:01,092
This 9 kg mass will accelerate downward

144
00:07:01,092 --> 00:07:05,086
with a magnitude of 4.75
meters per second squared.

145
00:07:05,086 --> 00:07:06,976
Remember if you're going to then go

146
00:07:06,976 --> 00:07:09,309
try to find out what one of these internal

147
00:07:09,309 --> 00:07:12,572
forces are, we neglected them because

148
00:07:12,572 --> 00:07:14,818
we treated this as a single mass.

149
00:07:14,818 --> 00:07:16,107
But you could ask the question,

150
00:07:16,107 --> 00:07:17,895
what is the size of this tension?

151
00:07:17,895 --> 00:07:20,049
Often that's like a part two because

152
00:07:20,049 --> 00:07:21,950
we might want to know what the tension

153
00:07:21,950 --> 00:07:23,970
is in this problem, if we do that now

154
00:07:23,970 --> 00:07:27,058
we can look at the 9 kg mass individually

155
00:07:27,058 --> 00:07:30,654
so I can say for just the 9 kg mass alone,

156
00:07:30,654 --> 00:07:33,490
what is the tension on it
and what are the force?

157
00:07:33,490 --> 00:07:36,038
We can find the forces on it simply by

158
00:07:36,038 --> 00:07:38,578
saying the acceleration of the 9 kg mass

159
00:07:38,578 --> 00:07:41,995
is the net force on the 9 kg mass divided

160
00:07:43,028 --> 00:07:45,608
by the mass of the 9 kg mass.

161
00:07:45,608 --> 00:07:48,534
Now this is just for the 9 kg mass since

162
00:07:48,534 --> 00:07:50,626
I'm done treating this as a system.

163
00:07:50,626 --> 00:07:53,570
This trick of treating
this two-mass system

164
00:07:53,570 --> 00:07:54,954
as a single object is just a way

165
00:07:54,954 --> 00:07:57,160
to quickly get the magnitude
of the acceleration.

166
00:07:57,160 --> 00:07:59,171
Now that I have that and I want to find

167
00:07:59,171 --> 00:08:02,936
an internal force I'm looking
at just this 9 kg box.

168
00:08:02,936 --> 00:08:07,165
And I can say that my
acceleration is not 4.75

169
00:08:07,165 --> 00:08:11,332
but -4.75 if we want to
treat downwards as negative

170
00:08:12,804 --> 00:08:16,035
and upwards as positive
then I have to plug

171
00:08:16,035 --> 00:08:17,649
this magnitude of acceleration in as

172
00:08:17,649 --> 00:08:20,383
a negative acceleration since the 9 kg

173
00:08:20,383 --> 00:08:22,992
mass is accelerating downward and that's

174
00:08:22,992 --> 00:08:27,658
going to equal what forces
are on the 9 kg mass:

175
00:08:27,658 --> 00:08:29,517
I called downward negative so that

176
00:08:29,517 --> 00:08:30,748
tension upwards is positive,

177
00:08:30,748 --> 00:08:34,030
but minus the force of gravity on

178
00:08:34,030 --> 00:08:37,362
the 9 kg mass which is 9 kg times

179
00:08:37,363 --> 00:08:41,196
9.8 meters per second
squared divided by 9 kg.

180
00:08:42,503 --> 00:08:44,210
I don't divide by the whole mass,

181
00:08:44,210 --> 00:08:45,632
because I'm done treating this system

182
00:08:45,632 --> 00:08:47,136
as if it were a single
mass and I'm now looking

183
00:08:47,136 --> 00:08:49,797
at an individual mass only so we go back

184
00:08:49,797 --> 00:08:53,280
to our old normal rules
for newton's second law

185
00:08:53,280 --> 00:08:55,678
where up is positive and down is negative

186
00:08:55,678 --> 00:08:58,847
and I only look at
forces on this 9 kg mass

187
00:08:58,847 --> 00:09:00,594
I don't worry about any
of these now because

188
00:09:00,594 --> 00:09:03,866
they are not directly
exerted on the 9 kg mass

189
00:09:03,866 --> 00:09:07,502
and at this point I'm only
looking at the 9 kg mass.

190
00:09:07,502 --> 00:09:09,688
So if I solve this now I
can solve for the tension

191
00:09:09,688 --> 00:09:12,771
and the tension I get is 45.5 newtons

192
00:09:16,137 --> 00:09:18,738
which is less than 9 times 9.8

193
00:09:18,738 --> 00:09:21,002
it's got to be less because this object is

194
00:09:21,002 --> 00:09:23,675
accelerating down so we know the net force

195
00:09:23,675 --> 00:09:26,012
has to point down, that means this tension

196
00:09:26,012 --> 00:09:28,644
has to be less than the force of gravity

197
00:09:28,644 --> 00:09:30,542
on the 9 kg block.

198
00:09:30,542 --> 00:09:33,277
So recapping, treating a system of masses

199
00:09:33,277 --> 00:09:35,329
as if they were a single object is a great

200
00:09:35,329 --> 00:09:37,198
way to quickly get the acceleration

201
00:09:37,198 --> 00:09:39,714
of the masses in that system.

202
00:09:39,714 --> 00:09:41,554
Once you find that acceleration

203
00:09:41,554 --> 00:09:44,430
you can then find any
internal force that you want

204
00:09:44,430 --> 00:09:47,605
by using Newton's second
law for an individual box.

205
00:09:47,605 --> 00:09:49,925
You're done treating as
a system and you just

206
00:09:49,925 --> 00:09:51,875
look at the individual box alone

207
00:09:51,875 --> 00:09:54,365
like we did here and
that allows you to find

208
00:09:54,365 --> 00:00:00,000
an internal force like
the force of tension.