WEBVTT
00:00:03.920 --> 00:00:06.320
Let's now extend our
concept of momentum
00:00:06.320 --> 00:00:08.240
to a system of particles.
00:00:08.240 --> 00:00:10.430
Again, we need to choose
a reference frame.
00:00:10.430 --> 00:00:12.170
So we'll have a ground frame.
00:00:12.170 --> 00:00:16.730
And let's consider N particles.
00:00:16.730 --> 00:00:18.290
Now when we have a
lot of particles,
00:00:18.290 --> 00:00:20.540
we need some type of notation.
00:00:20.540 --> 00:00:22.820
So let's use the symbol j.
00:00:22.820 --> 00:00:26.210
And it will goes
from 1 to N. And then
00:00:26.210 --> 00:00:30.770
our arbitrary j
particle will be moving.
00:00:30.770 --> 00:00:33.260
This particle will have mass mj.
00:00:33.260 --> 00:00:36.770
And it will be moving
with a velocity vj.
00:00:36.770 --> 00:00:40.190
Now recall in our system, we
have many other particles.
00:00:40.190 --> 00:00:41.550
We can call that one 1.
00:00:41.550 --> 00:00:43.760
This is one n.
00:00:43.760 --> 00:00:47.040
We have lots of different
particles in the system.
00:00:47.040 --> 00:00:49.940
And this just represents
an arbitrary particle
00:00:49.940 --> 00:00:51.500
in that system.
00:00:51.500 --> 00:00:54.950
And the momentum
of the jth particle
00:00:54.950 --> 00:00:59.810
is just the mass, mj,
times the velocity, vj.
00:00:59.810 --> 00:01:03.890
And again, we're assuming
some fixed reference frame.
00:01:03.890 --> 00:01:07.430
So the total momentum
of this system,
00:01:07.430 --> 00:01:10.400
we now have to add
up the momentum
00:01:10.400 --> 00:01:16.340
of all the particles, all the
way up to the nth particle.
00:01:16.340 --> 00:01:18.710
Now, when we make
a sum like this,
00:01:18.710 --> 00:01:22.020
there is a standard
mathematical summation notation,
00:01:22.020 --> 00:01:23.450
which we'll write like this.
00:01:23.450 --> 00:01:30.920
We'll do the sum, this capital
sigma sin of j goes from 1 to j
00:01:30.920 --> 00:01:38.720
goes to N of the momentum
of the jth particle.
00:01:38.720 --> 00:01:48.430
And that represents the sum
j goes from 1 to n of mj vj.
00:01:48.430 --> 00:01:52.070
And this is what we call
the momentum of the system.
00:01:52.070 --> 00:01:53.386
This is a vector sum.
00:01:56.720 --> 00:02:01.550
And now let's see how
Newton's second law applies
00:02:01.550 --> 00:02:03.830
to the momentum of the system.
00:02:03.830 --> 00:02:06.950
Suppose that acting
on our particles--
00:02:06.950 --> 00:02:12.340
for instance, here's our jth
particle-- we have a force
00:02:12.340 --> 00:02:16.370
Fj acting on the jth particle.
00:02:16.370 --> 00:02:21.320
Then we know that
from Newton's law
00:02:21.320 --> 00:02:28.329
that the force will be also
the sum of the forces on all
00:02:28.329 --> 00:02:36.020
of the particles, F1, F2,
plus dot, dot, dot, plus FN.
00:02:36.020 --> 00:02:40.170
So once again, we can
write this as a sum
00:02:40.170 --> 00:02:45.140
j goes from 1 to N of the
force on the jth particle.
00:02:45.140 --> 00:02:48.170
And that's the
force on the summing
00:02:48.170 --> 00:02:52.200
over all the forces on all
the particles in the system.
00:02:52.200 --> 00:02:56.490
But now, we can apply
Newton's second law.
00:02:56.490 --> 00:03:01.610
So Newton's second
law is the statement
00:03:01.610 --> 00:03:04.760
that the force on
the jth particle
00:03:04.760 --> 00:03:11.810
causes the momentum of the
jth particle to change.
00:03:11.810 --> 00:03:16.235
And when we write that now,
the total force on the system,
00:03:16.235 --> 00:03:24.470
j goes from 1 to N, is just the
sum of the change in momentum.
00:03:24.470 --> 00:03:28.360
Because every single term--
let's just look at that.
00:03:28.360 --> 00:03:37.730
T1 plus dP2/dt plus dot,
dot, dot, plus dPN/dt,
00:03:37.730 --> 00:03:39.440
that's what we mean by the sum.
00:03:39.440 --> 00:03:49.010
We can rewrite this as
d/dt of P1 plus P2 plus P3
00:03:49.010 --> 00:03:53.410
plus dot, dot, dot, plus PN.
00:03:53.410 --> 00:04:00.190
And what we see is
that the total force is
00:04:00.190 --> 00:04:09.740
the derivative of the sum j goes
from 1 to N of the momentum.
00:04:09.740 --> 00:04:14.030
But recall, this
sum we've defined
00:04:14.030 --> 00:04:16.200
as the momentum of the system.
00:04:16.200 --> 00:04:19.399
So our conclusion
is the total force
00:04:19.399 --> 00:04:25.570
causes the momentum of
the system to change.
00:04:25.570 --> 00:04:28.070
Now so far, all
we've done is we've
00:04:28.070 --> 00:04:31.550
recast Newton's second
law in this form.
00:04:31.550 --> 00:04:34.550
Our next step is to
analyze the forces
00:04:34.550 --> 00:04:37.010
on the individual
particles we have
00:04:37.010 --> 00:04:39.480
and apply Newton's third law.
00:04:39.480 --> 00:04:41.620
So we'll do that next.