Angular momentum of an extended object

WEBVTT
Kind: captions
Language: en

00:00:00.406 --> 00:00:02.255
- [Voiceover] So we saw in previous videos

00:00:02.255 --> 00:00:05.699
that a ball of mass m rotating in a circle

00:00:05.699 --> 00:00:08.224
of radius r at a speed v has

00:00:08.224 --> 00:00:10.449
what we call angular momentum,

00:00:10.449 --> 00:00:11.282
and the symbol we use

00:00:11.282 --> 00:00:14.221
for angular momentum is a capital L,

00:00:14.221 --> 00:00:15.889
and the amount of angular momentum

00:00:15.889 --> 00:00:17.666
that it would have would be the mass

00:00:17.666 --> 00:00:19.580
of the ball times the speed of the ball,

00:00:19.580 --> 00:00:22.274
so that means this is
basically just the magnitude

00:00:22.274 --> 00:00:24.401
of the momentum, but then we multiply

00:00:24.401 --> 00:00:27.063
by the radius of the
circle it's traveling in,

00:00:27.063 --> 00:00:29.186
and that gives us the angular momentum

00:00:29.186 --> 00:00:30.944
of this ball going in a circle,

00:00:30.944 --> 00:00:32.837
which is great and good to know,

00:00:32.837 --> 00:00:34.809
but sometimes you don't have a ball going

00:00:34.809 --> 00:00:37.644
in a circle and you wanna
know the angular momentum.

00:00:37.644 --> 00:00:40.000
So, for instance, instead of this case,

00:00:40.000 --> 00:00:41.408
let's say you have this case,

00:00:41.408 --> 00:00:42.776
where instead of a ball going around

00:00:42.776 --> 00:00:45.814
in a circle, you've got a rod of mass m

00:00:45.814 --> 00:00:48.757
and radius R and the whole rod rotates

00:00:48.757 --> 00:00:50.196
around in a circle.

00:00:50.196 --> 00:00:52.913
Let's say the outside
edge travels at a speed v

00:00:52.913 --> 00:00:54.445
just like the ball did.

00:00:54.445 --> 00:00:55.278
So the question is,

00:00:55.278 --> 00:00:58.091
will this rod also have
an angular momentum

00:00:58.091 --> 00:01:00.647
that's equal to mvR and it won't.

00:01:00.647 --> 00:01:01.949
You can probably convince yourself of that

00:01:01.949 --> 00:01:04.368
because for the ball all
the mass was traveling

00:01:04.368 --> 00:01:06.856
at a speed v and all the mass was

00:01:06.856 --> 00:01:09.259
at the outside edge of the
circle that it traces out.

00:01:09.259 --> 00:01:11.918
In other words, all this mass is traveling

00:01:11.918 --> 00:01:13.652
at a radius of R.

00:01:13.652 --> 00:01:16.241
But for this rod, some
of the mass, in fact,

00:01:16.241 --> 00:01:18.879
only part of the mass,
only this outside edge

00:01:18.879 --> 00:01:22.303
of the mass, is actually
traveling at a radius R.

00:01:22.303 --> 00:01:25.539
That's the part that travels
at the full radius R.

00:01:25.539 --> 00:01:28.345
The rest of these pieces of
mass, like this one in here,

00:01:28.345 --> 00:01:29.890
traces out a circle.

00:01:29.890 --> 00:01:31.695
It definitely traces out a circle,

00:01:31.695 --> 00:01:33.814
but the circle it traces out is not equal

00:01:33.814 --> 00:01:35.195
to the radius R.

00:01:35.195 --> 00:01:37.002
It's got a diminished R value.

00:01:37.002 --> 00:01:39.418
So how do we determine
the angular momentum

00:01:39.418 --> 00:01:42.651
of an object whose mass
is distributed in a way

00:01:42.651 --> 00:01:44.795
that some of the mass is close to the axis

00:01:44.795 --> 00:01:47.277
and some of the mass is
far away from the axis.

00:01:47.277 --> 00:01:48.912
That's what we're gonna do in this video.

00:01:48.912 --> 00:01:51.344
That's the goal and the
approach physicists take

00:01:51.344 --> 00:01:53.778
to this is almost always the same.

00:01:53.778 --> 00:01:55.648
We say, well, I've got the formula

00:01:55.648 --> 00:01:58.269
for the angular momentum
of a single particle

00:01:58.269 --> 00:01:59.662
traveling at a single radius,

00:01:59.662 --> 00:02:02.758
so let's just imagine
our continuous object

00:02:02.758 --> 00:02:06.292
being composed of a bunch of single masses

00:02:06.292 --> 00:02:08.974
all traveling at a single radius.

00:02:08.974 --> 00:02:10.994
So if I break this continuous mass up

00:02:10.994 --> 00:02:13.258
into individual pieces, right?

00:02:13.258 --> 00:02:14.816
So if I imagine it being broken up

00:02:14.816 --> 00:02:16.682
into all these little pieces,

00:02:16.682 --> 00:02:19.729
then if I found the angular
momentum of each piece

00:02:19.729 --> 00:02:22.976
and added it up, I'd get
the total angular momentum

00:02:22.976 --> 00:02:24.401
for the whole object.

00:02:24.401 --> 00:02:25.704
So let's try this.

00:02:25.704 --> 00:02:28.875
So the angular momentum of
some piece of the object,

00:02:28.875 --> 00:02:30.337
let's say that little piece of mass,

00:02:30.337 --> 00:02:33.151
is gonna be, well the
mass of that little piece,

00:02:33.151 --> 00:02:34.539
I'm gonna write m.

00:02:34.539 --> 00:02:37.659
It's not the entire
mass of this entire rod,

00:02:37.659 --> 00:02:40.630
it'd just be the mass of that
small piece times the speed

00:02:40.630 --> 00:02:44.147
of that piece times the
radius that it is at.

00:02:44.147 --> 00:02:45.517
So to make this clear,
let me just write this as

00:02:45.517 --> 00:02:49.350
like piece one, so this
would be m1, v1 and R1

00:02:51.090 --> 00:02:53.653
and this would be the angular momentum

00:02:53.653 --> 00:02:55.557
of that small piece, and you could do this

00:02:55.557 --> 00:02:58.473
for mass two over here, and you would get

00:02:58.473 --> 00:02:59.380
that the angular momentum

00:02:59.380 --> 00:03:02.297
of mass two would be m2, v2 and R2.

00:03:05.049 --> 00:03:08.190
Now keep in mind that these
v's are all gonna be different

00:03:08.190 --> 00:03:09.519
so the speed out here

00:03:09.519 --> 00:03:11.518
at the outside edge is gonna be fastest.

00:03:11.518 --> 00:03:13.644
This speed's not gonna be as great,

00:03:13.644 --> 00:03:16.526
and this speed closer to
the middle is even smaller

00:03:16.526 --> 00:03:18.709
because they're tracing
out smaller circles

00:03:18.709 --> 00:03:20.024
in the same amount of time

00:03:20.024 --> 00:03:23.557
as these outside pieces
trace out larger circles

00:03:23.557 --> 00:03:24.859
in the same amount of time.

00:03:24.859 --> 00:03:26.246
So at this point, you might be worried.

00:03:26.246 --> 00:03:28.054
You might be like, "This
is gonna be really hard.

00:03:28.054 --> 00:03:29.408
We're gonna have to add all these up.

00:03:29.408 --> 00:03:31.008
They've all got different speeds.

00:03:31.008 --> 00:03:32.636
They're all at different radii.

00:03:32.636 --> 00:03:33.897
How are we gonna do this?"

00:03:33.897 --> 00:03:35.248
Well, you gotta have faith

00:03:35.248 --> 00:03:37.769
and something magical is about to happen.

00:03:37.769 --> 00:03:38.702
So let me show you what happens

00:03:38.702 --> 00:03:40.963
if we imagine adding all these up.

00:03:40.963 --> 00:03:42.218
I only draw two.

00:03:42.218 --> 00:03:44.153
You gotta imagine there's
an infinite amount

00:03:44.153 --> 00:03:45.914
of these so that makes
it seem even harder,

00:03:45.914 --> 00:03:48.808
but imagine breaking this
up into an infinite amount

00:03:48.808 --> 00:03:50.719
of these little discrete masses

00:03:50.719 --> 00:03:54.036
and considering each
individual angular momentum,

00:03:54.036 --> 00:03:56.277
they'd be very small
because this m1 would be

00:03:56.277 --> 00:03:59.654
an infinitesimal very small piece of mass,

00:03:59.654 --> 00:04:02.072
and let's add them all
up and see what we get.

00:04:02.072 --> 00:04:05.989
So if we add up all of
the mvR's of every piece

00:04:07.139 --> 00:04:09.527
of mass on this rod, that would be

00:04:09.527 --> 00:04:12.937
the total angular momentum of the rod.

00:04:12.937 --> 00:04:14.252
So in other words, this is really

00:04:14.252 --> 00:04:17.919
just m1, v1, R1 m2, v2, R2 and so on.

00:04:22.568 --> 00:04:24.122
You'd have an infinite
amount of them, right?

00:04:24.122 --> 00:04:24.955
I can't write them all out

00:04:24.955 --> 00:04:25.983
'cause there's an infinite amount.

00:04:25.983 --> 00:04:27.190
But just imagine that.

00:04:27.190 --> 00:04:29.210
So what can we possibly do with this?

00:04:29.210 --> 00:04:31.076
How do we clean this up?

00:04:31.076 --> 00:04:32.575
When you're doing a physics problem,

00:04:32.575 --> 00:04:34.785
you don't want to solve an infinite series

00:04:34.785 --> 00:04:36.509
by writing each term out infinitely.

00:04:36.509 --> 00:04:38.054
We want a clever way to deal with this,

00:04:38.054 --> 00:04:40.479
and there's a really clever
way to deal with this.

00:04:40.479 --> 00:04:41.312
Watch this.

00:04:41.312 --> 00:04:44.883
So if we write this as
L equals the sum of mvR.

00:04:44.883 --> 00:04:48.917
One problem we have is that
each mass has a different v.

00:04:48.917 --> 00:04:51.567
If I can pull things
out of this summation,

00:04:51.567 --> 00:04:53.819
it would help me out 'cause
it would simplify things.

00:04:53.819 --> 00:04:55.031
I could just factor them out,

00:04:55.031 --> 00:04:57.520
but right now I can't factor out the R,

00:04:57.520 --> 00:05:00.446
'cause these all are
different radii from the axis.

00:05:00.446 --> 00:05:03.150
You always measure your
radius from the axis here,

00:05:03.150 --> 00:05:05.308
and they're all at different
radii from the axis,

00:05:05.308 --> 00:05:06.982
and they all have different speeds.

00:05:06.982 --> 00:05:09.708
But, remember, we like
writing quantities in terms

00:05:09.708 --> 00:05:13.426
of angular variables because
the angular variables

00:05:13.426 --> 00:05:16.063
are the same for every point on this mass.

00:05:16.063 --> 00:05:20.230
So every point on this rotating
rod has a different speed v

00:05:21.331 --> 00:05:24.711
but they all have the
same angular speed omega,

00:05:24.711 --> 00:05:25.836
so that's a key.

00:05:25.836 --> 00:05:27.890
That's often what we do and
we're gonna do that here.

00:05:27.890 --> 00:05:30.609
I'm gonna write this as summation of m,

00:05:30.609 --> 00:05:32.872
but instead of writing
v, I'm gonna write this

00:05:32.872 --> 00:05:34.739
as R times omega.

00:05:34.739 --> 00:05:38.229
So, remember, for something
rotating in the circle,

00:05:38.229 --> 00:05:42.634
the speed v is gonna be
equal to R times omega

00:05:42.634 --> 00:05:44.393
and that's what I'm gonna
substitute down here,

00:05:44.393 --> 00:05:47.070
so the speed at any
point here is the radii

00:05:47.070 --> 00:05:49.601
of that point times the angular speed

00:05:49.601 --> 00:05:51.819
of this rod rotating in a circle,

00:05:51.819 --> 00:05:54.112
and I still have to
multiply by the last R here,

00:05:54.112 --> 00:05:55.163
so this was v.

00:05:55.163 --> 00:05:57.599
We substituted in what v
was, but we have to multiply

00:05:57.599 --> 00:05:59.762
by R and what do we get?

00:05:59.762 --> 00:06:02.244
We get that L is gonna be the summation

00:06:02.244 --> 00:06:03.911
of mR squared omega.

00:06:05.402 --> 00:06:06.544
And this is great.

00:06:06.544 --> 00:06:09.921
The omega is the same for
every single mass in here.

00:06:09.921 --> 00:06:12.939
Every single mass travels
at the same angular speed,

00:06:12.939 --> 00:06:15.911
so we could factor this
out of the summation.

00:06:15.911 --> 00:06:18.375
Imagine all these terms
would have an omega.

00:06:18.375 --> 00:06:20.266
We can factor that out,
and I could just bring that

00:06:20.266 --> 00:06:22.134
outside of the summation.

00:06:22.134 --> 00:06:25.803
So I'll write this as the
summation of mR squared,

00:06:25.803 --> 00:06:28.285
and to make this clear, I'm
gonna put parentheses here.

00:06:28.285 --> 00:06:31.749
It's that summation and then
that whole thing times omega,

00:06:31.749 --> 00:06:33.805
'cause we're just factoring out omega.

00:06:33.805 --> 00:06:35.086
And you might not be impressed.

00:06:35.086 --> 00:06:36.385
You might be like, "All right.

00:06:36.385 --> 00:06:37.218
Big deal.

00:06:37.218 --> 00:06:39.481
We've still got an infinite sum in here.

00:06:39.481 --> 00:06:40.782
What the heck am I gonna do with that?"

00:06:40.782 --> 00:06:42.772
You don't have to do anything with that.

00:06:42.772 --> 00:06:44.017
This is where the magic happens.

00:06:44.017 --> 00:06:45.685
Look at what sum you've got.

00:06:45.685 --> 00:06:48.555
You've got the sum of all the mR squareds.

00:06:48.555 --> 00:06:50.310
Remember what mR squared was?

00:06:50.310 --> 00:06:53.899
Mr squared was the moment
of inertia of a point mass,

00:06:53.899 --> 00:06:57.096
and if I add up all the mR
squareds, I get the moment

00:06:57.096 --> 00:07:00.574
of inertia of the entire
mass, this entire object.

00:07:00.574 --> 00:07:02.420
I get its total moment of inertia.

00:07:02.420 --> 00:07:05.496
So what we found was a really handy way

00:07:05.496 --> 00:07:07.691
to write the angular
momentum of an object.

00:07:07.691 --> 00:07:11.457
It's just the moment of
inertia of an object, I,

00:07:11.457 --> 00:07:15.262
times the angular speed of that object.

00:07:15.262 --> 00:07:17.259
So this is a great formula,

00:07:17.259 --> 00:07:19.640
and it totally makes
sense for this reason.

00:07:19.640 --> 00:07:21.403
Think about regular momentum, right.

00:07:21.403 --> 00:07:25.494
Regular momentum, p, was just equal to mv.

00:07:25.494 --> 00:07:28.404
Well, if you then told me,
"Determine the angular momentum,"

00:07:28.404 --> 00:07:30.274
and I didn't wanna go
through this derivation,

00:07:30.274 --> 00:07:31.176
I mighta just been like,

00:07:31.176 --> 00:07:33.935
"All right, angular momentum, shoot."

00:07:33.935 --> 00:07:37.678
Well, I'm just gonna replace
mass with angular mass,

00:07:37.678 --> 00:07:39.888
and angular mass, the angular inertia,

00:07:39.888 --> 00:07:41.432
is just the moment of inertia,

00:07:41.432 --> 00:07:44.914
and I'll just replace the
speed with the angular speed,

00:07:44.914 --> 00:07:46.505
and look, I just get this formula.

00:07:46.505 --> 00:07:47.782
So it makes sense because

00:07:47.782 --> 00:07:49.675
if you replace all the linear quantities

00:07:49.675 --> 00:07:51.656
with their angular counterpart,

00:07:51.656 --> 00:07:54.088
you indeed just get the angular momentum

00:07:54.088 --> 00:07:55.836
of a rotating object.

00:07:55.836 --> 00:07:56.707
So this is how you do it.

00:07:56.707 --> 00:07:58.776
If you've got an object,
an extended object

00:07:58.776 --> 00:08:02.318
where the mass is distributed
about the entire object,

00:08:02.318 --> 00:08:03.659
if you just take the moment of inertia

00:08:03.659 --> 00:08:06.436
of that object and multiply
by its angular speed,

00:08:06.436 --> 00:08:08.233
you get its angular momentum.

00:08:08.233 --> 00:08:10.667
So, for instance, if this rod has a mass

00:08:10.667 --> 00:08:13.156
of let's say three kilograms,

00:08:13.156 --> 00:08:14.985
and that mass is evenly distributed.

00:08:14.985 --> 00:08:17.807
Let's say the radius of
this object is two meters.

00:08:17.807 --> 00:08:21.193
So that's the distance from
the axis to the outside edge.

00:08:21.193 --> 00:08:24.189
And let's say the angular
speed of this object was,

00:08:24.189 --> 00:08:27.207
let's say 10 radians per second,

00:08:27.207 --> 00:08:30.430
we can figure out the
angular momentum of this rod

00:08:30.430 --> 00:08:32.889
by saying that the angular
momentum is gonna equal

00:08:32.889 --> 00:08:34.676
the moment of inertia.

00:08:34.676 --> 00:08:36.057
Well the moment of inertia of a rod

00:08:36.057 --> 00:08:40.767
about one end is equal to 1/3 mL squared.

00:08:40.767 --> 00:08:43.128
That's the moment of inertia
of a rod about the end,

00:08:43.128 --> 00:08:47.076
and I then multiply by the
angular speed of the object.

00:08:47.076 --> 00:08:49.014
So if I plug in numbers,
I get the angular momentum

00:08:49.014 --> 00:08:51.479
of this rod is gonna be,
I'll use purple here,

00:08:51.479 --> 00:08:55.969
1/3 times three kilograms times the length

00:08:55.969 --> 00:08:57.750
of the object was two meters,

00:08:57.750 --> 00:08:59.683
and we square that, and then we multiply

00:08:59.683 --> 00:09:03.250
by the angular speed and that
was 10 radians per second,

00:09:03.250 --> 00:09:05.047
which give us an angular momentum

00:09:05.047 --> 00:09:08.464
of 40 kilogram meters squared per second.

00:09:09.517 --> 00:09:11.575
So recapping, if you've got a point mass

00:09:11.575 --> 00:09:13.875
where all the mass
rotates at the same radius

00:09:13.875 --> 00:09:16.038
and you wanna find the angular momentum,

00:09:16.038 --> 00:09:17.517
the easiest way to get it is probably

00:09:17.517 --> 00:09:19.408
with the formula mvR.

00:09:19.408 --> 00:09:22.722
However, if you have a mass
whose mass is distributed

00:09:22.722 --> 00:09:25.319
throughout the object
so that different points

00:09:25.319 --> 00:09:27.826
on the object are at different radii,

00:09:27.826 --> 00:09:30.144
the easiest way to get
the angular momentum

00:09:30.144 --> 00:09:31.827
of that object is most likely

00:09:31.827 --> 00:09:34.309
with the formula I omega,

00:09:34.309 --> 00:09:35.324
where I is the moment

00:09:35.324 --> 00:09:36.673
of inertia of the object

00:09:36.673 --> 00:09:39.496
and omega is the angular
velocity of the object.


More on Newton's third law

WEBVTT
Kind: captions
Language: en

00:00:00.601 --> 00:00:01.434
- [Voiceover] We should talk a little more

00:00:01.434 --> 00:00:02.572
about Newtons's Third Law,

00:00:02.572 --> 00:00:04.482
because there are some deep misconceptions

00:00:04.482 --> 00:00:06.493
that many people have about this law.

00:00:06.493 --> 00:00:08.148
It seems simple, but it's not nearly

00:00:08.148 --> 00:00:09.732
as simple as you might think.

00:00:09.732 --> 00:00:11.358
So people often phrase it as,

00:00:11.358 --> 00:00:14.193
for every action there's an
equal and opposite reaction.

00:00:14.193 --> 00:00:16.510
But that's just way
too vague to be useful.

00:00:16.510 --> 00:00:18.115
So a version that's a little better,

00:00:18.115 --> 00:00:21.821
says that for every
force, there's an equal

00:00:21.821 --> 00:00:23.612
and opposite force.

00:00:23.612 --> 00:00:24.849
So this is a little better.

00:00:24.849 --> 00:00:26.933
The equal sign means that these forces

00:00:26.933 --> 00:00:28.567
are equal in magnitude.

00:00:28.567 --> 00:00:31.391
And this negative sign
means they're just different

00:00:31.391 --> 00:00:33.272
by the direction of the vector.

00:00:33.272 --> 00:00:35.771
So these are vectors, so
this says that this pink

00:00:35.771 --> 00:00:38.087
vector F, has the opposite direction,

00:00:38.087 --> 00:00:41.947
but equal in magnitude
to this green vector F.

00:00:41.947 --> 00:00:44.594
But to show you why this is
still a little bit too vague,

00:00:44.594 --> 00:00:46.254
consider this, if this
is all you knew about

00:00:46.254 --> 00:00:48.347
Newtons's Third Law, that for every force,

00:00:48.347 --> 00:00:50.167
there's an equal and opposite force,

00:00:50.167 --> 00:00:52.117
you might wonder, if you were clever,

00:00:52.117 --> 00:00:54.230
you might be like, wait a minute,

00:00:54.230 --> 00:00:58.063
if for every force F,
right, there's got to be

00:00:59.319 --> 00:01:02.152
a force that's equal and opposite.

00:01:03.249 --> 00:01:04.418
Well why doesn't that just mean

00:01:04.418 --> 00:01:07.149
that every force in the universe cancels?

00:01:07.149 --> 00:01:09.782
Shouldn't every force just
cancel then, at that point?

00:01:09.782 --> 00:01:12.240
Doesn't that just mean that there's

00:01:12.240 --> 00:01:15.024
no acceleration that's even possible?

00:01:15.024 --> 00:01:17.918
Because if I go and exert
a force F on something,

00:01:17.918 --> 00:01:19.836
if there's gonna be a force negative F,

00:01:19.836 --> 00:01:22.803
doesn't that mean that no
matter what force I put forward,

00:01:22.803 --> 00:01:24.501
it's just gonna get cancelled?

00:01:24.501 --> 00:01:26.765
And the answer no, and the reason it's no

00:01:26.765 --> 00:01:28.512
is because these two forces

00:01:28.512 --> 00:01:30.474
are exerted on different objects.

00:01:30.474 --> 00:01:31.844
So you have to be careful.

00:01:31.844 --> 00:01:33.225
So the reason I say that this statement

00:01:33.225 --> 00:01:35.846
of Newtons's Third Law is
still a little bit too vague,

00:01:35.846 --> 00:01:38.244
is because this is really
on different objects.

00:01:38.244 --> 00:01:41.190
So if this is the force on object A,

00:01:41.190 --> 00:01:45.233
exerted by object B,
then this force over here

00:01:45.233 --> 00:01:49.400
has to be the force on object
B, exerted by object A.

00:01:53.238 --> 00:01:55.554
In other words, these
forces down here are exerted

00:01:55.554 --> 00:01:56.712
on different objects.

00:01:56.712 --> 00:01:58.988
I'm gonna move this over to this side.

00:01:58.988 --> 00:02:00.930
I'm gonna move this over to here.

00:02:00.930 --> 00:02:02.230
Let's draw two different objects

00:02:02.230 --> 00:02:04.912
to show explicitly what I mean.

00:02:04.912 --> 00:02:07.388
So if there was some object A,

00:02:07.388 --> 00:02:09.919
so I put some object A in here.

00:02:09.919 --> 00:02:11.451
Just wanna make sure there's an object A.

00:02:11.451 --> 00:02:13.413
Let's say this is object A, and it had

00:02:13.413 --> 00:02:16.937
this green force exerted on it, F.

00:02:16.937 --> 00:02:19.651
So this object right here is A.

00:02:19.651 --> 00:02:23.166
Well, there's gonna be
another object, object B.

00:02:23.166 --> 00:02:26.689
We'll just make it another circle.

00:02:26.689 --> 00:02:29.424
So we'll make it look like this.

00:02:29.424 --> 00:02:31.007
So here's object B.

00:02:31.992 --> 00:02:34.015
And it's gonna have this pink force,

00:02:34.015 --> 00:02:35.824
F, negative F exerted on it.

00:02:35.824 --> 00:02:37.681
So I'm gonna call this object B.

00:02:37.681 --> 00:02:41.867
Now we're okay, now we know
these forces can't cancel,

00:02:41.867 --> 00:02:43.625
and the reason these forces can't cancel,

00:02:43.625 --> 00:02:45.666
is cause they're on two different objects.

00:02:45.666 --> 00:02:46.925
But when you just say
that Newtons's Third Law,

00:02:46.925 --> 00:02:48.710
is that every force has an
equal and opposite force,

00:02:48.710 --> 00:02:51.311
it's not clear that it has
to be on different objects.

00:02:51.311 --> 00:02:53.180
But it does have to be
on different objects.

00:02:53.180 --> 00:02:55.630
So these Newtons force law pairs,

00:02:55.630 --> 00:02:57.711
often times is called force pairs,

00:02:57.711 --> 00:03:00.210
or Newton's third law partner forces,

00:03:00.210 --> 00:03:02.587
are always on different objects.

00:03:02.587 --> 00:03:05.371
So the convention I'm using
is that the first letter

00:03:05.371 --> 00:03:09.548
represents the object
that the force is on.

00:03:09.548 --> 00:03:11.862
So this A represents
that this green force F

00:03:11.862 --> 00:03:16.029
this green force F, is on
A and it's exerted by B.

00:03:18.183 --> 00:03:20.946
And this shows that it's exerted on B,

00:03:20.946 --> 00:03:23.172
because the first
letter's on the first one,

00:03:23.172 --> 00:03:26.589
and it's exerted by the second object, A.

00:03:27.723 --> 00:03:30.486
So this pink force is exerted on B.

00:03:30.486 --> 00:03:32.425
This green force is exerted on A.

00:03:32.425 --> 00:03:34.741
They're equal and opposite,
they do not cancel,

00:03:34.741 --> 00:03:38.492
they cannot cancel because
they're not on the same object.

00:03:38.492 --> 00:03:40.036
So that's why these don't cancel.

00:03:40.036 --> 00:03:41.702
And they are the same magnitude,

00:03:41.702 --> 00:03:43.429
even if the two objects
are not the same size.

00:03:43.429 --> 00:03:45.144
This is another misconception,

00:03:45.144 --> 00:03:48.090
if object A is a planet, a big planet.

00:03:48.090 --> 00:03:50.885
Or maybe a star, this is
yellow, it looks like a star.

00:03:50.885 --> 00:03:53.018
Let's say this is some big star,

00:03:53.018 --> 00:03:55.486
and this is some smaller
planet orbiting that star.

00:03:55.486 --> 00:03:59.424
This is not to scale, unless
this planet was enormous.

00:03:59.424 --> 00:04:02.167
So this is some planet,
but this planet could be

00:04:02.167 --> 00:04:05.540
hundreds, thousands of
times, millions of times

00:04:05.540 --> 00:04:08.892
less massive than this
star but it would still

00:04:08.892 --> 00:04:10.010
exert the same force.

00:04:10.010 --> 00:04:11.951
So if this star is pulling on the planet

00:04:11.951 --> 00:04:15.040
with this pink force negative F,

00:04:15.040 --> 00:04:17.324
then this planet has to
be pulling on the star

00:04:17.324 --> 00:04:19.144
with this green force
F and they have to have

00:04:19.144 --> 00:04:23.318
the same magnitude, even if
they are different sizes.

00:04:23.318 --> 00:04:24.711
So people quote Newtons's Third Law,

00:04:24.711 --> 00:04:27.047
but sometimes they
don't really believe it.

00:04:27.047 --> 00:04:28.571
If I told you this planet
was a million times

00:04:28.571 --> 00:04:31.709
less massive than this star,
people would want to say

00:04:31.709 --> 00:04:36.048
that well, then the star obviously
pulls more on the planet,

00:04:36.048 --> 00:04:37.743
than the planet pulls on the star.

00:04:37.743 --> 00:04:40.172
But that's not true according
to Newtons's Third Law.

00:04:40.172 --> 00:04:43.330
And Newtons's Third Law says
that they have to be the same,

00:04:43.330 --> 00:04:44.570
even if they're different sizes.

00:04:44.570 --> 00:04:46.561
So if this was the earth
and this was the moon,

00:04:46.561 --> 00:04:48.267
the earth pulls on the moon, just as much

00:04:48.267 --> 00:04:50.270
as the moon pulls on the earth.

00:04:50.270 --> 00:04:51.945
And you might still object, you might say,

00:04:51.945 --> 00:04:54.577
wait that makes no sense, I
know the star just basically

00:04:54.577 --> 00:04:58.600
sits there and the planet gets
whipped around in a circle.

00:04:58.600 --> 00:05:00.409
How come this planet's
getting whipped around

00:05:00.409 --> 00:05:02.054
and the star's just staying put?

00:05:02.054 --> 00:05:04.867
That's because, just because
the forces are equal,

00:05:04.867 --> 00:05:06.492
that doesn't mean that
the result is equal.

00:05:06.492 --> 00:05:08.138
In other words, the forces could be equal,

00:05:08.138 --> 00:05:11.226
but the accelerations
don't have to be equal.

00:05:11.226 --> 00:05:15.940
Acceleration is gonna be the
net force divided by the mass.

00:05:15.940 --> 00:05:18.024
So even if the force is the same,

00:05:18.024 --> 00:05:19.426
you divide by that mass, you'll get

00:05:19.426 --> 00:05:22.697
a different acceleration
and that's why the result of

00:05:22.697 --> 00:05:24.688
the force does not have to be the same,

00:05:24.688 --> 00:05:27.399
even though the forces
do have to be the same,

00:05:27.399 --> 00:05:29.410
because of Newtons's Third Law.

00:05:29.410 --> 00:05:31.320
Another misconception
people sometimes make,

00:05:31.320 --> 00:05:33.860
is they think there might
be a delay in the creation

00:05:33.860 --> 00:05:36.391
of this Newtons's Third Law partner force.

00:05:36.391 --> 00:05:39.020
And people think, maybe if
I exert this first force

00:05:39.020 --> 00:05:42.048
fast enough, I can catch
the universe sleeping,

00:05:42.048 --> 00:05:43.888
and there might be some sort of delay

00:05:43.888 --> 00:05:45.603
in the creation of this other force.

00:05:45.603 --> 00:05:48.581
But that's not true, Newtons's
Third Law is universal.

00:05:48.581 --> 00:05:49.719
No matter what the situation,

00:05:49.719 --> 00:05:52.532
no matter what the acceleration
or non acceleration,

00:05:52.532 --> 00:05:54.900
or motion or no motion, whether one object

00:05:54.900 --> 00:05:57.562
is bigger or smaller, if
their Newtons's Third Law

00:05:57.562 --> 00:06:00.447
partner forces, they are
equal they are opposite

00:06:00.447 --> 00:06:02.060
and they are always equal and opposite,

00:06:02.060 --> 00:06:03.372
at every given moment in time.

00:06:03.372 --> 00:06:06.368
So even if I came in all guns a blazing,

00:06:06.368 --> 00:06:09.143
Chuck Norris style, trying
to dropkick some wall.

00:06:09.143 --> 00:06:12.869
That does not look like the
correct form for a drop kick.

00:06:12.869 --> 00:06:15.319
But even if I came in,
flying at this wall,

00:06:15.319 --> 00:06:18.854
as soon as I start to make
contact with the wall,

00:06:18.854 --> 00:06:20.143
I'm gonna exert a force on the wall,

00:06:20.143 --> 00:06:22.215
and the wall has to exert a force back.

00:06:22.215 --> 00:06:26.441
So I'd exert a force on
the wall to the right.

00:06:26.441 --> 00:06:30.383
And this would be the force
on the wall, by my foot.

00:06:30.383 --> 00:06:32.069
There'd have to be an
equal and opposite force

00:06:32.069 --> 00:06:36.275
instantly transmitted
backwards, on my foot.

00:06:36.275 --> 00:06:39.221
So this would be the force
on my foot, by the wall.

00:06:39.221 --> 00:06:42.240
This happens instantaneously,
there is no delay.

00:06:42.240 --> 00:06:43.743
You can't kick this wall fast enough,

00:06:43.743 --> 00:06:46.962
for this other force to not
be generated instantaneously.

00:06:46.962 --> 00:06:49.664
As soon as your foot starts
to exert any force on the wall

00:06:49.664 --> 00:06:51.798
what so ever, the wall
is gonna start exerting

00:06:51.798 --> 00:06:54.317
that same force back on your foot.

00:06:54.317 --> 00:06:56.085
So Newtons's Third Law is universal,

00:06:56.085 --> 00:06:58.392
but people still have trouble identifying

00:06:58.392 --> 00:07:00.566
these third law partner forces.

00:07:00.566 --> 00:07:01.774
So one of the best ways to do it,

00:07:01.774 --> 00:07:03.259
is by listing both objects,

00:07:03.259 --> 00:07:05.413
as soon as you list both objects,

00:07:05.413 --> 00:07:06.856
well to figure out where
the partner force is,

00:07:06.856 --> 00:07:09.079
you can just reverse these labels.

00:07:09.079 --> 00:07:11.711
So I know over here, if one of my forces

00:07:11.711 --> 00:07:14.211
is the force on the wall by my foot,

00:07:14.211 --> 00:07:16.292
to find the partner force to this force,

00:07:16.292 --> 00:07:18.852
I can just reverse the
labels and say it's gotta be

00:07:18.852 --> 00:07:22.915
the force on my foot, by the
wall, which I drew over here.

00:07:22.915 --> 00:07:24.329
So this is a great way to identify

00:07:24.329 --> 00:07:25.913
the third law partner forces,

00:07:25.913 --> 00:07:27.446
cause it's not always obvious

00:07:27.446 --> 00:07:29.417
what force is the partner force.

00:07:29.417 --> 00:07:30.920
So to show you how this can be tricky,

00:07:30.920 --> 00:07:32.557
consider this example.

00:07:32.557 --> 00:07:34.508
Say we got the ground and a table.

00:07:34.508 --> 00:07:38.031
So this example drives
people crazy for some reason.

00:07:38.031 --> 00:07:40.217
If I've got a box sitting on a table,

00:07:40.217 --> 00:07:42.257
we'll call it box A.

00:07:42.257 --> 00:07:44.260
Box A is gonna have forces exerted on it.

00:07:44.260 --> 00:07:47.673
One of those forces is gonna
be the gravitational force.

00:07:47.673 --> 00:07:49.360
So the force of gravity is
gonna pull straight down

00:07:49.360 --> 00:07:51.595
on box A, and if I were to ask you,

00:07:51.595 --> 00:07:54.764
what force is the third law
partner force to this force

00:07:54.764 --> 00:07:58.076
of gravity, I'm willing to
bet a lot of people might say,

00:07:58.076 --> 00:08:00.941
well there's an upwards force on box A,

00:08:00.941 --> 00:08:02.578
exerted by the table.

00:08:02.578 --> 00:08:04.000
And that's true.

00:08:04.000 --> 00:08:06.081
And if this box A is just sitting here,

00:08:06.081 --> 00:08:08.478
not accelerating, these
two forces are going

00:08:08.478 --> 00:08:09.953
to be equal and opposite.

00:08:09.953 --> 00:08:12.806
So it's even more tempting
to say that these two forces

00:08:12.806 --> 00:08:15.183
are equal and opposite
because of the third law,

00:08:15.183 --> 00:08:16.484
but that's not true.

00:08:16.484 --> 00:08:18.068
These two forces are equal and opposite

00:08:18.068 --> 00:08:19.592
because of the second law.

00:08:19.592 --> 00:08:21.929
The second law says if
there's no acceleration,

00:08:21.929 --> 00:08:23.688
then the net force has to be zero,

00:08:23.688 --> 00:08:25.301
the forces have to cancel.

00:08:25.301 --> 00:08:26.308
And that's what's happening here.

00:08:26.308 --> 00:08:27.516
These forces are equal and opposite,

00:08:27.516 --> 00:08:29.762
they're canceling on box A.

00:08:29.762 --> 00:08:31.945
Which is a way to know
that they are not third law

00:08:31.945 --> 00:08:34.293
partner forces, cause
third law partner forces

00:08:34.293 --> 00:08:36.894
are always exerted on different objects.

00:08:36.894 --> 00:08:40.165
They can never cancel if they're
third law partner forces.

00:08:40.165 --> 00:08:41.697
So what's going on over here?

00:08:41.697 --> 00:08:43.781
We've got two forces that are canceling,

00:08:43.781 --> 00:08:45.874
that are equal and opposite,
but they're not third law

00:08:45.874 --> 00:08:49.044
partner forces, they're partner
forces are somewhere else.

00:08:49.044 --> 00:08:50.832
I haven't drawn their partner forces yet.

00:08:50.832 --> 00:08:53.331
So let's try to figure out what
they're partner forces are.

00:08:53.331 --> 00:08:55.220
So let's get rid of this,
let's come back to here,

00:08:55.220 --> 00:08:58.126
let's slow it down to figure
out what the partner force is,

00:08:58.126 --> 00:09:00.207
name the two objects interacting.

00:09:00.207 --> 00:09:02.868
So this force of gravity,
I shouldn't be vague,

00:09:02.868 --> 00:09:05.571
I should call it the force on object A,

00:09:05.571 --> 00:09:09.625
our box A exerted by, well
you can't just say gravity.

00:09:09.625 --> 00:09:11.300
Gravity is not an object.

00:09:11.300 --> 00:09:12.804
So the object that is exerting

00:09:12.804 --> 00:09:15.668
this gravitational force
on A, is the earth.

00:09:15.668 --> 00:09:17.578
So this force really,
this gravitational force,

00:09:17.578 --> 00:09:19.865
if I wanna be careful,
is the force on object A

00:09:19.865 --> 00:09:21.174
exerted by the earth.

00:09:21.174 --> 00:09:23.827
Now it's easy to figure out
where the partner force is.

00:09:23.827 --> 00:09:25.035
The partner force can be found just

00:09:25.035 --> 00:09:26.712
by reversing these labels.

00:09:26.712 --> 00:09:28.683
So instead of the force on A by the earth,

00:09:28.683 --> 00:09:30.390
there's gotta be an
equal and opposite force,

00:09:30.390 --> 00:09:33.269
which is the force on the earth, by box A.

00:09:33.269 --> 00:09:35.768
So opposite means it has to point up.

00:09:35.768 --> 00:09:37.698
So it has to be an upward force.

00:09:37.698 --> 00:09:40.328
And that upward force has
to be exerted on the earth,

00:09:40.328 --> 00:09:42.734
by box A, and this is kind of weird,

00:09:42.734 --> 00:09:44.984
because you may not have realized it,

00:09:44.984 --> 00:09:48.751
but if the earth is pulling
down on a box, or you,

00:09:48.751 --> 00:09:50.905
that means you are
pulling up on the earth.

00:09:50.905 --> 00:09:52.367
And this might seem ridiculous,

00:09:52.367 --> 00:09:55.639
I mean if you jump up, you
jump up, you fall back down,

00:09:55.639 --> 00:09:57.577
you move around, but the
earth just sits there.

00:09:57.577 --> 00:09:59.540
If your forces are equal,
how come the earth doesn't

00:09:59.540 --> 00:10:01.765
move around like you do.

00:10:01.765 --> 00:10:03.452
And again, it's because just because

00:10:03.452 --> 00:10:05.594
the forces are the same, the acceleration

00:10:05.594 --> 00:10:06.865
doesn't have to be the same.

00:10:06.865 --> 00:10:08.897
The mass of the earth is so big,

00:10:08.897 --> 00:10:11.892
compared to your mass, there's
basically no acceleration.

00:10:11.892 --> 00:10:13.985
Even though the forces on you

00:10:13.985 --> 00:10:16.655
and the forces on the earth are the same.

00:10:16.655 --> 00:10:19.532
So these two are third law partner forces.

00:10:19.532 --> 00:10:22.222
These two are joined together forever.

00:10:22.222 --> 00:10:25.758
They have to be equal,
no matter what happens,

00:10:25.758 --> 00:10:27.891
these two forces will always be equal.

00:10:27.891 --> 00:10:29.812
I don't care if this box is accelerating

00:10:29.812 --> 00:10:33.255
or not accelerating, or that
there's motion or no motion.

00:10:33.255 --> 00:10:35.376
Whether it's hitting a
wall, sitting on a table,

00:10:35.376 --> 00:10:38.436
falling through space,
these two forces must

00:10:38.436 --> 00:10:41.512
always be equal and opposite,
because of the third law.

00:10:41.512 --> 00:10:42.833
So how about this other force,

00:10:42.833 --> 00:10:45.727
this force that the table was exerting.

00:10:45.727 --> 00:10:48.652
So this is, the force on A by the table.

00:10:48.652 --> 00:10:50.147
So if I wanna label it correctly,

00:10:50.147 --> 00:10:54.150
I'd call it the force on
box A, exerted by the table.

00:10:54.150 --> 00:10:56.425
Now finding the third law
partner force is easy,

00:10:56.425 --> 00:10:57.978
I can just reverse these labels,

00:10:57.978 --> 00:10:59.664
and I'd get that there must be,

00:10:59.664 --> 00:11:01.168
instead of an upwards force, a downwards

00:11:01.168 --> 00:11:03.008
force on the table, by A.

00:11:03.008 --> 00:11:05.527
So I'm gonna have another
force here on the table.

00:11:05.527 --> 00:11:07.742
It's gonna be a downward force.

00:11:07.742 --> 00:11:09.693
Downward force on the table by A,

00:11:09.693 --> 00:11:12.871
that's the third law
partner force to this upward

00:11:12.871 --> 00:11:14.923
force that the table is exerting.

00:11:14.923 --> 00:11:18.734
These two forces are also
third law partner forces.

00:11:18.734 --> 00:11:20.632
these forces are going to be equal

00:11:20.632 --> 00:11:22.736
and opposite no matter what happens.

00:11:22.736 --> 00:11:25.000
This force on box A by the table.

00:11:25.000 --> 00:11:27.235
And this force on the table by box A

00:11:27.235 --> 00:11:30.425
must be equal no matter what happens,

00:11:30.425 --> 00:11:32.680
but the force on box A by the table,

00:11:32.680 --> 00:11:34.439
does not have to be equal and opposite

00:11:34.439 --> 00:11:36.541
to the force on A by the earth.

00:11:36.541 --> 00:11:38.096
It happens to be equal and opposite,

00:11:38.096 --> 00:11:40.128
in a case where there's no acceleration.

00:11:40.128 --> 00:11:43.489
If we stuck this whole
situation into an elevator,

00:11:43.489 --> 00:11:47.939
or a rocket that had some
huge acceleration upwards,

00:11:47.939 --> 00:11:50.052
even if there's acceleration upwards,

00:11:50.052 --> 00:11:52.470
these partner forces have to be equal.

00:11:52.470 --> 00:11:53.912
So the force on A by the table,

00:11:53.912 --> 00:11:56.472
and the force on the table
by A will have to be equal.

00:11:56.472 --> 00:11:58.910
Similarly the force on the earth by A,

00:11:58.910 --> 00:12:00.861
and the force on A by the
earth have to be equal.

00:12:00.861 --> 00:12:03.746
But no longer will these
two forces have to be equal,

00:12:03.746 --> 00:12:05.473
cause they're not partner forces.

00:12:05.473 --> 00:12:08.155
They might be equal and
opposite in some circumstances,

00:12:08.155 --> 00:12:10.105
but they don't always have
to be equal and opposite.

00:12:10.105 --> 00:12:11.730
If we're accelerating upwards,

00:12:11.730 --> 00:12:13.417
this upward force on the box,

00:12:13.417 --> 00:12:15.916
must be bigger than the
downwards force on the box.

00:12:15.916 --> 00:12:17.533
So these won't be equal.

00:12:17.533 --> 00:12:19.950
Recapping quickly, Newtons's
Third Law is a statement

00:12:19.950 --> 00:12:22.124
about the forces on two different objects.

00:12:22.124 --> 00:12:23.819
And because it's about
two different objects,

00:12:23.819 --> 00:12:25.993
those forces can never cancel.

00:12:25.993 --> 00:12:28.220
To find the Newtons's
Third Law partner force,

00:12:28.220 --> 00:12:30.362
just reverse the label
after you've identified

00:12:30.362 --> 00:12:32.405
the two objects that are interacting.

00:12:32.405 --> 00:12:36.001
The third law partner forces
have to be equal in magnitude,

00:12:36.001 --> 00:12:38.448
even if one object is
larger than the other,

00:12:38.448 --> 00:12:41.150
or has more charge or
any property that might

00:12:41.150 --> 00:12:44.677
seem like it would convey more
force, than another object.

00:12:44.677 --> 00:12:46.363
If those are the two objects interacting,

00:12:46.363 --> 00:12:47.977
their forces must be of equal magnitude

00:12:47.977 --> 00:12:51.738
and opposite directions,
the forces instantaneously

00:12:51.738 --> 00:12:53.921
generated this partner forces.

00:12:53.921 --> 00:12:58.127
And be careful, some forces
might seem like partner forces,

00:12:58.127 --> 00:12:59.630
and might be equal and opposite,

00:12:59.630 --> 00:13:02.425
but they're not necessarily
third law partner forces.

00:13:02.425 --> 00:13:05.411
They made just be equal and
opposite for other reasons.


Mild and medium tension

WEBVTT
Kind: captions
Language: en

00:00:00.020 --> 00:00:02.640
- [Voiceover] You bought a huge canister,

00:00:02.640 --> 00:00:05.210
aluminum can, of super hot red peppers,

00:00:05.210 --> 00:00:08.770
three kilograms' worth, and
you hung them from two strings

00:00:08.770 --> 00:00:10.910
from the ceiling 'cause
you don't want anyone

00:00:10.910 --> 00:00:13.010
to get your super hot red peppers,

00:00:13.010 --> 00:00:14.500
and you wanted to know,

00:00:14.500 --> 00:00:16.350
what's the tension in
both of these strings?

00:00:16.350 --> 00:00:17.830
If this is the angle

00:00:17.830 --> 00:00:20.960
that the strings make with the ceiling,

00:00:20.960 --> 00:00:23.280
what are these two tensions?

00:00:23.280 --> 00:00:24.590
This problem's hard.

00:00:24.590 --> 00:00:27.910
This problem's spicy, this
is a spicy tension problem.

00:00:27.910 --> 00:00:29.950
Let's start with something
a little more mild

00:00:29.950 --> 00:00:31.370
and we'll work our way up to this

00:00:31.370 --> 00:00:32.770
to see how this works.

00:00:32.770 --> 00:00:35.760
We're gonna do that because
even though this one

00:00:35.760 --> 00:00:38.320
seems very difficult, and easier problems

00:00:38.320 --> 00:00:39.680
seem much different from it,

00:00:39.680 --> 00:00:42.510
the process that you use
to figure out the answer

00:00:42.510 --> 00:00:43.940
is always the same.

00:00:43.940 --> 00:00:45.610
So I'm gonna try to show you

00:00:45.610 --> 00:00:46.740
that you shouldn't get distracted.

00:00:46.740 --> 00:00:49.510
Even though the details
are different, the process,

00:00:49.510 --> 00:00:52.190
the overall strategy is the same.

00:00:52.190 --> 00:00:54.050
Let's start with
something a little easier.

00:00:54.050 --> 00:00:56.970
Let's start with this, a nice, red apple.

00:00:56.970 --> 00:00:59.160
Let's say this apple is three kilograms

00:00:59.160 --> 00:01:01.220
so we can keep the same number.

00:01:01.220 --> 00:01:03.610
Three kilogram apple,
hanging from a string.

00:01:03.610 --> 00:01:06.200
We wanna know what's
the tension in the rope.

00:01:06.200 --> 00:01:07.600
Well, this one's easy.

00:01:07.600 --> 00:01:09.320
The way you solve it is the way you solve

00:01:09.320 --> 00:01:10.320
all of these problems.

00:01:10.320 --> 00:01:12.170
So even though it's easy,
we're gonna go through

00:01:12.170 --> 00:01:13.890
the entire process so
you can see how it works,

00:01:13.890 --> 00:01:15.230
and it won't take long.

00:01:15.230 --> 00:01:16.270
We draw the forces.

00:01:16.270 --> 00:01:19.760
The force of gravity is
exerted on this apple

00:01:19.760 --> 00:01:21.950
because it's always exerted
on everything near the Earth,

00:01:21.950 --> 00:01:23.180
and it's mg.

00:01:23.180 --> 00:01:24.520
Now, we've got a tension force.

00:01:24.520 --> 00:01:26.280
And the tension does not push.

00:01:26.280 --> 00:01:27.520
You can't push with a rope.

00:01:27.520 --> 00:01:30.000
You can pull with a rope,
so this tension force

00:01:30.000 --> 00:01:32.800
points upward, I'll call it T.

00:01:32.800 --> 00:01:33.910
That's always the first step.

00:01:33.910 --> 00:01:35.390
You draw your force diagram,

00:01:35.390 --> 00:01:37.690
and now you use Newton's Second Law

00:01:37.690 --> 00:01:39.660
for either the horizontal direction

00:01:39.660 --> 00:01:42.460
or the vertical direction,
of both if you have to.

00:01:42.460 --> 00:01:44.280
So we're gonna use Newton's Second Law,

00:01:44.280 --> 00:01:45.600
which looks like this.

00:01:45.600 --> 00:01:48.100
And the acceleration
in a certain direction

00:01:48.100 --> 00:01:50.100
equals the net force in that direction

00:01:50.100 --> 00:01:51.250
divided by the mass.

00:01:51.250 --> 00:01:52.490
Which direction do we pick?

00:01:52.490 --> 00:01:53.560
It's pretty obvious here.

00:01:53.560 --> 00:01:54.720
We'll pick the vertical direction

00:01:54.720 --> 00:01:58.040
because there are no forces
in the horizontal direction.

00:01:58.040 --> 00:01:59.340
And I'll plug in the acceleration.

00:01:59.340 --> 00:02:02.040
If this apple's just
hanging here and not moving,

00:02:02.040 --> 00:02:04.060
then it's not accelerating,
it's just sitting here.

00:02:04.060 --> 00:02:06.720
Unless this was like an
elevator or a rocket ship

00:02:06.720 --> 00:02:08.300
that had acceleration upward.

00:02:08.300 --> 00:02:10.580
You could do that, you'd
plug in that acceleration.

00:02:10.580 --> 00:02:13.070
But if it's just hanging
from the ceiling at rest,

00:02:13.070 --> 00:02:15.140
this acceleration's gonna be zero.

00:02:15.140 --> 00:02:17.220
That's gonna equal the net force.

00:02:17.220 --> 00:02:18.310
We've got tension upward.

00:02:18.310 --> 00:02:20.310
To figure out what goes here,

00:02:20.310 --> 00:02:22.250
like we draw this force
diagram for a reason.

00:02:22.250 --> 00:02:23.820
This isn't just like

00:02:23.820 --> 00:02:26.060
happy fun painting time over here.

00:02:26.060 --> 00:02:27.320
This is a strategy.

00:02:27.320 --> 00:02:30.250
We draw these forces
'cause this lets us know

00:02:30.250 --> 00:02:33.350
what we plug into Newton's
Second Law for the net force.

00:02:33.350 --> 00:02:35.100
If it's not on here, I
don't put it up here.

00:02:35.100 --> 00:02:36.950
And if it is on here, I
have to put it in here

00:02:36.950 --> 00:02:38.050
if it's in that direction.

00:02:38.050 --> 00:02:41.050
So tension is vertical, and
this is the vertical direction

00:02:41.050 --> 00:02:42.130
for the net force.

00:02:42.130 --> 00:02:43.320
So I put tension in here.

00:02:43.320 --> 00:02:44.850
I'm gonna call upward positive.

00:02:44.850 --> 00:02:46.890
That means I make this tension positive.

00:02:46.890 --> 00:02:48.650
And how about mg over here?

00:02:48.650 --> 00:02:51.067
mg points down so I minus mg.

00:02:52.130 --> 00:02:53.970
Then I divide by the mass.

00:02:53.970 --> 00:02:56.490
Well, now I just solve for the tension.

00:02:56.490 --> 00:02:57.680
So this is the process.

00:02:57.680 --> 00:03:00.210
You draw your force diagram,
you use Newton's Second Law,

00:03:00.210 --> 00:03:02.110
and then you try to solve
for the force that you want.

00:03:02.110 --> 00:03:03.690
In this case, it's the tension.

00:03:03.690 --> 00:03:05.760
So I multiply both sides by m.

00:03:05.760 --> 00:03:07.660
It's still zero on the left-hand side,

00:03:07.660 --> 00:03:11.460
so zero equals t minus mg.

00:03:11.460 --> 00:03:13.340
And if I solve this for t,

00:03:13.340 --> 00:03:16.920
I get something that might
not be all that surprising.

00:03:16.920 --> 00:03:19.820
I just get that the
tension is equal to mg.

00:03:19.820 --> 00:03:21.800
And you might be like, "Well, duh.

00:03:22.740 --> 00:03:24.980
"That was like way more
trouble than it had to be."

00:03:24.980 --> 00:03:26.340
I mean, you just knew it, it was mg.

00:03:26.340 --> 00:03:28.220
This tension just has to balance gravity,

00:03:28.220 --> 00:03:31.340
so it's just mg, why did we
go through all this trouble?

00:03:31.340 --> 00:03:34.460
And the reason is, I
mean, it is equal to mg,

00:03:34.460 --> 00:03:37.320
but it won't always be equal to mg.

00:03:37.320 --> 00:03:40.670
So if you wanna know what to
do when it's not equal to mg,

00:03:40.670 --> 00:03:43.060
you have to know how to
use this process well.

00:03:43.060 --> 00:03:45.200
Why would it not be equal to mg?

00:03:45.200 --> 00:03:46.200
Well, imagine this.

00:03:46.200 --> 00:03:48.400
Imagine I had, say, two tensions up here.

00:03:48.400 --> 00:03:50.120
Or let's start with this,
let's say I just pull down

00:03:50.120 --> 00:03:51.310
on the rope.

00:03:51.310 --> 00:03:54.630
I just come over here and
someone pulls down on the apple.

00:03:54.630 --> 00:03:56.140
Let's say someone just
pulls down the apple

00:03:56.140 --> 00:03:57.890
with an extra five Newtons.

00:03:57.890 --> 00:04:00.470
Right, pulling this apple,
which pulls this string,

00:04:00.470 --> 00:04:01.590
which makes it tighter.

00:04:01.590 --> 00:04:02.950
What would I do in that case?

00:04:02.950 --> 00:04:05.330
Well, I've got one more force
here in my force diagram.

00:04:05.330 --> 00:04:06.890
I just add that.

00:04:06.890 --> 00:04:08.050
So that's five Newtons.

00:04:08.050 --> 00:04:10.220
It's pointing downward, so
when I come over to here

00:04:10.220 --> 00:04:12.720
to my net force, I've got
to subtract five Newtons

00:04:12.720 --> 00:04:14.730
'cause that's pointing downward.

00:04:14.730 --> 00:04:16.760
And now I do my algebra just as before.

00:04:16.760 --> 00:04:18.710
I multiply both sides by m.

00:04:18.710 --> 00:04:22.530
I'll have another minus five Newtons here.

00:04:22.530 --> 00:04:25.090
And then when I solve for t
I'll add mg to both sides,

00:04:25.090 --> 00:04:27.880
and I'll add five Newtons to both sides.

00:04:27.880 --> 00:04:29.410
And so this would be my force.

00:04:29.410 --> 00:04:31.170
So with numbers, what are we gonna get?

00:04:31.170 --> 00:04:34.460
The mass was three, so
we'll have three kilograms

00:04:35.520 --> 00:04:37.390
times the acceleration due to gravity

00:04:37.390 --> 00:04:40.050
is 9.8, but to make the numbers nice,

00:04:40.050 --> 00:04:41.850
let's just round to 10 for now,

00:04:41.850 --> 00:04:43.650
10 meters per second squared.

00:04:43.650 --> 00:04:46.900
That way we don't get lost
in details and decimals.

00:04:46.900 --> 00:04:49.030
So plus five Newtons.

00:04:49.030 --> 00:04:51.510
It's mg when it's just hanging there,

00:04:51.510 --> 00:04:53.800
but if there's extra
forces, it won't be mg.

00:04:53.800 --> 00:04:57.040
In this case, with a five
Newton force downward,

00:04:57.040 --> 00:04:58.980
it's gonna be 30 plus five,

00:04:58.980 --> 00:05:01.320
so it's gonna be 35 Newtons.

00:05:01.320 --> 00:05:03.820
So it's equal to mg in
the easiest possible case,

00:05:03.820 --> 00:05:07.270
but if there's extra forces,
it won't be equal to that.

00:05:07.270 --> 00:05:10.100
Also, if there was another rope pulling up

00:05:10.100 --> 00:05:11.590
in the same spot as the first rope,

00:05:11.590 --> 00:05:13.800
now I'd have two tensions pulling up.

00:05:13.800 --> 00:05:17.590
Now I've got another tension
force pulling up this way.

00:05:17.590 --> 00:05:18.630
So what would I do now?

00:05:18.630 --> 00:05:19.860
I'd have to add another T.

00:05:19.860 --> 00:05:21.490
I'd have plus T.

00:05:21.490 --> 00:05:24.470
Well T plus T is just 2T.

00:05:24.470 --> 00:05:27.850
So I'd have a 2T here, I'd have a 2T here.

00:05:27.850 --> 00:05:30.820
To solve for T, I'd just
have to do 35 Newtons

00:05:30.820 --> 00:05:31.820
divided by two.

00:05:31.820 --> 00:05:34.810
You could start making these
harder, and harder, and harder,

00:05:34.810 --> 00:05:36.960
and you could do that
by adding extra ropes,

00:05:36.960 --> 00:05:38.610
or adding forces down.

00:05:38.610 --> 00:05:40.860
And you might convince
yourself erroneously

00:05:40.860 --> 00:05:42.820
you have to do something
new, but you don't.

00:05:42.820 --> 00:05:45.000
You still just draw your force diagram,

00:05:45.000 --> 00:05:46.730
then you go to Newton's Second Law,

00:05:46.730 --> 00:05:48.410
you put those forces in carefully,

00:05:48.410 --> 00:05:50.350
you solve for what you wanna know.

00:05:50.350 --> 00:05:52.950
And if this is confusing,
why is it 35 over two?

00:05:52.950 --> 00:05:53.950
It kind of makes sense.

00:05:53.950 --> 00:05:56.300
The total downward force is 35 Newtons

00:05:56.300 --> 00:05:59.990
'cause this is 30 Newtons of weight here.

00:05:59.990 --> 00:06:01.920
We've got 35 Newtons in downward.

00:06:01.920 --> 00:06:04.320
The total upward force has to be 35.

00:06:04.320 --> 00:06:06.470
And if these strings are
attached at the same point,

00:06:06.470 --> 00:06:08.270
they're both gonna bear
the same amount of weight.

00:06:08.270 --> 00:06:10.450
They have to total up to 35,

00:06:10.450 --> 00:06:13.950
so you just get 35 over
two amongst each of them

00:06:13.950 --> 00:06:16.310
'cause they each bear half the weight.

00:06:16.310 --> 00:06:18.000
So that was one of the easiest problems.

00:06:18.000 --> 00:06:20.090
We turned it into a little
bit of a harder problem.

00:06:20.090 --> 00:06:22.450
It gets even harder if we add an angle.

00:06:22.450 --> 00:06:24.020
Remember, those jalapenos
were at an angle.

00:06:24.020 --> 00:06:25.220
We gotta do this.

00:06:25.220 --> 00:06:26.630
We gotta do one at an angle.

00:06:26.630 --> 00:06:29.200
Say you got a chalkboard
hanging from two strings.

00:06:29.200 --> 00:06:32.200
One string's horizontal, one
string is up here at an angle.

00:06:32.200 --> 00:06:33.610
What do you do now?

00:06:33.610 --> 00:06:36.450
It's easy to convince yourself
you gotta try something new,

00:06:36.450 --> 00:06:38.560
or go for a new strategy, but you don't.

00:06:38.560 --> 00:06:39.950
You solve this the same way.

00:06:39.950 --> 00:06:41.510
We're gonna draw our forces.

00:06:41.510 --> 00:06:43.540
We've still got a force of gravity down.

00:06:43.540 --> 00:06:46.850
So this force of gravity is just mg.

00:06:46.850 --> 00:06:50.450
And for consistency's
sake, let's say the mass

00:06:50.450 --> 00:06:53.610
of this chalkboard is
also three kilograms.

00:06:54.690 --> 00:06:55.860
All right, what else do I have?

00:06:55.860 --> 00:06:59.960
I'm gonna have a tension that
points up and to the right.

00:07:00.970 --> 00:07:02.710
It doesn't point this way.

00:07:02.710 --> 00:07:04.830
Tension doesn't push, tension pulls.

00:07:04.830 --> 00:07:06.890
So this tension's gotta pull this way.

00:07:06.890 --> 00:07:08.990
We'll call this T one.

00:07:08.990 --> 00:07:10.950
This is T one.

00:07:10.950 --> 00:07:13.290
And over on my force diagram,
it would look like this.

00:07:13.290 --> 00:07:17.310
I'd have a T one that
points something like that.

00:07:17.310 --> 00:07:19.020
So here's my T one.

00:07:19.930 --> 00:07:22.280
I'll put it here, T one.

00:07:22.280 --> 00:07:23.900
And I've got one more force now.

00:07:23.900 --> 00:07:25.830
I've got this horizontal force here.

00:07:25.830 --> 00:07:27.450
Again, it does not push.

00:07:27.450 --> 00:07:29.360
This is a rope, it can only pull.

00:07:29.360 --> 00:07:31.040
So it pulls to the left.

00:07:31.040 --> 00:07:32.630
I'll call this T two.

00:07:32.630 --> 00:07:35.180
So on my force diagram, I would have

00:07:35.180 --> 00:07:37.700
T two, and that's it.

00:07:37.700 --> 00:07:38.860
Those are all my forces.

00:07:38.860 --> 00:07:40.040
I don't have a normal force.

00:07:40.040 --> 00:07:41.740
Sometimes people wanna draw,

00:07:41.740 --> 00:07:44.070
people are so used to there
always being normal force.

00:07:44.070 --> 00:07:45.470
They're like, "Normal force, right?"

00:07:45.470 --> 00:07:46.640
And it's like, "No."

00:07:46.640 --> 00:07:49.400
There's no surface
touching this chalkboard.

00:07:49.400 --> 00:07:50.940
It's just hanging by strings.

00:07:50.940 --> 00:07:53.260
The only force keeping it up
would be the vertical component

00:07:53.260 --> 00:07:54.940
of this T one, so this is it.

00:07:54.940 --> 00:07:56.780
These are the only forces we've got.

00:07:56.780 --> 00:07:57.830
What do you do after that?

00:07:57.830 --> 00:07:58.970
Same as the apple problem.

00:07:58.970 --> 00:08:00.620
We go to Newton's Second Law,

00:08:00.620 --> 00:08:04.250
and we say that the acceleration
will be the net force

00:08:04.250 --> 00:08:05.820
divided by the mass.

00:08:05.820 --> 00:08:07.240
Which direction do we pick?

00:08:07.240 --> 00:08:08.780
It's not quite as obvious here.

00:08:08.780 --> 00:08:11.590
We've got forces vertically
and horizontally.

00:08:11.590 --> 00:08:13.030
So here's my advice.

00:08:13.030 --> 00:08:14.520
Look for something that you know.

00:08:14.520 --> 00:08:17.150
in this case, I know the
mass is three kilograms,

00:08:17.150 --> 00:08:18.890
and I know the acceleration
due to gravity,

00:08:18.890 --> 00:08:20.960
or the magnitude of it, 9.8,

00:08:20.960 --> 00:08:22.090
but in this case, 10.

00:08:22.090 --> 00:08:23.400
Since I know this force, remember,

00:08:23.400 --> 00:08:25.680
this force is just 30 Newtons

00:08:25.680 --> 00:08:28.250
'cause three times 9.8,

00:08:28.250 --> 00:08:30.550
or three times 10 meters
per second squared

00:08:30.550 --> 00:08:31.650
is just 30 Newtons.

00:08:31.650 --> 00:08:34.220
Since I know this force,
it's a vertical force,

00:08:34.220 --> 00:08:35.840
I know something about that direction.

00:08:35.840 --> 00:08:37.580
I'm just gonna start with that direction

00:08:37.580 --> 00:08:39.820
'cause I already know something about it.

00:08:39.820 --> 00:08:41.640
So I'm gonna do a in the y direction,

00:08:41.640 --> 00:08:42.660
F in the y direction.

00:08:42.660 --> 00:08:44.140
We'll start with the vertical direction.

00:08:44.140 --> 00:08:46.430
If you make a mistake and
you pick the wrong direction,

00:08:46.430 --> 00:08:47.870
it's not the end of the world.

00:08:47.870 --> 00:08:48.970
Just pick the other direction.

00:08:48.970 --> 00:08:50.650
There's only two to worry about,

00:08:50.650 --> 00:08:52.810
so if you screw one up,
just go on to the next one.

00:08:52.810 --> 00:08:54.170
It's not that big of a deal.

00:08:54.170 --> 00:08:55.960
All right, acceleration vertically again.

00:08:55.960 --> 00:08:57.780
Let's say this is at rest,

00:08:57.780 --> 00:08:59.980
just hanging from these strings.

00:08:59.980 --> 00:09:01.890
So we don't have to worry
about any acceleration.

00:09:01.890 --> 00:09:03.200
Although, if there was, you would just

00:09:03.200 --> 00:09:04.600
plug that acceleration in here.

00:09:04.600 --> 00:09:07.400
It's not that much harder of a problem.

00:09:07.400 --> 00:09:10.060
Zero equals, all right, what do we got?

00:09:10.060 --> 00:09:12.770
We need to put our vertical forces up top.

00:09:12.770 --> 00:09:13.780
I know this one.

00:09:13.780 --> 00:09:16.490
30 Newtons downward.

00:09:16.490 --> 00:09:18.480
So I'm gonna have negative 30 Newtons

00:09:18.480 --> 00:09:20.060
'cause those 30 Newtons point down,

00:09:20.060 --> 00:09:22.720
and I'm gonna consider
downward as a negative,

00:09:22.720 --> 00:09:24.400
upward as positive.

00:09:24.400 --> 00:09:25.740
This is really just mg.

00:09:25.740 --> 00:09:28.120
I could've wrote it as negative mg.

00:09:28.120 --> 00:09:29.150
What else do I have?

00:09:29.150 --> 00:09:31.820
I've got T one; T one points up,

00:09:31.820 --> 00:09:34.100
but I can't add all of T one here

00:09:34.100 --> 00:09:36.060
'cause it doesn't all point up.

00:09:36.060 --> 00:09:37.500
I can only add all of T one

00:09:37.500 --> 00:09:40.040
if T one pointed straight upward.

00:09:40.040 --> 00:09:41.040
But it doesn't.

00:09:41.040 --> 00:09:42.370
Part of it points up.

00:09:42.370 --> 00:09:45.320
So this part of T one points to the right,

00:09:45.320 --> 00:09:46.330
pulls to the right.

00:09:46.330 --> 00:09:47.800
This part of T one pulls up.

00:09:47.800 --> 00:09:50.530
It's this component right
here that's gonna be

00:09:50.530 --> 00:09:52.970
the component that actually
causes this chalkboard

00:09:52.970 --> 00:09:56.160
to stay up, that keeps
it from falling down

00:09:56.160 --> 00:09:58.170
because that's the part
that's fighting gravity.

00:09:58.170 --> 00:10:00.750
We'll call this T one in the y direction.

00:10:00.750 --> 00:10:04.120
We'll call this T one in the x direction.

00:10:04.120 --> 00:10:08.500
So we need to add plus T
one in the y direction.

00:10:08.500 --> 00:10:10.640
And that's it, those
are the only two forces

00:10:10.640 --> 00:10:11.780
that are vertical.

00:10:11.780 --> 00:10:13.410
T one x is not vertical.

00:10:13.410 --> 00:10:15.790
That's horizontal, and
this T two is horizontal.

00:10:15.790 --> 00:10:17.600
So I've included T one y and mg.

00:10:17.600 --> 00:10:19.440
Those are the only two
forces that are vertical.

00:10:19.440 --> 00:10:21.190
So now I divide by them my mass.

00:10:22.080 --> 00:10:24.190
I can multiply both sides by mass.

00:10:24.190 --> 00:10:27.000
m times zero is still zero.

00:10:27.000 --> 00:10:31.270
I get zero equals negative 30 Newtons

00:10:31.270 --> 00:10:34.070
plus T one y.

00:10:34.070 --> 00:10:35.720
And now we have to figure out, okay,

00:10:35.720 --> 00:10:37.090
T one y,

00:10:37.090 --> 00:10:38.940
T one y has to equal what?

00:10:38.940 --> 00:10:41.020
I can solve this for T one y.

00:10:41.020 --> 00:10:43.310
I add 30 to both sides.

00:10:43.310 --> 00:10:44.900
I'm gonna get T one y

00:10:44.900 --> 00:10:48.040
equals positive 30 Newtons.

00:10:48.040 --> 00:10:49.490
And that makes sense.

00:10:49.490 --> 00:10:52.410
I mean, this T one y is the only component

00:10:52.410 --> 00:10:53.760
that's balancing out gravity.

00:10:53.760 --> 00:10:54.760
We know it has to balance

00:10:54.760 --> 00:10:57.460
because there's no
acceleration vertically.

00:10:57.460 --> 00:10:59.780
So this T one y has to
be the exact same size

00:10:59.780 --> 00:11:01.080
as the force of gravity.

00:11:01.080 --> 00:11:02.740
I drew it, it's not proportional here.

00:11:02.740 --> 00:11:03.770
Sorry about that, I should've drawn it

00:11:03.770 --> 00:11:05.850
with this component
exactly the same length

00:11:05.850 --> 00:11:07.720
as this component 'cause
they to be the same,

00:11:07.720 --> 00:11:09.310
they have to cancel.

00:11:09.310 --> 00:11:10.790
But that just tells me T one y.

00:11:10.790 --> 00:11:12.810
I wanna know what T one is.

00:11:12.810 --> 00:11:15.240
How do I solve for what T one is,

00:11:15.240 --> 00:11:17.640
and what T two is?

00:11:17.640 --> 00:11:18.830
These are what I wanna figure out.

00:11:18.830 --> 00:11:19.840
What are the tensions?

00:11:19.840 --> 00:11:22.440
I don't just want the
component, I want the tension.

00:11:22.440 --> 00:11:26.500
And so now I say that
this component, T one y,

00:11:26.500 --> 00:11:28.840
is gonna be related to the total T one,

00:11:28.840 --> 00:11:31.500
and it's related through this angle here.

00:11:31.500 --> 00:11:33.170
So I can say that T one y,

00:11:33.170 --> 00:11:35.820
whatever this angle is right here.

00:11:35.820 --> 00:11:37.710
Remember, we can use trigonometry

00:11:37.710 --> 00:11:41.360
and we can say that sine theta

00:11:41.360 --> 00:11:45.330
is gonna be the opposite
side over the hypotenuse.

00:11:45.330 --> 00:11:47.830
And in this case the
opposite side to this angle,

00:11:47.830 --> 00:11:50.700
opposite is T one y.

00:11:50.700 --> 00:11:52.880
It's gonna be T one y divided by,

00:11:52.880 --> 00:11:55.140
the hypotenuse side is the total tension.

00:11:55.140 --> 00:11:57.270
So that's always the total
magnitude of the force.

00:11:57.270 --> 00:12:00.460
In this case, we're calling that T one.

00:12:00.460 --> 00:12:02.130
So I wanna solve for T one.

00:12:02.130 --> 00:12:04.710
So if I multiply both sides by T one,

00:12:04.710 --> 00:12:06.550
I'll get T one times sine theta,

00:12:06.550 --> 00:12:08.890
and then I divide both
sides by sine theta.

00:12:08.890 --> 00:12:11.900
I'll end up with T one equals

00:12:11.900 --> 00:12:16.330
T one in the y direction
divided by sine theta.

00:12:16.330 --> 00:12:17.960
I know T one in the y direction.

00:12:17.960 --> 00:12:20.490
That was 30 degree, or
sorry, not 30 degrees.

00:12:20.490 --> 00:12:22.160
That was 30 Newtons.

00:12:22.160 --> 00:12:25.900
So I've got 30 Newtons,
that's my force, upward.

00:12:25.900 --> 00:12:28.320
This vertical component right
here had to be 30 Newtons

00:12:28.320 --> 00:12:29.830
'cause it had to balance gravity,

00:12:29.830 --> 00:12:32.640
divided by sine of the angle.

00:12:32.640 --> 00:12:34.230
But what is this angle?

00:12:34.230 --> 00:12:36.260
We know this angle's 30.

00:12:36.260 --> 00:12:37.570
And you could probably convince yourself,

00:12:37.570 --> 00:12:39.400
if I draw a triangle this way,

00:12:39.400 --> 00:12:41.110
let's try to figure out,

00:12:41.110 --> 00:12:43.960
we wanna figure out what
this angle is right here

00:12:43.960 --> 00:12:45.460
'cause that's what this angle is here.

00:12:45.460 --> 00:12:47.490
So if this is 30 and that's 90,

00:12:47.490 --> 00:12:49.520
then this has to be 60.

00:12:49.520 --> 00:12:52.840
And if that's 60, and this is 90,

00:12:52.840 --> 00:12:54.220
then this has to be 30.

00:12:54.220 --> 00:12:57.220
So this angle is 30 degrees right here.

00:12:57.220 --> 00:12:58.930
So that's 30 degrees.

00:12:58.930 --> 00:13:01.400
So this angle right here,

00:13:01.400 --> 00:13:04.100
which is this angle
right here, has to be 30.

00:13:04.100 --> 00:13:05.210
So when I'm taking my sine,

00:13:05.210 --> 00:13:07.500
I'm taking my sine of 30 degrees.

00:13:07.500 --> 00:13:10.880
And I get 30 Newtons
divided by sine of 30,

00:13:10.880 --> 00:13:12.980
and sine of 30 is 1/2.

00:13:12.980 --> 00:13:17.080
So .5, so I get that this is 60 Newtons.

00:13:19.190 --> 00:13:20.610
And that might seem crazy.

00:13:20.610 --> 00:13:22.640
You might be like, "Wait a minute.

00:13:22.640 --> 00:13:25.180
"T one is 60 Newtons?

00:13:25.180 --> 00:13:26.340
"60 Newtons?

00:13:26.340 --> 00:13:29.270
"The weight of this
chalkboard is only 30 Newtons.

00:13:29.270 --> 00:13:32.340
"How in the world can
the tension in this rope

00:13:32.340 --> 00:13:34.020
"be 60 Newtons?"

00:13:34.880 --> 00:13:38.110
I mean, if we just hung
it by a single string,

00:13:38.110 --> 00:13:40.540
if we just hung this
chalkboard by a single string

00:13:40.540 --> 00:13:41.740
over the center of mass,

00:13:41.740 --> 00:13:43.600
you'd just get a tension of 30 Newtons.

00:13:43.600 --> 00:13:44.960
How can this be 60 Newtons?

00:13:44.960 --> 00:13:48.170
And the reason is, this
part's gotta be 30 Newtons.

00:13:48.170 --> 00:13:50.450
We know that 'cause it
has to balance gravity.

00:13:50.450 --> 00:13:53.120
But that's only part of the total tension.

00:13:53.120 --> 00:13:56.700
So if the total tension, if
part of the total tension is 30,

00:13:56.700 --> 00:13:58.870
all of the tension's
gotta be more than 30.

00:13:58.870 --> 00:14:01.290
And in this case, it's 60 Newtons.

00:14:01.290 --> 00:14:03.130
So that's why it's larger in this case,

00:14:03.130 --> 00:14:04.310
'cause it's at an angle.

00:14:04.310 --> 00:14:07.080
So this component has to equal gravity,

00:14:07.080 --> 00:14:08.930
and this total amount has
to be bigger than that

00:14:08.930 --> 00:14:11.750
so that its component is equal to gravity.

00:14:11.750 --> 00:14:13.640
Right, how do we figure out T two?

00:14:13.640 --> 00:14:16.070
Well, you don't invent a new strategy.

00:14:16.070 --> 00:14:18.470
We keep going, we're just
gonna say that the acceleration

00:14:18.470 --> 00:14:20.290
in the horizontal direction

00:14:20.290 --> 00:14:22.470
is the net force in the
horizontal direction

00:14:22.470 --> 00:14:24.170
divided by the mass, so we still

00:14:24.170 --> 00:14:25.820
stick with Newton's Second Law

00:14:25.820 --> 00:14:28.160
even when we wanna find this other force.

00:14:28.160 --> 00:14:30.040
This force is horizontal,
so it makes sense

00:14:30.040 --> 00:14:31.830
that we're gonna use Newton's Second Law

00:14:31.830 --> 00:14:33.240
for the horizontal direction.

00:14:33.240 --> 00:14:35.580
Again, if this chalkboard
is not accelerating,

00:14:35.580 --> 00:14:37.620
the acceleration is zero,
so I'll draw a line here

00:14:37.620 --> 00:14:40.300
to keep my calculations separate.

00:14:41.920 --> 00:14:44.270
Equals net force in the x direction.

00:14:44.270 --> 00:14:46.470
Okay, now I'm gonna have
T one in the x direction.

00:14:46.470 --> 00:14:48.880
So this is gonna be T one in the x.

00:14:48.880 --> 00:14:51.810
So I'll have T one in the x direction.

00:14:51.810 --> 00:14:53.490
That's positive 'cause it points right,

00:14:53.490 --> 00:14:55.550
and I'm gonna consider rightward positive.

00:14:55.550 --> 00:14:58.120
Minus T two, all of T two.

00:14:58.120 --> 00:14:59.470
I don't have to break T two up.

00:14:59.470 --> 00:15:02.760
T two points completely in
the horizontal direction.

00:15:02.760 --> 00:15:04.710
And I divide that by the mass.

00:15:04.710 --> 00:15:06.930
And well, I can multiply
both sides by mass.

00:15:06.930 --> 00:15:09.450
I'd get zero equals

00:15:09.450 --> 00:15:11.540
T one in the x direction

00:15:12.750 --> 00:15:16.130
puh, minus, excuse me, minus T two.

00:15:16.130 --> 00:15:19.180
So if I solve this for T two,

00:15:19.180 --> 00:15:22.070
I'm gonna get that T two, if
I add T two to both sides,

00:15:22.070 --> 00:15:24.470
I get that T two just equals

00:15:24.470 --> 00:15:26.550
T one in the x direction.

00:15:26.550 --> 00:15:28.760
But how big is T one in the x direction?

00:15:28.760 --> 00:15:31.430
We know T one is 60 Newtons.

00:15:31.430 --> 00:15:33.560
We know T one in the y direction

00:15:33.560 --> 00:15:35.670
of this piece here was 30 Newtons.

00:15:35.670 --> 00:15:37.550
How big is this piece?

00:15:37.550 --> 00:15:39.250
Well we can use, instead of sine now,

00:15:39.250 --> 00:15:41.320
we can use cosine.

00:15:41.320 --> 00:15:45.040
So if I use cosine, I can
get that cosine theta,

00:15:45.040 --> 00:15:48.220
which is 30 degrees here,
'cause this angle here's 30.

00:15:48.220 --> 00:15:51.190
Cosine of 30 would be the adjacent.

00:15:51.190 --> 00:15:54.100
That's this T one x, so it's adjacent

00:15:54.100 --> 00:15:55.500
over the hypotenuse.

00:15:55.500 --> 00:15:58.290
The hypotenuse is T
one, and we know T one.

00:15:58.290 --> 00:16:00.190
T one was 60.

00:16:00.190 --> 00:16:01.640
So I can solve for T one x,

00:16:01.640 --> 00:16:04.210
and I get that T one in the x direction,

00:16:04.210 --> 00:16:06.940
if I multiply both sides by 60 Newtons,

00:16:06.940 --> 00:16:09.420
this is 60 Newtons, I get
that T one in the x direction

00:16:09.420 --> 00:16:11.540
would be 60 Newtons

00:16:11.540 --> 00:16:14.210
times cosine of 30.

00:16:14.210 --> 00:16:17.080
And the cosine of 30 is,

00:16:17.080 --> 00:16:19.130
cosine of 30 is root three over two.

00:16:19.130 --> 00:16:21.030
So I get 60 Newtons

00:16:21.030 --> 00:16:23.970
times root three over two,

00:16:23.970 --> 00:16:27.130
which means that T one
in the x direction is,

00:16:27.130 --> 00:16:28.830
60 over two would be 30,

00:16:28.830 --> 00:16:32.350
so this is 30 root three Newtons.

00:16:32.350 --> 00:16:33.870
And that's what I can bring up here.

00:16:33.870 --> 00:16:35.460
This is T one x.

00:16:35.460 --> 00:16:39.110
So since that's T one x, I can
say that T one x right here

00:16:39.110 --> 00:16:42.780
is 30 root three Newtons.

00:16:42.780 --> 00:16:45.770
And by Newton's Second Law
in the horizontal direction,

00:16:45.770 --> 00:16:48.430
that's what T two had to equal.

00:16:48.430 --> 00:16:51.720
So T two equals 30 root three Newtons.

00:16:51.720 --> 00:16:52.720
So we figured it out.

00:16:52.720 --> 00:16:55.640
T two equals 30 root three Newtons,

00:16:55.640 --> 00:16:57.280
and that should be surprising.

00:16:57.280 --> 00:16:59.080
This force here, in order to make it

00:16:59.080 --> 00:17:01.450
so that there's no
acceleration horizontally,

00:17:01.450 --> 00:17:04.030
just has to equal this force here.

00:17:04.030 --> 00:17:06.220
Those are the only two horizontal forces.

00:17:06.220 --> 00:17:09.210
We knew T one x was 30 root three.

00:17:09.210 --> 00:17:10.350
That's what we found.

00:17:10.350 --> 00:17:13.620
So that means T two also
has to be 30 root three

00:17:13.620 --> 00:17:15.520
to make it so that these
forces are balanced

00:17:15.520 --> 00:17:17.180
in the horizontal direction.

00:17:17.180 --> 00:17:18.180
All right, so we did it.

00:17:18.180 --> 00:17:20.050
We figured out T one, 60 Newtons.

00:17:20.050 --> 00:17:23.260
We figured out T two,
30 root three Newtons.

00:17:23.260 --> 00:17:25.450
Now we're ready, now we could figure out

00:17:25.450 --> 00:17:27.580
the super hot jalapeno problem.

00:17:27.580 --> 00:17:29.300
We'll do that in the next video.


Treating systems (the easy way)

WEBVTT
Kind: captions
Language: en

00:00:00.631 --> 00:00:01.664
- [Voiceover] So in the previous video

00:00:01.664 --> 00:00:04.175
we solved this problem the hard way.

00:00:04.175 --> 00:00:05.850
Maybe you watched it, maybe you didn't,

00:00:05.850 --> 00:00:07.604
maybe you just skipped right
to here and you're like,

00:00:07.604 --> 00:00:08.855
"I don't even wanna know the hard way.

00:00:08.855 --> 00:00:11.037
"Just show me the easy way please."

00:00:11.037 --> 00:00:12.769
Well, that's what we're
gonna talk about now.

00:00:12.769 --> 00:00:14.015
Turns out there's a trick

00:00:14.015 --> 00:00:17.837
and the trick is after you
solve this problem the hard way

00:00:17.837 --> 00:00:20.576
with a five kilogram mass
and a three kilogram mass,

00:00:20.576 --> 00:00:22.030
when you find the acceleration,

00:00:22.030 --> 00:00:23.103
what you get is this.

00:00:23.103 --> 00:00:25.300
That the acceleration of
the five kilogram mass

00:00:25.300 --> 00:00:29.320
is just 29.4 divided by eight kilograms.

00:00:29.320 --> 00:00:31.908
But when you do enough of these,
you might start realizing,

00:00:31.908 --> 00:00:34.580
"Wait a minute. 29.4 Newtons.

00:00:34.580 --> 00:00:37.122
"That was just the force of gravity

00:00:37.122 --> 00:00:39.180
"pulling on this three kilogram mass."

00:00:39.180 --> 00:00:40.074
In other words,

00:00:40.074 --> 00:00:42.012
the only force that was really propelling

00:00:42.012 --> 00:00:45.272
this whole entire system forward.

00:00:45.272 --> 00:00:48.248
Or at least the only external
force propelling it forward.

00:00:48.248 --> 00:00:50.090
And then eight kilograms down here.

00:00:50.090 --> 00:00:51.980
You're gonna be like,
"Wait. Eight kilograms?

00:00:51.980 --> 00:00:55.402
"That's just five kilograms
plus three kilograms.

00:00:55.402 --> 00:00:57.442
"Is that just a coincidence

00:00:57.442 --> 00:01:00.137
"or is this telling us
something deep and fundamental?"

00:01:00.137 --> 00:01:02.080
And it's not a coincidence.

00:01:02.080 --> 00:01:03.893
Turns out you'll always get this.

00:01:03.893 --> 00:01:06.905
That what you'll end up with
after solving this hard way,

00:01:06.905 --> 00:01:08.775
you'll get in the very end,

00:01:08.775 --> 00:01:12.485
you'll get all the external
forces added up here

00:01:12.485 --> 00:01:14.629
where forces that make it go

00:01:14.629 --> 00:01:17.487
like this force of gravity
end up being positive

00:01:17.487 --> 00:01:20.275
and forces that try to resist the motion.

00:01:20.275 --> 00:01:23.274
So if there was friction, that
would be an external force

00:01:23.274 --> 00:01:26.343
that tries to resist
motion, would be up top

00:01:26.343 --> 00:01:29.531
and then you get the
total mass on the bottom.

00:01:29.531 --> 00:01:30.661
And this makes sense.

00:01:30.661 --> 00:01:33.208
The acceleration of this entire system,

00:01:33.208 --> 00:01:36.016
if we think about it as a single object---

00:01:36.016 --> 00:01:39.437
So if you imagine this
was just one single object

00:01:39.437 --> 00:01:40.668
and you asked yourself,

00:01:40.668 --> 00:01:44.098
"What's the total acceleration
of this entire system?"

00:01:44.098 --> 00:01:46.885
Well, it's only gonna depend
on the external forces

00:01:46.885 --> 00:01:49.928
and in this case, the only
external force making it go

00:01:49.928 --> 00:01:52.469
was this force of gravity right here.

00:01:52.469 --> 00:01:53.302
You might object.

00:01:53.302 --> 00:01:54.135
You might be like, "Wait.

00:01:54.135 --> 00:01:55.783
"What about this tension right here?

00:01:55.783 --> 00:01:59.191
"Isn't the tension pulling
on this five kilogram mass

00:01:59.191 --> 00:02:00.986
"making this system go?"

00:02:00.986 --> 00:02:03.976
It is but since it's
an internal force now,

00:02:03.976 --> 00:02:07.461
if we're treating this entire
system as our one object,

00:02:07.461 --> 00:02:09.894
since this tension is
trying to make it go,

00:02:09.894 --> 00:02:11.126
you've got another tension over here

00:02:11.126 --> 00:02:12.980
resisting the motion on this mass,

00:02:12.980 --> 00:02:14.280
trying to make it stop.

00:02:14.280 --> 00:02:16.236
That's what internal forces do.

00:02:16.236 --> 00:02:17.538
There's always equal and opposite

00:02:17.538 --> 00:02:19.377
on one part of the object than the other

00:02:19.377 --> 00:02:20.859
so you can't propel yourself forward

00:02:20.859 --> 00:02:22.388
with an internal force.

00:02:22.388 --> 00:02:24.736
So these end up cancelling
out essentially.

00:02:24.736 --> 00:02:26.030
The only force you have in this case

00:02:26.030 --> 00:02:27.805
was the force of gravity on top,

00:02:27.805 --> 00:02:29.442
only external forces,

00:02:29.442 --> 00:02:31.894
and the total mass on the bottom.

00:02:31.894 --> 00:02:33.113
And that's trick.

00:02:33.113 --> 00:02:35.233
That's the trick to quickly find

00:02:35.233 --> 00:02:37.417
the acceleration of some system

00:02:37.417 --> 00:02:38.890
that might be complicated

00:02:38.890 --> 00:02:40.454
if you had to do it in multiple equations

00:02:40.454 --> 00:02:41.852
and multiple unknowns

00:02:41.852 --> 00:02:44.729
but much, much easier
once you realize this.

00:02:44.729 --> 00:02:46.417
So the trick, sometimes it's called

00:02:46.417 --> 00:02:50.649
just "Treating systems
as a single object".

00:02:50.649 --> 00:02:51.847
Let me just show you really quick.

00:02:51.847 --> 00:02:54.858
If that made no sense, let me
just show you what this means.

00:02:54.858 --> 00:02:56.652
If we just get rid of this.

00:02:56.652 --> 00:02:58.767
So what I'm claiming is this.

00:02:58.767 --> 00:03:00.245
If you ever have a system

00:03:00.245 --> 00:03:02.692
where multiple objects
are required to move

00:03:02.692 --> 00:03:06.052
with the exact same magnitude
of acceleration, right?

00:03:06.052 --> 00:03:08.338
Because maybe they're
tied together by rope

00:03:08.338 --> 00:03:10.326
or maybe they're pushing on each other.

00:03:10.326 --> 00:03:12.660
Maybe there's many boxes in a row

00:03:12.660 --> 00:03:14.954
and these boxes all have to be pushed

00:03:14.954 --> 00:03:16.252
at the same acceleration

00:03:16.252 --> 00:03:19.427
because they can't get
pushed through each other.

00:03:19.427 --> 00:03:22.884
Right, if there's some
condition where multiple objects

00:03:22.884 --> 00:03:26.191
must have the same
magnitude of acceleration,

00:03:26.191 --> 00:03:30.540
then you can simply find the
acceleration of that system

00:03:30.540 --> 00:03:32.045
as if it were a single object.

00:03:32.045 --> 00:03:34.281
I'm writing, "SYS" for system.

00:03:34.281 --> 00:03:36.223
By just using Newton's second law,

00:03:36.223 --> 00:03:38.204
but instead of looking
at an individual object

00:03:38.204 --> 00:03:40.034
for a given direction,

00:03:40.034 --> 00:03:43.731
we're just gonna do all
of the external forces,

00:03:43.731 --> 00:03:46.999
all of the external forces on our system,

00:03:46.999 --> 00:03:49.043
treat it as if it were a single object,

00:03:49.043 --> 00:03:52.233
divide it by the total mass of our system.

00:03:52.233 --> 00:03:54.843
And so when you plug in
these external forces---

00:03:54.843 --> 00:03:56.025
These are forces that are external

00:03:56.025 --> 00:03:58.755
so external means not
internal to the system.

00:03:58.755 --> 00:04:02.524
So if I think of this five kilogram box

00:04:02.524 --> 00:04:06.555
and this three kilogram
box as a single mass,

00:04:06.555 --> 00:04:08.509
tension would be an internal force

00:04:08.509 --> 00:04:13.440
because it's applied internally
between these two objects,

00:04:13.440 --> 00:04:15.973
between objects inside of our system.

00:04:15.973 --> 00:04:18.485
But the force of gravity
on the three kilogram mass,

00:04:18.485 --> 00:04:21.021
that's an external force
'cause that's the Earth

00:04:21.021 --> 00:04:23.275
pulling down on the three kilogram mass

00:04:23.275 --> 00:04:26.498
and the Earth is not part of our system.

00:04:26.498 --> 00:04:29.607
Similarly, the normal
force is an external force

00:04:29.607 --> 00:04:33.203
but it's exactly cancelled
by the gravitational force.

00:04:33.203 --> 00:04:35.114
So even though those are external,

00:04:35.114 --> 00:04:36.212
they're not gonna make it in here.

00:04:36.212 --> 00:04:37.072
I mean, you can put 'em in there

00:04:37.072 --> 00:04:38.509
but they're just gonna cancel anyway.

00:04:38.509 --> 00:04:41.515
We only look at forces in
the direction of motion

00:04:41.515 --> 00:04:44.249
and if it's a force that causes motion,

00:04:44.249 --> 00:04:46.155
we're gonna make that a positive force.

00:04:46.155 --> 00:04:48.282
If it's a force in the direction of motion

00:04:48.282 --> 00:04:49.936
like this force of gravity is,

00:04:49.936 --> 00:04:51.884
we make those positive forces.

00:04:51.884 --> 00:04:55.708
So forces will be plugged
in positive into here

00:04:55.708 --> 00:04:58.649
if they make the system go.

00:04:58.649 --> 00:05:00.608
And that might seem weird.

00:05:00.608 --> 00:05:01.441
You might be like, "Wait.

00:05:01.441 --> 00:05:03.717
"How do I decide if it
makes the system go?"

00:05:03.717 --> 00:05:06.081
Well, just ask yourself,
"Is that force directed

00:05:06.081 --> 00:05:09.357
"in the same direction as
the motion of the system?"

00:05:09.357 --> 00:05:12.167
So, we're just saying the
system is gonna accelerate

00:05:12.167 --> 00:05:14.251
if there's forces that make it go

00:05:14.251 --> 00:05:17.695
and we're gonna plug in negative forces,

00:05:17.695 --> 00:05:20.321
the forces that make the system stop

00:05:20.321 --> 00:05:22.976
or resist the motion of system.

00:05:22.976 --> 00:05:27.037
So maybe I should say,
"Resist motion of the system."

00:05:27.037 --> 00:05:28.611
In this case, for this one down here,

00:05:28.611 --> 00:05:30.610
I don't have any of those.

00:05:30.610 --> 00:05:33.193
So resist motion of the system.

00:05:35.100 --> 00:05:35.933
I don't have any of those.

00:05:35.933 --> 00:05:39.215
I could have if I had a force of friction.

00:05:39.215 --> 00:05:42.994
Then there'd be a external
force that resist the motion.

00:05:42.994 --> 00:05:45.835
I would plug in that
external force as a negative

00:05:45.835 --> 00:05:47.814
'cause it resist the motion.

00:05:47.814 --> 00:05:49.719
So even though this might sound weird,

00:05:49.719 --> 00:05:51.929
it makes sense if you think about it.

00:05:51.929 --> 00:05:54.805
The acceleration of our system
treated as a single object

00:05:54.805 --> 00:05:56.671
is only gonna depend on the forces

00:05:56.671 --> 00:05:58.547
that try to make the system go

00:05:58.547 --> 00:06:00.846
and the forces that try
to make the system stop

00:06:00.846 --> 00:06:02.269
or resist the motion.

00:06:02.269 --> 00:06:05.254
So if we add those accordingly
with positives and negatives,

00:06:05.254 --> 00:06:06.614
we divide it by the total mass

00:06:06.614 --> 00:06:09.395
which gives the total measure
of the inertia of our system,

00:06:09.395 --> 00:06:11.387
we'll get the acceleration of our system.

00:06:11.387 --> 00:06:13.190
It makes sense and it works.

00:06:13.190 --> 00:06:14.237
Turns out it always works

00:06:14.237 --> 00:06:16.653
and it saves a ridiculous amount of time.

00:06:16.653 --> 00:06:19.110
For instance, if we
wanted to do this problem,

00:06:19.110 --> 00:06:21.124
if you just gave me this
problem straight away

00:06:21.124 --> 00:06:22.928
and you were told, "Do
this however you want.",

00:06:22.928 --> 00:06:24.006
I would use this trick.

00:06:24.006 --> 00:06:27.302
And I would say that the
acceleration of this system

00:06:27.302 --> 00:06:29.497
which is composed of
this five kilogram mass

00:06:29.497 --> 00:06:31.377
and our three kilogram mass

00:06:31.377 --> 00:06:32.513
is just gonna be equal to---

00:06:32.513 --> 00:06:35.443
I'd ask myself, "What
force makes this system go?

00:06:35.443 --> 00:06:37.587
"What force drives this system?"

00:06:37.587 --> 00:06:40.705
And it's this force of gravity
on the three kilogram mass

00:06:40.705 --> 00:06:42.569
that's driving this system, right?

00:06:42.569 --> 00:06:44.510
If you took this force away,

00:06:44.510 --> 00:06:46.206
if you eliminated that force,

00:06:46.206 --> 00:06:47.657
nothing's gonna happen here.

00:06:47.657 --> 00:06:49.869
This is the force making the system go

00:06:49.869 --> 00:06:54.036
so I'd put it in my three
kilograms times nine point eight.

00:06:55.643 --> 00:06:56.476
And at this point,

00:06:56.476 --> 00:06:58.654
you might be like, "Well,
okay, that gravity made it go.

00:06:58.654 --> 00:07:00.698
"Should I include this gravity, too?"

00:07:00.698 --> 00:07:04.347
But no, that gravity is
perpendicular to the motion for one

00:07:04.347 --> 00:07:06.890
so this gravity isn't making the system,

00:07:06.890 --> 00:07:09.420
that's just causing this
mass to sit on the table

00:07:09.420 --> 00:07:12.308
and for two, it's cancelled
by that normal force.

00:07:12.308 --> 00:07:13.756
So those cancel anyway,

00:07:13.756 --> 00:07:16.010
even though they're external forces.

00:07:16.010 --> 00:07:18.391
This is it, this is the only
one that drives the system.

00:07:18.391 --> 00:07:21.386
So I put that in here and
I divide by my total mass

00:07:21.386 --> 00:07:23.428
'cause that tells me how
much my system resists

00:07:23.428 --> 00:07:25.245
through inertia,

00:07:25.245 --> 00:07:27.483
changes in velocity,
and this is what I get.

00:07:27.483 --> 00:07:28.534
I get the same thing I got before,

00:07:28.534 --> 00:07:32.085
I get back my three point six eight

00:07:32.085 --> 00:07:35.037
meters per second squared,
and I get in one line.

00:07:35.037 --> 00:07:37.646
I mean, this trick is
amazing and it works,

00:07:37.646 --> 00:07:38.958
and it works in every example

00:07:38.958 --> 00:07:40.825
where two masses or more masses

00:07:40.825 --> 00:07:44.209
are forced to move with
the same acceleration.

00:07:44.209 --> 00:07:46.864
So this is great. This'll
save you a ton of time.

00:07:46.864 --> 00:07:48.661
This is supposed to be a three here.

00:07:48.661 --> 00:07:50.238
And to show you how useful it is,

00:07:50.238 --> 00:07:51.704
let say there was friction,

00:07:51.704 --> 00:07:53.867
let's say there was a
coefficient of friction

00:07:53.867 --> 00:07:56.124
of zero point three.

00:07:56.124 --> 00:07:57.787
Well, now I'd have a frictional force

00:07:57.787 --> 00:08:00.509
so there'd be an external
frictional force here.

00:08:00.509 --> 00:08:02.640
It'd be applied this five kilogram mass.

00:08:02.640 --> 00:08:04.508
I'd have to subtract it up here.

00:08:04.508 --> 00:08:06.203
So if I get rid of this---

00:08:06.203 --> 00:08:08.535
So it's not gonna be three
point six eight anymore.

00:08:08.535 --> 00:08:12.699
I'm gonna have a force of
friction that I have to subtract.

00:08:12.699 --> 00:08:16.630
So minus mu K so the force of friction---

00:08:16.630 --> 00:08:18.309
I'll just put force of friction.

00:08:18.309 --> 00:08:21.232
And so to solve for the force of friction,

00:08:21.232 --> 00:08:23.403
the force of friction
is gonna be equal to---

00:08:23.403 --> 00:08:25.610
Well, I know three times
nine point eight is---

00:08:25.610 --> 00:08:26.599
Let me just write this in here,

00:08:26.599 --> 00:08:30.766
29.4 Newtons minus the force
of friction it's given by.

00:08:32.012 --> 00:08:34.851
So there's a formula
for force of friction.

00:08:34.851 --> 00:08:38.184
The force of friction is always mu K FN.

00:08:39.193 --> 00:08:41.411
So the force of friction
on this five kilogram mass

00:08:41.411 --> 00:08:44.370
is gonna be mu K which is point three.

00:08:44.370 --> 00:08:48.067
So it's gonna be zero point
three times the normal force,

00:08:48.067 --> 00:08:50.492
not the normal force on our entire system.

00:08:50.492 --> 00:08:52.451
I don't include this three kilogram mass.

00:08:52.451 --> 00:08:55.318
It's only the normal force
on this five kilogram mass

00:08:55.318 --> 00:08:58.332
that's contributing to this
force of friction here.

00:08:58.332 --> 00:09:00.445
So even though we're treating
this system as a whole,

00:09:00.445 --> 00:09:03.102
we still have to find individual forces

00:09:03.102 --> 00:09:06.434
on this individual boxes correctly.

00:09:06.434 --> 00:09:08.819
So it won't be the entire
mass that goes here.

00:09:08.819 --> 00:09:10.491
The normal force on the five kilogram mass

00:09:10.491 --> 00:09:14.658
is just gonna be five kilograms
times nine point eight

00:09:17.510 --> 00:09:19.399
meters per second squared.

00:09:19.399 --> 00:09:21.535
I divide by my total mass down here

00:09:21.535 --> 00:09:26.257
because the entire mass is
resisting motion through inertia.

00:09:26.257 --> 00:09:28.880
And if I solve this from my
acceleration of the system,

00:09:28.880 --> 00:09:33.110
I get one point eight four
meters per seconds squared.

00:09:33.110 --> 00:09:35.871
So this is less, less than
our three point six eight

00:09:35.871 --> 00:09:36.849
and that makes sense.

00:09:36.849 --> 00:09:38.642
Now, there's a resistive force,

00:09:38.642 --> 00:09:40.324
a resistive external force,

00:09:40.324 --> 00:09:42.508
tryna prevent the system from moving.

00:09:42.508 --> 00:09:43.824
But you have to be careful.

00:09:43.824 --> 00:09:45.583
What I'm really finding here,

00:09:45.583 --> 00:09:49.100
I'm really finding the
magnitude of the acceleration.

00:09:49.100 --> 00:09:50.508
This is just giving me the magnitude.

00:09:50.508 --> 00:09:51.350
If I'm playing this game

00:09:51.350 --> 00:09:54.007
where positive forces
are ones that make it go

00:09:54.007 --> 00:09:56.802
and negative forces are
ones that resist motion,

00:09:56.802 --> 00:09:58.389
external forces that is,

00:09:58.389 --> 00:10:00.643
I'm just getting the
magnitude of the acceleration.

00:10:00.643 --> 00:10:03.489
Individual boxes will have that magnitude

00:10:03.489 --> 00:10:04.529
of the acceleration

00:10:04.529 --> 00:10:07.104
but they may have positive
or negative accelerations.

00:10:07.104 --> 00:10:08.039
In other words,

00:10:08.039 --> 00:10:11.289
this five kilogram mass
accelerating to the right,

00:10:11.289 --> 00:10:13.339
gonna have a positive acceleration.

00:10:13.339 --> 00:10:15.877
In other words, the acceleration
of the five kilogram mass

00:10:15.877 --> 00:10:18.197
will be positive one point eight four

00:10:18.197 --> 00:10:20.181
and the acceleration of
the three kilogram mass

00:10:20.181 --> 00:10:21.677
since it's accelerating downward

00:10:21.677 --> 00:10:24.291
will be negative one point eight four

00:10:24.291 --> 00:10:25.124
meters per seconds squared.