1
00:00:03,470 --> 00:00:05,960
We'd like to consider
a system of particles.

2
00:00:05,960 --> 00:00:08,200
Let's say we have object one.

3
00:00:08,200 --> 00:00:12,440
We'll call this the
j-th object, object n.

4
00:00:12,440 --> 00:00:15,460
And somewhere in this system of
particles is a center of mass.

5
00:00:15,460 --> 00:00:20,240
Now, we know that
the external force

6
00:00:20,240 --> 00:00:24,680
is causing the momentum
of the system to change,

7
00:00:24,680 --> 00:00:26,510
and what we'd now
like to show is

8
00:00:26,510 --> 00:00:28,520
that we can reduce
this system to

9
00:00:28,520 --> 00:00:32,630
an effective single particle.

10
00:00:32,630 --> 00:00:37,220
The way we'll do that is recall
that the momentum of the system

11
00:00:37,220 --> 00:00:40,670
is given by the sum of
the individual momentums.

12
00:00:40,670 --> 00:00:42,950
We'll call that the
j-th particle where

13
00:00:42,950 --> 00:00:45,370
we're summing j from 1 to n.

14
00:00:45,370 --> 00:00:47,330
And that's the sum
of the momentums

15
00:00:47,330 --> 00:00:49,040
of the various particles.

16
00:00:49,040 --> 00:00:52,220
Now when you
differentiate the momentum

17
00:00:52,220 --> 00:00:54,830
of the system with
respect to time,

18
00:00:54,830 --> 00:00:59,490
we're just differentiating
the velocity.

19
00:00:59,490 --> 00:01:01,670
And so that becomes
the acceleration

20
00:01:01,670 --> 00:01:03,230
of the j-th particle.

21
00:01:03,230 --> 00:01:07,130
Now we saw when we
define the center of mass

22
00:01:07,130 --> 00:01:08,660
that the acceleration
of the center

23
00:01:08,660 --> 00:01:11,960
of mass times the total
mass of the system

24
00:01:11,960 --> 00:01:16,530
was equal to the sum of mj aj.

25
00:01:16,530 --> 00:01:18,890
j goes from 1 to n.

26
00:01:18,890 --> 00:01:22,460
So we see that
another way to think

27
00:01:22,460 --> 00:01:26,480
about how the momentum of the
system of particles is changing

28
00:01:26,480 --> 00:01:32,750
is simply the total mass
times the acceleration

29
00:01:32,750 --> 00:01:34,820
of the center of mass.

30
00:01:34,820 --> 00:01:39,770
And our combination of
Newton's second and third law

31
00:01:39,770 --> 00:01:42,920
now becomes that only
the external forces

32
00:01:42,920 --> 00:01:45,590
cause the momentum of the
system to change so that's

33
00:01:45,590 --> 00:01:50,960
equal to the total mass times
the acceleration of the center

34
00:01:50,960 --> 00:01:52,039
of mass.

35
00:01:52,039 --> 00:01:55,340
Now what does this
equation really mean?

36
00:01:55,340 --> 00:01:58,140
So let's draw our
pictures again.

37
00:01:58,140 --> 00:02:00,140
Here's our system.

38
00:02:00,140 --> 00:02:02,810
We have particles 1, et cetera.

39
00:02:02,810 --> 00:02:06,110
That's the j-th
particle, n particles.

40
00:02:06,110 --> 00:02:09,259
And in here is the
center of mass.

41
00:02:09,259 --> 00:02:12,680
Now I'm going to outline
my system like this

42
00:02:12,680 --> 00:02:14,240
and what this
equation is telling

43
00:02:14,240 --> 00:02:20,220
us is that we can just focus
by putting all of the mass

44
00:02:20,220 --> 00:02:23,750
and total at the center of mass.

45
00:02:23,750 --> 00:02:27,770
And that center of
mass is going to move

46
00:02:27,770 --> 00:02:32,180
according to some trajectory.

47
00:02:32,180 --> 00:02:35,780
And all we have
to think about is

48
00:02:35,780 --> 00:02:47,850
this is a point particle
of total mass m, m total.

49
00:02:47,850 --> 00:02:52,350
So what we've done is we've
done a very important reduction.

50
00:02:52,350 --> 00:02:55,050
We've taken a complicated
system of particles

51
00:02:55,050 --> 00:02:58,230
and reduced it to a
single point particle

52
00:02:58,230 --> 00:03:03,780
of total mass m located at cm.

53
00:03:03,780 --> 00:03:06,390
And the dynamics of
that total particle

54
00:03:06,390 --> 00:03:11,010
is if there is external
force acting on this system,

55
00:03:11,010 --> 00:03:14,520
we place this external
force at the center of mass.

56
00:03:14,520 --> 00:03:18,030
And now we can calculate
the acceleration

57
00:03:18,030 --> 00:03:23,760
of the center of mass is just
that external force divided

58
00:03:23,760 --> 00:03:25,650
by the total mass.

59
00:03:25,650 --> 00:03:29,520
And that's how we can reduce
this complicated system

60
00:03:29,520 --> 00:03:36,360
of particles to a translational
motion of the center of mass.

61
00:03:36,360 --> 00:03:41,489
Now we still cannot describe
the individual interactions

62
00:03:41,489 --> 00:03:43,780
in the system, but we're not
trying to do that anymore.

63
00:03:43,780 --> 00:03:46,500
We're not trying to trace
how each particle moves.

64
00:03:46,500 --> 00:03:49,170
We're just looking at as our
system is a point particle

65
00:03:49,170 --> 00:03:52,110
and talking about how
that point-like object is

66
00:03:52,110 --> 00:03:54,300
translating the space.

67
00:03:54,300 --> 00:03:56,880
And this is a powerful
tool that we use again

68
00:03:56,880 --> 00:03:58,647
and again and again.