Intuition on static and kinetic friction comparisons

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.680
I mentioned in the
last several videos

00:00:02.680 --> 00:00:07.210
that the coefficient of kinetic
friction tends to be less,

00:00:07.210 --> 00:00:09.280
sometimes it'll be
roughly equal to,

00:00:09.280 --> 00:00:11.080
the coefficient of
static friction.

00:00:11.080 --> 00:00:12.970
But this might lead
you to-- at least,

00:00:12.970 --> 00:00:14.470
a question that
I've had in my mind,

00:00:14.470 --> 00:00:17.760
and I still have to
some degree-- is why?

00:00:17.760 --> 00:00:20.680
Why is the coefficient of
kinetic friction lower?

00:00:20.680 --> 00:00:22.930
Or why can it be lower?

00:00:22.930 --> 00:00:26.842
And the current best theory--
one I can visualize in my head,

00:00:26.842 --> 00:00:28.550
and based on the
reading that I've done--

00:00:28.550 --> 00:00:31.850
is the difference between-- so
let's think about it this way.

00:00:31.850 --> 00:00:35.450
So if we look at it at a kind
of a regular human level,

00:00:35.450 --> 00:00:36.620
maybe we have a block.

00:00:39.190 --> 00:00:41.390
So this is the static case.

00:00:41.390 --> 00:00:43.570
So let's think about
the static case.

00:00:43.570 --> 00:00:45.110
Let me draw it like this.

00:00:45.110 --> 00:00:47.010
So I'll draw the
static case over here.

00:00:47.010 --> 00:00:50.780
So I have a block that
is stationary on top

00:00:50.780 --> 00:00:53.480
of-- let me do the surface
in a different color-- on top

00:00:53.480 --> 00:00:56.300
of some type of surface
right over here.

00:00:56.300 --> 00:00:59.220
And over here, I'm going
to have a block moving

00:00:59.220 --> 00:01:05.120
at a constant velocity
relative to some surface,

00:01:05.120 --> 00:01:07.090
relative to the same surface.

00:01:07.090 --> 00:01:08.710
And so let me draw it out.

00:01:08.710 --> 00:01:11.145
So this is moving at
some constant velocity.

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And so the
interesting thing here

00:01:17.730 --> 00:01:20.760
is, assuming that these are
the same masses, that these are

00:01:20.760 --> 00:01:22.940
the same surfaces,
is, why should

00:01:22.940 --> 00:01:25.950
the coefficient of
friction here-- why

00:01:25.950 --> 00:01:30.262
should the coefficient of static
friction-- so here, since this

00:01:30.262 --> 00:01:31.720
is stationary,
what's under play is

00:01:31.720 --> 00:01:33.178
the coefficient of
static friction.

00:01:33.178 --> 00:01:36.180
Why should that be larger
than the coefficient

00:01:36.180 --> 00:01:37.945
of kinetic friction over here?

00:01:37.945 --> 00:01:39.820
Why should that be large
than the coefficient

00:01:39.820 --> 00:01:40.880
of kinetic friction?

00:01:40.880 --> 00:01:42.660
Or another way to
think about it is,

00:01:42.660 --> 00:01:47.520
you would need to apply
more force to overcome

00:01:47.520 --> 00:01:50.520
the static friction here, and
start to get this accelerating,

00:01:50.520 --> 00:01:54.120
than you would need to
apply to get this already

00:01:54.120 --> 00:01:56.700
moving body to accelerate.

00:01:56.700 --> 00:02:00.631
Because there would be kind of
a less of a responsive friction

00:02:00.631 --> 00:02:01.130
force.

00:02:01.130 --> 00:02:03.260
So let's think about
that a little bit.

00:02:03.260 --> 00:02:08.090
So what I'm going to do is
zoom in into the atomic level.

00:02:08.090 --> 00:02:10.190
And so when you zoom
in to the atomic level,

00:02:10.190 --> 00:02:12.920
almost nothing is
completely smooth.

00:02:12.920 --> 00:02:18.420
So the surface over here might
look something like this.

00:02:18.420 --> 00:02:21.820
So I'm going to draw the
molecules that make up

00:02:21.820 --> 00:02:26.410
the surface, the
best to my ability.

00:02:26.410 --> 00:02:28.930
So the molecules, when
you zoom up really close

00:02:28.930 --> 00:02:32.510
for the surface, might
look something like this.

00:02:32.510 --> 00:02:35.940
So we're really zooming
into the atomic level,

00:02:35.940 --> 00:02:37.150
unimaginably small level.

00:02:37.150 --> 00:02:38.820
Much smaller than
that box I just drew.

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But I'm just trying
to look at what's

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happening with the atoms
where they contact,

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or the molecules
where they contact?

00:02:44.070 --> 00:02:46.575
And the box's molecules might
look something like this.

00:02:51.030 --> 00:02:52.360
They aren't completely smooth.

00:02:55.460 --> 00:02:57.340
And hopefully this
video also emphasizes

00:02:57.340 --> 00:02:59.800
that all of these forces and
all of this contact that we're

00:02:59.800 --> 00:03:01.470
talking about in these
videos-- and it's actually

00:03:01.470 --> 00:03:03.330
interesting
philosophically-- nothing

00:03:03.330 --> 00:03:05.780
is ever really in
contact with each other.

00:03:05.780 --> 00:03:10.730
You really just have atoms
that are repulsing each other,

00:03:10.730 --> 00:03:14.100
because their electrons
the electromagnetic force

00:03:14.100 --> 00:03:16.070
of repulsion between
them is not allowing

00:03:16.070 --> 00:03:17.810
them to get any closer together.

00:03:17.810 --> 00:03:20.055
So that's all-- when
you push something,

00:03:20.055 --> 00:03:26.269
it's just the electrons in your
hand pushing on the electron--

00:03:26.269 --> 00:03:28.810
or the electronic clouds in your
hand pushing on the electron

00:03:28.810 --> 00:03:31.700
clouds of, say, the
pen you're holding,

00:03:31.700 --> 00:03:34.710
or the key on your
keyboard, or the mug,

00:03:34.710 --> 00:03:37.260
so that it repulses
it and causes

00:03:37.260 --> 00:03:38.740
it to go in the other direction.

00:03:38.740 --> 00:03:41.010
So there's never
any of this thing

00:03:41.010 --> 00:03:45.450
like, what we imagine in
our heads, real contact.

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And if you really want
to blow your mind--

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and watch the chemistry
videos if you want

00:03:49.530 --> 00:03:52.056
understand this-- is
that most of these atoms

00:03:52.056 --> 00:03:53.555
are actually free
spaced themselves.

00:03:53.555 --> 00:03:58.180
That the electron
cloud-- or I guess

00:03:58.180 --> 00:04:00.670
where most of the probability
of finding the electron--

00:04:00.670 --> 00:04:03.590
is huge compared to the
size of the electron,

00:04:03.590 --> 00:04:05.120
or the size of the nucleus.

00:04:05.120 --> 00:04:07.460
So it's kind of just a
lot of free space pushing

00:04:07.460 --> 00:04:10.220
on a lot of other free space
through the electromagnetic

00:04:10.220 --> 00:04:10.780
force.

00:04:10.780 --> 00:04:13.270
But anyway, we're talking
about friction here.

00:04:13.270 --> 00:04:15.660
So if you were to really zoom
in here, when this thing is

00:04:15.660 --> 00:04:20.040
stationary, the surfaces
aren't actually even.

00:04:20.040 --> 00:04:22.890
And so you could imagine that
these molecules that you have,

00:04:22.890 --> 00:04:24.790
sometimes when it's
sitting stationary,

00:04:24.790 --> 00:04:26.810
they might be kind of
fit into each other.

00:04:26.810 --> 00:04:29.810
They've kind of slid in to
maybe these little ruts here

00:04:29.810 --> 00:04:30.690
and there.

00:04:30.690 --> 00:04:32.920
And so if you're trying
to move this object,

00:04:32.920 --> 00:04:36.680
if you're trying to accelerate
it to the left with some force,

00:04:36.680 --> 00:04:39.210
you have to overcome,
essentially,

00:04:39.210 --> 00:04:41.800
either-- for example,
this part right over here

00:04:41.800 --> 00:04:44.710
either has to somehow break
off, or the whole thing

00:04:44.710 --> 00:04:48.170
has to be shifted up a couple
of atoms or a couple molecules.

00:04:48.170 --> 00:04:51.280
Or maybe, this part over
here has to be broken off,

00:04:51.280 --> 00:04:53.460
or has to be shifted
down one atom-- you

00:04:53.460 --> 00:04:54.990
wouldn't notice these things.

00:04:54.990 --> 00:04:57.072
You wouldn't notice the
shifting of a block,

00:04:57.072 --> 00:04:58.280
or the shifting of the floor.

00:04:58.280 --> 00:05:01.370
You wouldn't notice it by
the width of a molecule,

00:05:01.370 --> 00:05:02.829
or diameter of an
atom or molecule.

00:05:02.829 --> 00:05:05.119
But that's essentially what
you're going to have to do.

00:05:05.119 --> 00:05:06.930
Or you have to rip
them off entirely

00:05:06.930 --> 00:05:09.830
in order to start
this thing moving.

00:05:09.830 --> 00:05:12.580
Once something is
already moving--

00:05:12.580 --> 00:05:16.270
and this is at least
how I think about it--

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it doesn't have a chance to
settle into these little ruts.

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So let me draw something
that's already moving.

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And I'll try to draw
a similar surface.

00:05:26.560 --> 00:05:28.530
So I'm trying to
draw the surface that

00:05:28.530 --> 00:05:32.460
looks, essentially, just
like the one I drew.

00:05:32.460 --> 00:05:35.310
So maybe it looks like that.

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This is supposed to
be the same surface.

00:05:37.040 --> 00:05:42.620
But once it's moving, it's not
sitting in these ruts anymore.

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The whole thing is moving.

00:05:43.710 --> 00:05:46.110
So it's kind of
sliding across the top.

00:05:46.110 --> 00:05:48.730
And so now it might look
something like this.

00:05:48.730 --> 00:05:50.040
I'll try my best to draw it.

00:05:53.889 --> 00:05:55.680
Maybe this has been
shifted up a little bit

00:05:55.680 --> 00:05:57.420
so that it could start sliding.

00:05:57.420 --> 00:06:01.490
You've overcame the
static friction.

00:06:01.490 --> 00:06:09.000
So now it is-- I'm trying to
draw the same surface here,

00:06:09.000 --> 00:06:11.160
give or take-- so
now it's moving.

00:06:11.160 --> 00:06:13.570
It doesn't have a chance
to really settle in.

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It has to kind of
bounce along the top.

00:06:16.010 --> 00:06:17.820
And so that's the
best understanding.

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And so the real force
of the friction here,

00:06:23.341 --> 00:06:25.590
as it's moving along it still
might every now and then

00:06:25.590 --> 00:06:27.510
bounce into a little
ruts here and there.

00:06:27.510 --> 00:06:32.320
But you also have any type
of chemical bonds that

00:06:32.320 --> 00:06:34.710
form between the
atoms temporarily

00:06:34.710 --> 00:06:36.660
that keep breaking and forming.

00:06:36.660 --> 00:06:41.350
And in order to keep this
thing, I guess, moving,

00:06:41.350 --> 00:06:43.730
or especially if you
want to accelerate it,

00:06:43.730 --> 00:06:45.870
you're going to have to
keep breaking these bonds.

00:06:45.870 --> 00:06:47.580
And so that's essentially
the force of friction

00:06:47.580 --> 00:06:48.680
that you're overcoming.

00:06:48.680 --> 00:06:51.450
Here you might have
those same bonds.

00:06:51.450 --> 00:06:53.230
And not only do you
have the same bonds,

00:06:53.230 --> 00:06:56.060
but you also have to
overcome these ruts,

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or these little ragged
parts that have a time

00:07:00.050 --> 00:07:01.960
to settle in to
these little nooks

00:07:01.960 --> 00:07:03.725
that you have to
overcome even more.

00:07:03.725 --> 00:07:04.725
So that's the intuition.

00:07:04.725 --> 00:07:07.620
And you know this is actually
still an area of research.

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So it's not like this
cut and dry thing.

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And it's a fun thing
to think about what's

00:07:11.036 --> 00:07:12.320
happening at the atomic level.

00:07:12.320 --> 00:07:14.240
But this is the
general intuition

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of why the coefficient
of static friction

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is higher than the coefficient
of kinetic friction.


Force of friction keeping velocity constant

WEBVTT
Kind: captions
Language: en

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I want to make a
quick clarification

00:00:02.260 --> 00:00:06.400
to the last video, and then
think about what's friction

00:00:06.400 --> 00:00:09.070
up to when the block
is actually moving.

00:00:09.070 --> 00:00:10.570
So in the last
video, we started off

00:00:10.570 --> 00:00:12.030
with the block being stationary.

00:00:12.030 --> 00:00:14.190
We knew that the
parallel component

00:00:14.190 --> 00:00:16.030
of the force of
gravity on that block

00:00:16.030 --> 00:00:19.734
was 49 newtons downwards,
down the slope.

00:00:19.734 --> 00:00:21.150
And when the block
was stationary,

00:00:21.150 --> 00:00:22.952
we said there must be
an offsetting force.

00:00:22.952 --> 00:00:24.660
And we said that's
the force of friction,

00:00:24.660 --> 00:00:27.600
and it must be 49
newtons upwards.

00:00:27.600 --> 00:00:31.010
And so they completely
net out in that direction.

00:00:31.010 --> 00:00:34.600
Now, what we said is we're going
to keep applying a little bit

00:00:34.600 --> 00:00:38.740
more force until we
can budge this block

00:00:38.740 --> 00:00:40.310
to start accelerating downwards.

00:00:40.310 --> 00:00:44.070
And I said I kept applying
a little bit more force,

00:00:44.070 --> 00:00:46.340
a little bit more force,
until I get to 1 newton,

00:00:46.340 --> 00:00:48.569
and then the block
started to budge.

00:00:48.569 --> 00:00:50.360
So at that point, when
it started to budge,

00:00:50.360 --> 00:00:53.890
I'm applying this 1 newton
over here, right over here.

00:00:53.890 --> 00:00:56.860
There was already
49 newtons of force,

00:00:56.860 --> 00:00:59.750
or the component of
gravity, in this direction.

00:00:59.750 --> 00:01:02.750
So combined, we're
providing 50 newtons

00:01:02.750 --> 00:01:06.070
to just start budging
it, to just overcome

00:01:06.070 --> 00:01:07.240
the force of friction.

00:01:07.240 --> 00:01:08.740
The one thing I
want to clarify here

00:01:08.740 --> 00:01:10.780
is this whole time
the force of friction

00:01:10.780 --> 00:01:13.320
was not constant at 49 newtons.

00:01:13.320 --> 00:01:16.420
When I wasn't messing
with this block,

00:01:16.420 --> 00:01:19.390
and the parallel component of
the force was 49 newtons, then

00:01:19.390 --> 00:01:21.670
the force of friction
was 49 newtons.

00:01:21.670 --> 00:01:23.640
When I started to press
on it a little bit,

00:01:23.640 --> 00:01:25.140
apply a little bit
of force, maybe I

00:01:25.140 --> 00:01:27.480
applied a tenth of a
newton on top of that,

00:01:27.480 --> 00:01:30.550
then the force of friction
was 49 and 1/10 newton,

00:01:30.550 --> 00:01:32.810
because it was still
providing enough force so

00:01:32.810 --> 00:01:34.690
that this block was not moving.

00:01:34.690 --> 00:01:36.930
Then maybe I applied
half a newton.

00:01:36.930 --> 00:01:39.060
And so the total force
in the downward direction

00:01:39.060 --> 00:01:40.930
would have been 49
and 1/2 newtons.

00:01:40.930 --> 00:01:44.010
But if it still was not moving,
then the force of friction

00:01:44.010 --> 00:01:45.469
was still completely
overcoming it.

00:01:45.469 --> 00:01:47.135
So the force of
friction, at that point,

00:01:47.135 --> 00:01:48.990
must have been 49
and 1/2 newtons,

00:01:48.990 --> 00:01:52.370
all the way up to the combined
force in the downward direction

00:01:52.370 --> 00:01:55.360
being 49.999999 newtons.

00:01:55.360 --> 00:01:59.160
And then the force of friction
was still 49.99999 newtons,

00:01:59.160 --> 00:02:01.740
all the way until I hit 50
newtons and then the block

00:02:01.740 --> 00:02:04.600
started to budge, which tells us
that the force of friction now,

00:02:04.600 --> 00:02:07.310
all of a sudden, or at least
the force of static friction all

00:02:07.310 --> 00:02:09.330
of a sudden now
couldn't keep up and it

00:02:09.330 --> 00:02:11.480
started to accelerate downwards.

00:02:11.480 --> 00:02:14.430
So in that static scenario,
the force of friction

00:02:14.430 --> 00:02:18.680
changed as I applied
more or less force

00:02:18.680 --> 00:02:20.684
in this downward direction.

00:02:20.684 --> 00:02:22.350
Now with that out of
the way, let's take

00:02:22.350 --> 00:02:23.570
a different scenario.

00:02:23.570 --> 00:02:27.600
Let me just redraw that same
block, just since all of this

00:02:27.600 --> 00:02:29.930
is getting messy.

00:02:29.930 --> 00:02:35.140
So we have the same block, and
as we said in the last video,

00:02:35.140 --> 00:02:37.750
we're now assuming that
this is wood on wood.

00:02:37.750 --> 00:02:40.460
So this is the wedge.

00:02:40.460 --> 00:02:42.990
This is the block
right over here.

00:02:42.990 --> 00:02:46.980
We know that the component
of gravity that is parallel

00:02:46.980 --> 00:02:51.460
to the plane right
there is 49 newtons.

00:02:51.460 --> 00:02:53.579
We know that this is 49 newtons.

00:02:53.579 --> 00:02:55.120
We know the component
of gravity that

00:02:55.120 --> 00:02:57.036
is perpendicular to the
plane-- we figured out

00:02:57.036 --> 00:03:00.755
this two videos ago-- is 49
square roots of 3 newtons.

00:03:03.430 --> 00:03:05.620
We know that this block
is not accelerating

00:03:05.620 --> 00:03:08.440
in this normal
direction, so there

00:03:08.440 --> 00:03:11.830
must be some force counteracting
gravity in that direction.

00:03:11.830 --> 00:03:15.740
And that's the normal force
of the wedge on the block.

00:03:15.740 --> 00:03:18.450
So that is going
in that direction

00:03:18.450 --> 00:03:22.090
at 49 square roots of 3 newtons.

00:03:22.090 --> 00:03:25.330
And now instead of assuming
that this block is stationary,

00:03:25.330 --> 00:03:28.540
let's assume that it's moving
with a constant velocity.

00:03:28.540 --> 00:03:30.170
So now we're dealing
with-- let me

00:03:30.170 --> 00:03:31.540
do that in a different color.

00:03:31.540 --> 00:03:33.640
So now we're dealing
with a scenario

00:03:33.640 --> 00:03:35.500
where the block has
a constant velocity.

00:03:39.480 --> 00:03:41.920
And for the sake of
this video, we'll

00:03:41.920 --> 00:03:47.100
assume that that constant
velocity is downward.

00:03:47.100 --> 00:03:51.390
And so the constant velocity,
v, is equal to-- I don't know.

00:03:51.390 --> 00:03:58.060
Let's say it is 5 meters
per second down the wedge,

00:03:58.060 --> 00:03:59.246
or down the ramp.

00:03:59.246 --> 00:04:01.620
Or I guess we could say in
the direction that is parallel

00:04:01.620 --> 00:04:02.830
to the surface of the ramp.

00:04:02.830 --> 00:04:05.370
So it's in this direction
right over here.

00:04:05.370 --> 00:04:07.380
So that's the constant velocity.

00:04:07.380 --> 00:04:09.920
So what are all
the forces at play?

00:04:09.920 --> 00:04:10.962
And be very careful here.

00:04:10.962 --> 00:04:13.045
There might be a temptation
that says, OK, there's

00:04:13.045 --> 00:04:13.760
a net force here.

00:04:13.760 --> 00:04:14.320
We're moving.

00:04:14.320 --> 00:04:16.570
So maybe that's the net force
that's causing the move.

00:04:16.570 --> 00:04:19.730
But remember, this
is super important.

00:04:19.730 --> 00:04:21.370
This is Newton's first law.

00:04:21.370 --> 00:04:24.730
If you have a net force, if
you have an unbalanced force,

00:04:24.730 --> 00:04:27.230
it will cause it to accelerate.

00:04:27.230 --> 00:04:29.290
And we are not
accelerating here.

00:04:29.290 --> 00:04:31.630
We have a constant velocity.

00:04:31.630 --> 00:04:36.210
We are not accelerating here.

00:04:36.210 --> 00:04:39.760
So if you're not accelerating
in that direction,

00:04:39.760 --> 00:04:41.760
then that means that the
force in that direction

00:04:41.760 --> 00:04:44.150
must be balanced.

00:04:44.150 --> 00:04:48.390
So there must be
some force acting

00:04:48.390 --> 00:04:51.060
in the exactly
opposite direction that

00:04:51.060 --> 00:04:54.450
keeps this thing from
accelerating downwards.

00:04:54.450 --> 00:04:57.110
And so it must be
exactly 49 newtons

00:04:57.110 --> 00:04:59.160
in the opposite direction.

00:04:59.160 --> 00:05:02.740
And as you can imagine, this
is the force of friction.

00:05:02.740 --> 00:05:06.852
This right over here is
the force of friction.

00:05:06.852 --> 00:05:09.185
And the difference between
this video and the last video

00:05:09.185 --> 00:05:11.320
is last time
friction was static.

00:05:11.320 --> 00:05:13.930
Even at 49 newtons,
the box was stationary.

00:05:13.930 --> 00:05:16.140
You had to keep nudging
until you get to 50 newtons,

00:05:16.140 --> 00:05:17.330
and then it started moving.

00:05:17.330 --> 00:05:19.610
Here we're just jumping
into this picture

00:05:19.610 --> 00:05:21.910
where we just see a
box that's moving down

00:05:21.910 --> 00:05:23.324
the slope at 5
meters per second.

00:05:23.324 --> 00:05:24.740
So we don't know
how much force it

00:05:24.740 --> 00:05:27.030
took to overcome
static friction.

00:05:27.030 --> 00:05:30.060
But we do know that there is
some force of friction that

00:05:30.060 --> 00:05:32.740
is keeping this box from
accelerating, that's keeping it

00:05:32.740 --> 00:05:35.960
at a constant velocity,
that is completely negating

00:05:35.960 --> 00:05:38.560
the parallel component
of the force of gravity,

00:05:38.560 --> 00:05:42.130
parallel to the
surface of this plane.

00:05:42.130 --> 00:05:44.970
So given this, let's
calculate another coefficient

00:05:44.970 --> 00:05:45.550
of friction.

00:05:45.550 --> 00:05:47.174
But this is going to
be the coefficient

00:05:47.174 --> 00:05:50.790
of kinetic friction, because now
we are moving down the block.

00:05:50.790 --> 00:05:53.130
And I'll do a video
on why sometimes

00:05:53.130 --> 00:05:54.630
a coefficient of
static friction can

00:05:54.630 --> 00:05:56.350
be different than
the coefficient

00:05:56.350 --> 00:05:58.170
of kinetic friction.

00:05:58.170 --> 00:06:01.810
So the coefficient of kinetic
friction-- we'll write it.

00:06:01.810 --> 00:06:07.440
So this is the Greek letter
mu, and we put this k here

00:06:07.440 --> 00:06:10.670
for kinetic, or we can kind
of say moving friction.

00:06:10.670 --> 00:06:14.870
It's going to be equal
to the force of friction,

00:06:14.870 --> 00:06:20.830
or I should say the magnitude
of the force of friction

00:06:20.830 --> 00:06:23.450
over the normal force.

00:06:26.520 --> 00:06:30.610
I should say the magnitude
of the normal force.

00:06:30.610 --> 00:06:32.390
And you can derive
this experimentally.

00:06:32.390 --> 00:06:34.550
One, if you just observe
this whole thing going on

00:06:34.550 --> 00:06:36.008
and you knew the
mass of the block,

00:06:36.008 --> 00:06:37.639
so you knew this
component of gravity

00:06:37.639 --> 00:06:38.930
that's going in this direction.

00:06:38.930 --> 00:06:41.090
If you knew this
angle was 30 degrees

00:06:41.090 --> 00:06:43.170
from the last situation,
you could figure out

00:06:43.170 --> 00:06:45.360
this coefficient of
kinetic friction.

00:06:45.360 --> 00:06:47.000
And what's cool
about this is this

00:06:47.000 --> 00:06:50.500
is in general going to be true
for any two materials that

00:06:50.500 --> 00:06:51.350
are like this.

00:06:51.350 --> 00:06:53.215
So maybe this is a
certain type of wood

00:06:53.215 --> 00:06:55.590
on a certain type of wood, or
a certain type of sandpaper

00:06:55.590 --> 00:06:58.630
on a certain type of sandpaper--
whatever you're talking about.

00:06:58.630 --> 00:07:01.030
And then you can use
that to make predictions

00:07:01.030 --> 00:07:04.370
if the incline was different,
or if the mass was different,

00:07:04.370 --> 00:07:06.610
or even if you were
on a different planet,

00:07:06.610 --> 00:07:09.050
or if someone was pressing
down on this block.

00:07:09.050 --> 00:07:11.560
That would change
the normal force.

00:07:11.560 --> 00:07:14.200
So given this right
here, let's figure out--

00:07:14.200 --> 00:07:17.430
for the sake of doing it-- the
coefficient of kinetic friction

00:07:17.430 --> 00:07:19.580
here.

00:07:19.580 --> 00:07:21.790
The force of friction
here, completely offsetting

00:07:21.790 --> 00:07:24.790
the parallel force of gravity
parallel to the surface,

00:07:24.790 --> 00:07:26.970
is 49 newtons.

00:07:26.970 --> 00:07:29.440
And the normal force
here, the force

00:07:29.440 --> 00:07:33.290
of contact between these
two things, this block

00:07:33.290 --> 00:07:35.760
and this wedge, is 49
square roots of 3 newtons.

00:07:39.760 --> 00:07:42.185
So we get 1 over the
square root of 3.

00:07:42.185 --> 00:07:43.560
And let me get
the calculator out

00:07:43.560 --> 00:07:45.500
to get an actual number here.

00:07:45.500 --> 00:07:49.890
So we have 1 divided by
the square root of 3,

00:07:49.890 --> 00:07:52.560
which gives us 0.5--
I'll just round.

00:07:52.560 --> 00:07:53.470
0.58.

00:07:53.470 --> 00:07:55.437
It is equal to 0.58.

00:07:55.437 --> 00:07:57.770
And there's no units here,
because the units cancel out.

00:07:57.770 --> 00:07:59.920
It's a unit-less measurement.

00:07:59.920 --> 00:08:02.920
Now the interesting thing here
is that the way I've set up

00:08:02.920 --> 00:08:06.000
this problem, the coefficient
of kinetic friction

00:08:06.000 --> 00:08:09.170
is lower, if we assume
the same materials,

00:08:09.170 --> 00:08:12.420
than the coefficient
of static friction was.

00:08:12.420 --> 00:08:15.400
And for some materials, they
might not be that different.

00:08:15.400 --> 00:08:17.920
But for other materials,
the kinetic friction

00:08:17.920 --> 00:08:20.470
can be lower than
static friction.

00:08:20.470 --> 00:08:22.894
You never see a situation
where the coefficient

00:08:22.894 --> 00:08:24.810
of static friction-- at
least that I know of--

00:08:24.810 --> 00:08:26.101
is lower than kinetic friction.

00:08:26.101 --> 00:08:28.880
But you do see situations
where the coefficient

00:08:28.880 --> 00:08:33.240
of kinetic friction is
lower than the coefficient

00:08:33.240 --> 00:08:34.780
of static friction.

00:08:34.780 --> 00:08:39.860
Once something is
moving, for some reason--

00:08:39.860 --> 00:08:43.650
and we'll theorize why
that might be-- friction

00:08:43.650 --> 00:08:47.007
is a little less potent than
when something is stationary.

00:08:47.007 --> 00:08:49.090
So we can say this generally,
that the coefficient

00:08:49.090 --> 00:08:54.890
of kinetic friction is less
than or equal to the coefficient

00:08:54.890 --> 00:08:56.280
of static friction.

00:08:56.280 --> 00:09:00.260
It's a little bit easier, or
friction provides a little less

00:09:00.260 --> 00:09:02.780
than or equal to the force
when something's moving

00:09:02.780 --> 00:09:05.637
than when something
is stationary.

00:09:05.637 --> 00:09:07.470
So I'll think about
that a little bit deeper

00:09:07.470 --> 00:09:09.440
in the next video.


Correction to force of friction keeping the block stationary

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:02.726
Just realized that I
typed in the wrong numbers

00:00:02.726 --> 00:00:04.100
at the end of the
last video when

00:00:04.100 --> 00:00:05.970
I was trying to put
them in the calculator.

00:00:05.970 --> 00:00:08.625
We were trying to determine the
coefficient of static friction,

00:00:08.625 --> 00:00:12.080
and it's 50 newtons divided by
49 square roots of 3 newtons.

00:00:12.080 --> 00:00:14.330
And in the last video, by
accident, in the calculator,

00:00:14.330 --> 00:00:15.870
instead of doing 49
square roots of 3,

00:00:15.870 --> 00:00:17.100
I wrote 40 square roots of 3.

00:00:17.100 --> 00:00:19.030
So let's actually
calculate it again.

00:00:19.030 --> 00:00:25.700
50 divided by 49 times--
let me write it this way,

00:00:25.700 --> 00:00:30.930
make sure we get-- 49
times the square root of 3

00:00:30.930 --> 00:00:35.260
gives us a coefficient of static
friction of, if we round it,

00:00:35.260 --> 00:00:37.380
0.59.

00:00:37.380 --> 00:00:44.380
So this is equal to 0.59.

00:00:44.380 --> 00:00:46.050
And so that's it.

00:00:46.050 --> 00:00:47.820
I just forgot to-- I
did the calculation

00:00:47.820 --> 00:00:50.590
wrong on the calculator
in the last video.


Force of friction keeping the block stationary

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:05.030
In the last video, we had a ten kilogram mass

00:00:05.070 --> 00:00:08.640
sitting on top of an inclined plane at a 30 degree angle

00:00:08.670 --> 00:00:11.220
And in order to figure out what would happen to this block

00:00:11.230 --> 00:00:16.020
we broke down the force of gravity on this block into

00:00:16.150 --> 00:00:19.860
the components that are parallel to the surface of the plane

00:00:19.900 --> 00:00:23.010
and perpendicular to the surface of the plane

00:00:23.020 --> 00:00:29.200
and for a perpendicular component, we got 49 times the square root of 3 N downwards

00:00:29.220 --> 00:00:31.950
That's 98 times this quantity over here, downwards

00:00:32.240 --> 00:00:37.850
But we said look! We don't see this block of ice accelerating downwards into this wedge

00:00:37.880 --> 00:00:39.630
because the wedge is supporting it

00:00:39.650 --> 00:00:44.540
So there must be a counteracting force that the wedge is exerting on the block

00:00:44.660 --> 00:00:49.990
And that counteracting force is the normal force of the wedge on the block of ice

00:00:50.010 --> 00:00:55.450
And that is exactly opposite to the force of gravity in this direction

00:00:55.450 --> 00:00:59.240
The normal force of gravity is the normal force of the wedge

00:00:59.360 --> 00:01:05.690
And these completely balance each other out in that normal, perpendicular direction

00:01:05.710 --> 00:01:09.620
And that is why this block is not accelerating either in that direction

00:01:09.880 --> 00:01:12.220
or in this direction over here

00:01:12.710 --> 00:01:16.740
But the one component of the force of gravity that did not seem to have any offset

00:01:16.740 --> 00:01:19.320
(at least the way we set up the problem in the last video)

00:01:19.320 --> 00:01:25.400
is the force/component that is parallel to the surface of the plane

00:01:25.740 --> 00:01:28.910
And we figured that out to be 49N, it was

00:01:29.060 --> 00:01:32.750
essentially the weight of the block times the sin of this angle

00:01:32.770 --> 00:01:35.020
And we said look, if there is no other forces

00:01:35.050 --> 00:01:37.710
then it would be accelerated in this direction

00:01:37.710 --> 00:01:40.870
And to figure out the rate of acceleration

00:01:40.990 --> 00:01:45.030
you take the force in that direction divided by the mass of the block

00:01:45.030 --> 00:01:47.880
and you would get 4.9 m/s^2

00:01:47.910 --> 00:01:50.970
Now let's say that wasn't happening. Let's say that you were to look at this system

00:01:50.970 --> 00:01:53.620
right over here and the block was just stationary

00:01:53.630 --> 00:01:56.460
And now, for the sake of argument, let's assume it is not ice on ice

00:01:56.460 --> 00:01:59.100
Let's assume that they are both made out of wood

00:01:59.130 --> 00:02:02.300
And now all of a sudden we have a situation where the block is stationary

00:02:02.610 --> 00:02:06.480
If it is stationary, what is necessarily the case?

00:02:06.860 --> 00:02:13.310
Well we already determined that if it is not accelerating in this normal direction

00:02:13.360 --> 00:02:15.490
there must be zero net forces on it

00:02:15.510 --> 00:02:17.420
But if it is stationary as a whole

00:02:17.450 --> 00:02:20.920
then there must be zero net forces in the parallel component too

00:02:20.950 --> 00:02:29.890
So there must be some force counteracting this 49 N that wants to take it down the slope

00:02:29.890 --> 00:02:36.110
So there must be some force counteracting

00:02:36.110 --> 00:02:39.920
the component of gravity that wants to accelerate it down the slope

00:02:39.930 --> 00:02:43.870
And the question is what is this force? We're dealing with a situation now

00:02:43.920 --> 00:02:49.880
where we're dealing with a stationary block, a block that is not accelerating

00:02:50.260 --> 00:02:52.370
So what is that force?

00:02:52.400 --> 00:02:56.430
I think you know from experience maybe what is the difference between

00:02:56.460 --> 00:03:01.000
a block of wood on top of wood and a block of ice on top of ice

00:03:01.100 --> 00:03:06.130
A block of ice on top of ice is much more slippery; there is no friction

00:03:06.240 --> 00:03:09.320
between ice and ice, but there is friction between wood and wood

00:03:09.320 --> 00:03:15.520
To make it a little bit more tangible, maybe we'd put some sandpaper on the surface over here

00:03:15.520 --> 00:03:18.070
And then it becomes a little bit clearer. The force

00:03:18.080 --> 00:03:25.530
that is keeping this block from sliding down in this situation is the force of friction

00:03:25.550 --> 00:03:29.690
and the force of friction will always act in a direction opposite

00:03:29.690 --> 00:03:33.090
to the motion if there was not any friction

00:03:33.330 --> 00:03:44.230
or the potential acceleration if there was not any action

00:03:44.250 --> 00:03:46.400
So what is the force of friction in this case?

00:03:46.500 --> 00:03:50.460
Well, this block is completely stationary. It's not accelerating down the ramp

00:03:50.650 --> 00:04:06.810
The force of friction over here is going to be 49 N, upwards, up the ramp

00:04:07.020 --> 00:04:10.820
Now I want think about, this is something that can be determined experimentally

00:04:11.110 --> 00:04:17.860
as long as you have some way of measuring force, you can do this experimentally

00:04:18.190 --> 00:04:22.090
But the interesting question here is

00:04:22.370 --> 00:04:27.320
how much do I have to push on this block

00:04:27.320 --> 00:04:32.140
until it starts to move down the ramp? How much do I have to push on it?

00:04:32.400 --> 00:04:35.330
Let's say you are able to experimentally determine

00:04:35.350 --> 00:04:55.230
that if you can apply another 1 N on this

00:04:55.250 --> 00:05:00.930
then all of a sudden, I can at least get the box to start accelerating down

00:05:01.170 --> 00:05:05.080
Not the rate which it would do naturally, but I can just start to nudge it down

00:05:05.080 --> 00:05:09.550
if I give it another push of one N in the parallel direction

00:05:10.000 --> 00:05:13.590
So what is the total force--so exactly one N

00:05:13.900 --> 00:05:18.740
So the total force at this point that's acting on it in order to just start to budge it

00:05:19.190 --> 00:05:21.780
I'll call this the budging force

00:05:22.510 --> 00:05:24.600
Remember here isn't a traditional class

00:05:24.600 --> 00:05:26.940
F sub B for the budging force

00:05:27.200 --> 00:05:29.790
The budging force is in the parallel direction

00:05:30.040 --> 00:05:35.820
If I'm applying 1 N in this direction and it already has 49 N due to

00:05:35.820 --> 00:05:40.920
the component of gravity in this direction, then my budging force is 50 N

00:05:40.950 --> 00:05:43.170
And so an interesting thing that you can determine

00:05:43.320 --> 00:05:46.890
based on the materials that are coming in contact with each other is

00:05:46.890 --> 00:05:51.080
just how much force you need to just start to overcome friction

00:05:51.080 --> 00:05:54.620
In this case, it's the budging force. That's a term that I made up

00:05:54.950 --> 00:05:59.860
And the interesting ratio which tends to hold for given materials pretty well

00:05:59.990 --> 00:06:02.770
is the ratio between the amount of force just to budge it

00:06:03.080 --> 00:06:06.810
and the amount of force between the two objects

00:06:06.810 --> 00:06:10.130
between how much force they are exerting on each other

00:06:10.280 --> 00:06:17.620
And in this case, the amount of force that is being exerted by this, by the wedge on the block

00:06:17.620 --> 00:06:21.090
is the normal force, is 49 sqrt of 3

00:06:21.100 --> 00:06:25.080
Maybe I should say the magnitude of the budging force

00:06:25.090 --> 00:06:29.100
over the magnitude of the force that is putting these two things in contact

00:06:29.480 --> 00:06:33.970
In this case, it is 49 square root of 3 N

00:06:34.750 --> 00:06:43.270
Over the magnitude of the normal force, and that is 49 sqrt of 3

00:06:43.300 --> 00:06:48.410
We call this the coefficient of static friction

00:06:48.430 --> 00:06:51.690
We're gonna use this a little bit more deeply in other problems

00:06:51.720 --> 00:06:55.010
but it tends to hold true for different materials so that

00:06:55.320 --> 00:06:59.560
in the future if you have a different mass, you have a different incline

00:06:59.600 --> 00:07:01.920
but you have the same materials

00:07:01.940 --> 00:07:05.270
given the normal force, you can figure out the budging force

00:07:05.270 --> 00:07:07.580
You can figure out exactly how much force you need to put

00:07:07.580 --> 00:07:10.870
if you know this, which you usually figure out experimentally

00:07:10.880 --> 00:07:16.240
So what would be the value in this case? You have 50 N over 49 square root of 3 N

00:07:16.410 --> 00:07:20.240
Let's get the calculator out

00:07:21.050 --> 00:07:31.840
So I have 50 divided by 40 times the square root of 3

00:07:32.520 --> 00:07:38.430
Gives me .72--I'll just round to two significant digits--0.72

00:07:38.440 --> 00:07:41.570
This is 0.72

00:07:41.580 --> 00:07:52.210
And then you can use this information. This is the coefficient of static friction

00:07:52.220 --> 00:07:55.330
We call it the coefficient of static friction because

00:07:55.350 --> 00:08:02.470
this deals with the ratio of the force of friction relative to the normal force

00:08:02.470 --> 00:08:07.130
I guess the force just to overcome the force of friction, just kind of get right over that

00:08:07.150 --> 00:08:12.330
the most friction that can be applied by kind of the abrasiveness of the 2 things

00:08:12.370 --> 00:08:14.510
when the object is stationary

00:08:14.520 --> 00:08:17.890
I'll do a whole video on how this is different

00:08:17.890 --> 00:08:20.830
when an object is stationary to when it's moving

00:08:20.850 --> 00:08:24.430
A lot of times, they're very very very close, but for certain materials

00:08:24.440 --> 00:08:28.160
you have a very--at least, a noticeably different coefficient

00:08:28.160 --> 00:08:33.280
of friction when the object is stationery as opposed to when it is moving

00:08:33.570 --> 00:08:36.730
So I'll leave you there. In the next few videos, we'll use coefficient of friction

00:08:36.730 --> 00:08:40.270
or calculate coefficient of friction to do some more problems


Ice accelerating down an incline

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.480
Let's say that I have
a ramp made of ice.

00:00:03.480 --> 00:00:07.180
Looks like maybe a wedge or
some type of an inclined plane

00:00:07.180 --> 00:00:08.474
made of ice.

00:00:08.474 --> 00:00:10.390
And we'll make everything
of ice in this video

00:00:10.390 --> 00:00:12.190
so that we have
negligible friction.

00:00:12.190 --> 00:00:14.400
So this right here is my ramp.

00:00:14.400 --> 00:00:15.560
It's made of ice.

00:00:15.560 --> 00:00:19.930
And this angle right over here,
let's just go with 30 degrees.

00:00:19.930 --> 00:00:21.850
And let's say on this
ramp made of ice,

00:00:21.850 --> 00:00:24.440
I have another block of ice.

00:00:24.440 --> 00:00:27.490
So this is a block of ice.

00:00:27.490 --> 00:00:31.520
It is a block of ice, it's
shiny like ice is shiny.

00:00:31.520 --> 00:00:37.230
And it has a mass
of 10 kilograms.

00:00:37.230 --> 00:00:39.030
And what I want to do
is think about what's

00:00:39.030 --> 00:00:41.810
going to happen to
this block of ice.

00:00:41.810 --> 00:00:44.042
So first of all, what are
the forces that we know

00:00:44.042 --> 00:00:44.750
are acting on it?

00:00:44.750 --> 00:00:47.070
Well if we're assuming
we're on Earth,

00:00:47.070 --> 00:00:49.460
and we will, and we're
near the surface,

00:00:49.460 --> 00:00:51.390
then there is the
force of gravity.

00:00:51.390 --> 00:00:56.050
There's the force of gravity
acting on this block of ice.

00:00:56.050 --> 00:00:59.960
And the force of
gravity is going

00:00:59.960 --> 00:01:05.280
to be equal to-- it's going to
be in the downward direction,

00:01:05.280 --> 00:01:08.660
and its magnitude is going to
be the mass of the block of ice

00:01:08.660 --> 00:01:11.450
times the gravitational
field times 9.8

00:01:11.450 --> 00:01:13.260
meters per second squared.

00:01:13.260 --> 00:01:16.040
So it's going to be
98 newtons downward.

00:01:16.040 --> 00:01:18.860
So this is 98 newtons downward.

00:01:18.860 --> 00:01:20.220
I just took 10 kilograms.

00:01:20.220 --> 00:01:21.530
Let me write it out.

00:01:21.530 --> 00:01:24.050
So the force due
to gravity is going

00:01:24.050 --> 00:01:28.410
to be equal to 10
kilograms times 9.8

00:01:28.410 --> 00:01:32.810
meters per second
squared downward.

00:01:32.810 --> 00:01:35.140
This 9.8 meters
per second squared

00:01:35.140 --> 00:01:38.480
downward, that is
the field vector

00:01:38.480 --> 00:01:41.920
for the gravitational field
of the surface of the earth,

00:01:41.920 --> 00:01:43.590
I guess is one way
to think about it.

00:01:43.590 --> 00:01:45.430
Sometimes you'll
see the negative 9.8

00:01:45.430 --> 00:01:46.880
meters per second squared.

00:01:46.880 --> 00:01:49.660
And then that negative is giving
you the direction implicitly

00:01:49.660 --> 00:01:51.300
because the
convention is normally

00:01:51.300 --> 00:01:53.440
that positive is upward
and negative is downward.

00:01:53.440 --> 00:01:55.580
We'll just go with
this right over here.

00:01:55.580 --> 00:01:58.370
So the magnitude of
this vector is 10 times

00:01:58.370 --> 00:02:02.230
9.8, which is 98 kilogram
meters per second squared, which

00:02:02.230 --> 00:02:03.920
is the same thing as newtons.

00:02:03.920 --> 00:02:05.580
So the magnitude
here is 98 newtons

00:02:05.580 --> 00:02:07.580
and it is pointing downwards.

00:02:07.580 --> 00:02:09.610
Now what we want to do
is break this vector up

00:02:09.610 --> 00:02:11.660
into the components
that are perpendicular

00:02:11.660 --> 00:02:14.370
and parallel to the
surface of this ramp.

00:02:14.370 --> 00:02:15.740
So let's do that.

00:02:15.740 --> 00:02:17.970
So first, let's think
about perpendicular

00:02:17.970 --> 00:02:19.920
to the surface of the ramp.

00:02:19.920 --> 00:02:22.550
So perpendicular to the
surface of the ramp.

00:02:22.550 --> 00:02:25.170
So this right over
here is a right angle.

00:02:25.170 --> 00:02:27.320
And we saw in the last
video, that whatever

00:02:27.320 --> 00:02:29.530
angle this over here
is, that is also

00:02:29.530 --> 00:02:31.120
going to be this
angle over here.

00:02:31.120 --> 00:02:36.340
So this angle over here is also
going to be a 30-degree angle.

00:02:36.340 --> 00:02:38.910
And we can use that
information to figure out

00:02:38.910 --> 00:02:42.110
the magnitude of this orange
vector right over here.

00:02:42.110 --> 00:02:44.960
And remember, this orange
vector is the component

00:02:44.960 --> 00:02:49.360
of the force of gravity that
is perpendicular to the plane.

00:02:49.360 --> 00:02:52.310
And then there's going to be
some component that is parallel

00:02:52.310 --> 00:02:53.020
to the plane.

00:02:53.020 --> 00:02:54.430
I'll draw that in yellow.

00:02:54.430 --> 00:02:57.620
Some component of
the force of gravity

00:02:57.620 --> 00:02:59.490
that is parallel to the plane.

00:02:59.490 --> 00:03:01.270
And clearly this
is a right angle,

00:03:01.270 --> 00:03:03.740
because this is
perpendicular to the plane.

00:03:03.740 --> 00:03:05.582
And this is parallel
to the plane.

00:03:05.582 --> 00:03:07.040
If it's perpendicular
to the plane,

00:03:07.040 --> 00:03:10.520
it's also perpendicular to
this vector right over here.

00:03:10.520 --> 00:03:12.180
So we can use some
basic trigonometry,

00:03:12.180 --> 00:03:14.013
like we did in the last
video, to figure out

00:03:14.013 --> 00:03:17.030
the magnitude of this orange
and this yellow vector

00:03:17.030 --> 00:03:18.200
right over here.

00:03:18.200 --> 00:03:21.610
This orange vector's
magnitude over the hypotenuse

00:03:21.610 --> 00:03:23.690
is going to be equal
to the cosine of 30.

00:03:23.690 --> 00:03:26.640
Or you could say that
the magnitude of this

00:03:26.640 --> 00:03:36.670
is 98 times the cosine
of 30 degrees newtons.

00:03:36.670 --> 00:03:39.370
98 times the cosine
of 30 degrees newtons.

00:03:39.370 --> 00:03:41.860
And if you want the whole
vector, it's in this direction.

00:03:41.860 --> 00:03:46.060
And the direction going into
the surface of the plane.

00:03:46.060 --> 00:03:47.902
And, based on the
simple trigonometry--

00:03:47.902 --> 00:03:49.860
and we go into this in
a little bit more detail

00:03:49.860 --> 00:03:53.510
in the last video-- we
know that the component

00:03:53.510 --> 00:03:58.220
of this vector that is parallel
to the surface of this plane

00:03:58.220 --> 00:04:04.230
is going to be 98
sine of 30 degrees.

00:04:04.230 --> 00:04:06.220
Sine of 30 degrees.

00:04:06.220 --> 00:04:07.840
Sine of 30 degrees.

00:04:07.840 --> 00:04:11.840
And this comes straight
out of this magnitude,

00:04:11.840 --> 00:04:14.830
which is opposite to the
angle over the hypotenuse.

00:04:14.830 --> 00:04:17.363
Opposite over hypotenuse is
equal to sine of an angle.

00:04:17.363 --> 00:04:18.779
And we did all the
work over here.

00:04:18.779 --> 00:04:20.195
I don't want to
keep repeating it.

00:04:20.195 --> 00:04:22.820
But I always want to emphasize
that this is coming straight

00:04:22.820 --> 00:04:25.970
out of basic
trigonometry, straight out

00:04:25.970 --> 00:04:28.060
of basic trigonometry.

00:04:28.060 --> 00:04:30.470
So once you do that, we know
the different components.

00:04:30.470 --> 00:04:31.730
We can calculate them.

00:04:31.730 --> 00:04:36.680
Cosine of 30 degrees is
square root of 3 over 2.

00:04:36.680 --> 00:04:38.234
Sine of 30 degrees is 1/2.

00:04:38.234 --> 00:04:40.150
That's just one of those
things that you learn

00:04:40.150 --> 00:04:43.400
and you can derive it yourself
using 30-60-90 triangles,

00:04:43.400 --> 00:04:45.050
or actually even
equilateral triangles.

00:04:45.050 --> 00:04:46.180
Or you could use a calculator.

00:04:46.180 --> 00:04:47.430
But it's also one
of those things

00:04:47.430 --> 00:04:49.304
that you memorize when
you take trigonometry.

00:04:49.304 --> 00:04:53.260
So no kind of magical
trick I did here.

00:04:53.260 --> 00:04:55.870
And so if you evaluate this,
98 times the square root

00:04:55.870 --> 00:05:00.339
of 3 over 2 newtons,
tells us that--

00:05:00.339 --> 00:05:02.130
let me write it in that
same orange color--

00:05:02.130 --> 00:05:04.970
the force, the
component of gravity

00:05:04.970 --> 00:05:07.390
that is perpendicular
to the plane.

00:05:07.390 --> 00:05:09.710
And this kind of implicitly
gives us this direction,

00:05:09.710 --> 00:05:10.890
it's perpendicular to the plane.

00:05:10.890 --> 00:05:12.330
But the force
component of gravity

00:05:12.330 --> 00:05:13.705
that's perpendicular
to the plane

00:05:13.705 --> 00:05:16.006
is equal to 98 times
square root of 3 over 2.

00:05:16.006 --> 00:05:18.760
98 divided by 2 is 49.

00:05:18.760 --> 00:05:22.570
So it's equal to 49 times
the square root of 3 newtons.

00:05:22.570 --> 00:05:25.730
And its direction is into
the surface of the plane,

00:05:25.730 --> 00:05:32.050
or downward or, let me just
write, into surface of plane.

00:05:32.050 --> 00:05:36.670
Surface of the plane, or
the surface of the ramp.

00:05:36.670 --> 00:05:38.227
And it's in this
direction over here.

00:05:38.227 --> 00:05:40.060
And I have to do this
because it's a vector.

00:05:40.060 --> 00:05:42.440
I have to tell you what
direction it's going in.

00:05:42.440 --> 00:05:45.809
And the component of the force
of gravity that is parallel.

00:05:45.809 --> 00:05:48.100
The component of the force
of gravity that is parallel,

00:05:48.100 --> 00:05:50.430
I drew it down here, but I
could shift it up over here.

00:05:50.430 --> 00:05:52.520
It's the same exact vector.

00:05:52.520 --> 00:05:55.460
The component of
gravity that is parallel

00:05:55.460 --> 00:05:59.090
to the surface of the plane
is 98 times sine of 30.

00:05:59.090 --> 00:06:03.800
That's 98 times 1/2,
which is 49 newtons.

00:06:03.800 --> 00:06:07.230
And it's going in that
direction, or parallel

00:06:07.230 --> 00:06:09.360
to the surface of the plane.

00:06:09.360 --> 00:06:12.260
Parallel, I always have
trouble spelling parallel.

00:06:12.260 --> 00:06:14.840
Parallel to-- don't even
know if I spelled it right--

00:06:14.840 --> 00:06:18.680
surface of the plane.

00:06:18.680 --> 00:06:20.650
So what's going to happen here?

00:06:20.650 --> 00:06:25.250
Well, if these were the
only forces acting on it.

00:06:25.250 --> 00:06:28.320
So if we had a net force
going into the surface

00:06:28.320 --> 00:06:31.700
of the plane of 49 square
roots of 3 newtons.

00:06:31.700 --> 00:06:35.100
If this was the only force
acting in this dimension

00:06:35.100 --> 00:06:37.510
or in the dimension that is
perpendicular to the surface

00:06:37.510 --> 00:06:39.320
of the plane, what would happen?

00:06:39.320 --> 00:06:41.630
Well, then the block
would just accelerate.

00:06:41.630 --> 00:06:44.190
At least just due to this force
it would accelerate downward.

00:06:44.190 --> 00:06:47.240
It would accelerate into
the surface of the plane.

00:06:47.240 --> 00:06:49.120
But we know it's not
going to accelerate.

00:06:49.120 --> 00:06:51.280
We know that there's this
big wedge of ice here

00:06:51.280 --> 00:06:54.530
that is keeping it from
accelerating in that direction.

00:06:54.530 --> 00:06:57.415
So at least in this dimension,
there will be no acceleration.

00:06:57.415 --> 00:06:58.790
When I talk about
this dimension,

00:06:58.790 --> 00:07:01.280
I'm talking about in
the direction that

00:07:01.280 --> 00:07:03.570
is perpendicular to the
surface of the plane.

00:07:03.570 --> 00:07:07.380
There will be no acceleration
because this wedge is here.

00:07:07.380 --> 00:07:12.030
So the wedge is exerting a force
that completely counteracts

00:07:12.030 --> 00:07:16.070
the force, the perpendicular
component of gravity.

00:07:16.070 --> 00:07:16.880
And that force.

00:07:16.880 --> 00:07:19.130
You might guess
what it's called.

00:07:19.130 --> 00:07:22.400
So the wedge is
exerting a force, just

00:07:22.400 --> 00:07:25.345
like that, that's going
to be 98 newtons upward.

00:07:29.250 --> 00:07:31.240
The wedge is going to
be exerting a force that

00:07:31.240 --> 00:07:36.450
is 49 square roots of 3,
because this right here

00:07:36.450 --> 00:07:38.900
is 49 square roots
of 3 newtons into.

00:07:38.900 --> 00:07:42.580
And so this is 49 square
roots of 3 newtons out

00:07:42.580 --> 00:07:48.390
of the surface,
out of the surface.

00:07:48.390 --> 00:07:50.750
And this is the normal force.

00:07:50.750 --> 00:07:53.660
It is the force
perpendicular to the surface

00:07:53.660 --> 00:07:56.310
that essentially, you could
kind of view as the contact

00:07:56.310 --> 00:08:01.260
force that the, in this case,
that the surface is exerting

00:08:01.260 --> 00:08:05.625
to keep this block of ice from
accelerating in that direction.

00:08:05.625 --> 00:08:07.500
We're not talking about
accelerating straight

00:08:07.500 --> 00:08:08.240
towards the center of the earth.

00:08:08.240 --> 00:08:10.365
We're talking about
accelerating in that direction.

00:08:10.365 --> 00:08:13.950
We broke up the force into kind
of the perpendicular direction

00:08:13.950 --> 00:08:15.750
and the parallel direction.

00:08:15.750 --> 00:08:18.580
So you have this
counteracting normal force.

00:08:18.580 --> 00:08:24.620
And that's why you don't
have the block plummeting

00:08:24.620 --> 00:08:27.940
or accelerating into the plane.

00:08:27.940 --> 00:08:30.130
Now what other
forces do we have?

00:08:30.130 --> 00:08:33.909
Well, we have the force that's
parallel to the surface.

00:08:33.909 --> 00:08:35.840
And if we assume that
there's no friction--

00:08:35.840 --> 00:08:38.909
and I can assume that there's
no friction in this video

00:08:38.909 --> 00:08:42.280
because we are assuming
that it is ice on ice-- what

00:08:42.280 --> 00:08:43.220
is going to happen?

00:08:43.220 --> 00:08:46.900
There's no counteracting
force to this 49 newtons.

00:08:46.900 --> 00:08:52.560
49 newtons parallel downwards,
I should say parallel downwards,

00:08:52.560 --> 00:08:54.280
to the surface of the plane.

00:08:54.280 --> 00:08:55.910
So what's going to happen?

00:08:55.910 --> 00:08:58.280
Well, it's going to
accelerate in that direction.

00:08:58.280 --> 00:09:02.060
You have force is equal to
mass times acceleration.

00:09:02.060 --> 00:09:06.640
Force is equal to mass
times acceleration.

00:09:06.640 --> 00:09:10.990
Or you divide both sides by
mass, you get force over mass

00:09:10.990 --> 00:09:13.820
is equal to acceleration.

00:09:13.820 --> 00:09:18.310
Over here, our
force is 49 newtons

00:09:18.310 --> 00:09:20.030
in that direction,
parallel downwards

00:09:20.030 --> 00:09:21.920
to the surface of the plane.

00:09:21.920 --> 00:09:24.050
And so if you
divide both by mass,

00:09:24.050 --> 00:09:27.460
if you divide both
of these by mass.

00:09:27.460 --> 00:09:31.150
So that's the same thing as
dividing it by 10 kilograms,

00:09:31.150 --> 00:09:33.676
dividing by 10 kilograms, that
will give you acceleration.

00:09:36.380 --> 00:09:39.130
That will give you
our acceleration.

00:09:39.130 --> 00:09:41.100
So acceleration is
49 newtons divided

00:09:41.100 --> 00:09:44.960
by 10 kilograms
in that direction,

00:09:44.960 --> 00:09:47.540
in this direction
right over there.

00:09:47.540 --> 00:09:50.720
And 49 divided by
10 is 4.9, and then

00:09:50.720 --> 00:09:54.900
newtons divided by kilograms
is meters per second squared.

00:09:54.900 --> 00:09:56.890
So then you get
your acceleration.

00:09:56.890 --> 00:10:02.810
Your acceleration is going to be
4.9 meters per second squared.

00:10:02.810 --> 00:10:05.760
And maybe I could say parallel.

00:10:05.760 --> 00:10:06.790
That's two bars.

00:10:06.790 --> 00:10:09.560
Or maybe I'll write parallel.

00:10:09.560 --> 00:10:16.845
Parallel downwards
to the surface.

00:10:16.845 --> 00:10:18.640
Now I'm going to
leave you there,

00:10:18.640 --> 00:10:20.790
and I'll let you think
about another thing

00:10:20.790 --> 00:10:22.850
that I'll address in
the next video is,

00:10:22.850 --> 00:10:24.980
what if you had this
just standing still?

00:10:24.980 --> 00:10:26.890
If it wasn't
accelerating downwards,

00:10:26.890 --> 00:10:29.120
if it wasn't accelerating
and sliding down,

00:10:29.120 --> 00:10:31.980
what would be the
force that's keeping it

00:10:31.980 --> 00:10:33.700
in a kind of a static state?

00:10:33.700 --> 00:10:36.530
We'll think about that
in the next video.


Inclined plane force components

WEBVTT
Kind: captions
Language: en

00:00:01.250 --> 00:00:03.280
Let's say I have some
type of a block here.

00:00:03.280 --> 00:00:06.580
And let's say this
block has a mass of m.

00:00:06.580 --> 00:00:10.100
So the mass of this
block is equal to m.

00:00:10.100 --> 00:00:13.250
And it's sitting on this-- you
could view this is an inclined

00:00:13.250 --> 00:00:16.500
plane, or a ramp, or
some type of wedge.

00:00:16.500 --> 00:00:19.957
And we want to think about what
might happen to this block.

00:00:19.957 --> 00:00:22.290
And we'll start thinking about
the different forces that

00:00:22.290 --> 00:00:24.770
might keep it in place
or not keep it in place

00:00:24.770 --> 00:00:26.270
and all of the rest.

00:00:26.270 --> 00:00:30.080
So the one thing we do know
is if this whole set up

00:00:30.080 --> 00:00:32.150
is near the surface
of the Earth--

00:00:32.150 --> 00:00:35.680
and we'll assume that it is
for the sake of this video--

00:00:35.680 --> 00:00:39.530
that there will be the
force of gravity trying

00:00:39.530 --> 00:00:42.782
to bring or attract this
mass towards the center

00:00:42.782 --> 00:00:44.990
of the Earth, and vice versa,
the center of the earth

00:00:44.990 --> 00:00:46.700
towards this mass.

00:00:46.700 --> 00:00:48.580
So we're going to have
some force of gravity.

00:00:48.580 --> 00:00:52.060
Let me start right at the center
of this mass right over here.

00:00:52.060 --> 00:00:57.070
And so you're going to
have the force of gravity.

00:00:57.070 --> 00:00:59.390
The force due to
gravity is going

00:00:59.390 --> 00:01:02.800
to be equal to the
gravitational field

00:01:02.800 --> 00:01:05.250
near the surface of the Earth.

00:01:05.250 --> 00:01:07.940
And so we'll call that g.

00:01:07.940 --> 00:01:10.715
We'll call that
g times the mass.

00:01:10.715 --> 00:01:11.590
Let me just write it.

00:01:11.590 --> 00:01:14.920
The mass times the
gravitational field

00:01:14.920 --> 00:01:17.320
near the surface of the Earth.

00:01:17.320 --> 00:01:19.010
And it's going to
be downwards, we

00:01:19.010 --> 00:01:22.270
know that, or at least towards
the surface of the Earth.

00:01:22.270 --> 00:01:24.570
Now, what else is going
to be happening here?

00:01:24.570 --> 00:01:26.120
Well, it gets a
little bit confusing,

00:01:26.120 --> 00:01:30.540
because you can't really say
that normal force is acting

00:01:30.540 --> 00:01:32.740
directly against this
force right over here.

00:01:32.740 --> 00:01:35.190
Because remember,
the normal force

00:01:35.190 --> 00:01:37.750
acts perpendicular to a surface.

00:01:37.750 --> 00:01:41.040
So over here, the surface is
not perpendicular to the force

00:01:41.040 --> 00:01:41.680
of gravity.

00:01:41.680 --> 00:01:44.330
So we have to think about it a
little bit differently than we

00:01:44.330 --> 00:01:47.100
do if this was sitting
on level ground.

00:01:47.100 --> 00:01:50.220
Well, the one thing we can do,
and frankly, that we should do,

00:01:50.220 --> 00:01:52.760
is maybe we can
break up this force,

00:01:52.760 --> 00:01:54.440
the force due to gravity.

00:01:54.440 --> 00:01:56.120
We can break it
up into components

00:01:56.120 --> 00:01:59.900
that are either
perpendicular to the surface

00:01:59.900 --> 00:02:02.386
or that are parallel
to the surface.

00:02:02.386 --> 00:02:04.010
And then we can use
those to figure out

00:02:04.010 --> 00:02:05.210
what's likely to happen.

00:02:05.210 --> 00:02:09.750
What are potentially the netting
forces, or balancing forces,

00:02:09.750 --> 00:02:10.459
over here?

00:02:10.459 --> 00:02:11.750
So let's see if we can do that.

00:02:11.750 --> 00:02:14.230
Let's see if we can
break this force vector,

00:02:14.230 --> 00:02:17.010
the force due to gravity,
into a component that

00:02:17.010 --> 00:02:23.570
is perpendicular to the
surface of this ramp.

00:02:23.570 --> 00:02:26.180
And also another
component that is parallel

00:02:26.180 --> 00:02:28.110
to the surface of this ramp.

00:02:28.110 --> 00:02:30.490
Let me do that in
a different color.

00:02:30.490 --> 00:02:33.980
That is parallel to the
surface of this ramp.

00:02:33.980 --> 00:02:37.370
And this is a little bit
unconventional notation,

00:02:37.370 --> 00:02:42.960
but I'll call this one over
here the force due to gravity

00:02:42.960 --> 00:02:44.582
that is perpendicular
to the ramp.

00:02:44.582 --> 00:02:47.040
That little upside down t, I'm
saying that's perpendicular.

00:02:47.040 --> 00:02:49.480
Because it shows a line
that's perpendicular to,

00:02:49.480 --> 00:02:53.020
I guess, this bottom line, this
horizontal line over there.

00:02:53.020 --> 00:02:54.540
And this blue thing
over here, I'm

00:02:54.540 --> 00:02:58.690
going to call this
the part of force

00:02:58.690 --> 00:03:01.500
due to gravity that is parallel.

00:03:01.500 --> 00:03:04.090
I'm just doing these
two upward vertical bars

00:03:04.090 --> 00:03:07.686
to show something that is
parallel to the surface.

00:03:07.686 --> 00:03:09.060
So this is the
component of force

00:03:09.060 --> 00:03:10.976
due to gravity that's
perpendicular, component

00:03:10.976 --> 00:03:12.880
of force that is parallel.

00:03:12.880 --> 00:03:14.580
So let's see if we
can use a little bit

00:03:14.580 --> 00:03:17.070
a geometry and
trigonometry, given

00:03:17.070 --> 00:03:24.080
that this wedge is at
a theta degree incline

00:03:24.080 --> 00:03:25.220
relative to the horizontal.

00:03:25.220 --> 00:03:27.580
If you were to measure
this angle right over here,

00:03:27.580 --> 00:03:28.630
you would get theta.

00:03:28.630 --> 00:03:30.770
So in future videos we'll
make it more concrete,

00:03:30.770 --> 00:03:33.399
like 30 degrees or 45
degrees or whatever.

00:03:33.399 --> 00:03:34.690
But let's just keep in general.

00:03:34.690 --> 00:03:36.560
If this is theta,
let's figure out

00:03:36.560 --> 00:03:40.600
what these components of
the gravitational force

00:03:40.600 --> 00:03:41.920
are going to be.

00:03:41.920 --> 00:03:44.680
Well, we can break out
our geometry over here.

00:03:44.680 --> 00:03:48.780
This, I'm assuming
is a right angle.

00:03:48.780 --> 00:03:50.560
And so if this is
a right angle, we

00:03:50.560 --> 00:03:55.540
know that the sum of the angles
in a triangle add up to 180.

00:03:55.540 --> 00:04:00.990
So if this angle, and this
90 degrees-- right angle

00:04:00.990 --> 00:04:03.640
says 90 degrees-- add
up to 180, then that

00:04:03.640 --> 00:04:08.230
means that this one and this one
need to add up to 90 degrees.

00:04:08.230 --> 00:04:13.360
Or, if this is theta,
this angle right over here

00:04:13.360 --> 00:04:17.006
is going to be 90 minus theta.

00:04:17.006 --> 00:04:18.839
Now, the other thing
that you may or may not

00:04:18.839 --> 00:04:20.980
remember from geometry
class is that if I

00:04:20.980 --> 00:04:28.175
have two parallel lines,
and I have a transversal.

00:04:28.175 --> 00:04:31.550
So I'm going to assume this
line is parallel to this line.

00:04:31.550 --> 00:04:33.600
And then I have a transversal.

00:04:33.600 --> 00:04:36.810
So let's say I have a
line that goes like this.

00:04:36.810 --> 00:04:40.340
We know from basic
geometry that this angle

00:04:40.340 --> 00:04:42.520
is going to be
equal to this angle.

00:04:42.520 --> 00:04:44.650
It comes from alternate
interior angles.

00:04:44.650 --> 00:04:47.770
And we prove it in
the geometry module,

00:04:47.770 --> 00:04:49.060
or in the geometry videos.

00:04:49.060 --> 00:04:50.820
But hopefully this makes a
little bit of intuitive sense,

00:04:50.820 --> 00:04:53.028
and you could even think
about how these angles would

00:04:53.028 --> 00:04:55.560
changes as the transversal
changes, and all of the rest.

00:04:55.560 --> 00:04:57.270
But the parallel
lines makes this angle

00:04:57.270 --> 00:05:00.010
similar to that angle, or
actually makes it identical,

00:05:00.010 --> 00:05:01.210
makes it congruent.

00:05:01.210 --> 00:05:03.910
This angle is going to be the
same measure as that angle.

00:05:03.910 --> 00:05:06.355
So can we apply that
anywhere over here?

00:05:10.510 --> 00:05:14.050
This line is perpendicular
to the surface of the Earth.

00:05:14.050 --> 00:05:17.240
Right over here that I'm
kind of shading in blue.

00:05:17.240 --> 00:05:19.790
And so is this force vector.

00:05:19.790 --> 00:05:22.660
It is also perpendicular to
the surface of the Earth.

00:05:22.660 --> 00:05:26.640
So this line over here and
this line over here in magenta

00:05:26.640 --> 00:05:29.100
are going to be parallel.

00:05:29.100 --> 00:05:30.040
I can even draw that.

00:05:30.040 --> 00:05:33.491
That line and that
line are both parallel.

00:05:33.491 --> 00:05:34.990
When you look at
it that way, you'll

00:05:34.990 --> 00:05:38.740
see that this big line over here
can be viewed as a transversal.

00:05:38.740 --> 00:05:42.150
Or you could have this
angle and this angle

00:05:42.150 --> 00:05:43.880
are going to be congruent.

00:05:43.880 --> 00:05:46.910
They're going to be
alternate interior angles.

00:05:46.910 --> 00:05:50.134
So this angle and this angle,
by the exact same idea here.

00:05:50.134 --> 00:05:52.050
It just looks a little
bit more confusing here

00:05:52.050 --> 00:05:53.660
because I have all
sorts of things.

00:05:53.660 --> 00:05:56.390
But this line and this
line are parallel.

00:05:56.390 --> 00:05:59.700
You can view this right
over here as a transversal.

00:05:59.700 --> 00:06:02.190
So this and this are
congruent angles.

00:06:02.190 --> 00:06:04.460
So this is 90 minus
theta degrees.

00:06:04.460 --> 00:06:08.130
This too will be 90
minus theta degrees.

00:06:08.130 --> 00:06:11.660
90 minus theta degrees.

00:06:11.660 --> 00:06:15.310
Now, given that, can we
figure out this angle?

00:06:15.310 --> 00:06:18.400
Well one thing, we're assuming
that this yellow force vector

00:06:18.400 --> 00:06:22.430
right here is perpendicular
to the surface of this plane

00:06:22.430 --> 00:06:25.090
or perpendicular to the
surface of this ramp.

00:06:25.090 --> 00:06:26.660
So that's perpendicular.

00:06:26.660 --> 00:06:29.260
This right here
is 90 minus theta.

00:06:29.260 --> 00:06:32.670
So what is this angle up
here going to be equal to?

00:06:32.670 --> 00:06:35.570
This angle, let
me do it in green.

00:06:35.570 --> 00:06:38.590
What is this angle up
here going to be equal to?

00:06:38.590 --> 00:06:42.180
So this angle plus 90
minus theta plus 90

00:06:42.180 --> 00:06:47.760
must be equal to 180, or this
angle plus 90 minus theta must

00:06:47.760 --> 00:06:49.600
be equal to-- let
me write this down.

00:06:49.600 --> 00:06:51.308
I don't want to do
too much in your head.

00:06:51.308 --> 00:06:52.110
So let's call it x.

00:06:52.110 --> 00:06:56.672
So x plus 90 minus theta.

00:06:56.672 --> 00:07:00.280
Plus this 90 degrees right over
here, plus this 90 degrees,

00:07:00.280 --> 00:07:04.220
needs to be equal
to 180 degrees.

00:07:04.220 --> 00:07:07.470
Let's see, we can subtract
180 degrees from both sides.

00:07:07.470 --> 00:07:10.690
So we subtract 90 twice,
you subtract 180 degrees

00:07:10.690 --> 00:07:13.820
and you get x minus
theta is equal to 0,

00:07:13.820 --> 00:07:16.380
or x is equal to theta.

00:07:16.380 --> 00:07:19.620
So whatever the inclination of
the plane is or of this ramp,

00:07:19.620 --> 00:07:22.940
that is also going to be
this angle right over here.

00:07:22.940 --> 00:07:25.070
And the value to
that is that now we

00:07:25.070 --> 00:07:28.010
can use our basic
trigonometry to figure out

00:07:28.010 --> 00:07:30.470
this component
and this component

00:07:30.470 --> 00:07:31.929
of the force of gravity.

00:07:31.929 --> 00:07:33.470
And to see that a
little bit clearer,

00:07:33.470 --> 00:07:36.890
let me shift this force
vector down over here.

00:07:36.890 --> 00:07:39.610
The parallel component,
let me shift it over here.

00:07:39.610 --> 00:07:41.520
And you can see the
perpendicular component

00:07:41.520 --> 00:07:44.960
plus the parallel component
is equal to the total force

00:07:44.960 --> 00:07:46.390
due to gravity.

00:07:46.390 --> 00:07:49.140
And you should also see that
this is a right triangle

00:07:49.140 --> 00:07:51.320
that I have set up over here.

00:07:51.320 --> 00:07:52.770
This is parallel to the plane.

00:07:52.770 --> 00:07:54.930
This is perpendicular
to the plane.

00:07:54.930 --> 00:07:56.620
And so we can use
basic trigonometry

00:07:56.620 --> 00:07:59.610
to figure out the magnitudes
of the perpendicular

00:07:59.610 --> 00:08:03.964
force due to gravity and the
parallel force due to gravity.

00:08:03.964 --> 00:08:05.380
Let's think about
it a little bit.

00:08:07.970 --> 00:08:08.880
I'll do it over here.

00:08:08.880 --> 00:08:11.630
The magnitude of
the perpendicular

00:08:11.630 --> 00:08:14.040
force due to gravity.

00:08:14.040 --> 00:08:15.950
Or I should say the
component of gravity

00:08:15.950 --> 00:08:19.434
that's perpendicular to
the ramp, the magnitude

00:08:19.434 --> 00:08:21.100
of that vector-- a
lot of fancy notation

00:08:21.100 --> 00:08:25.270
but it's really just the length
of this vector right over here.

00:08:25.270 --> 00:08:29.560
So the magnitude of
this over the hypotenuse

00:08:29.560 --> 00:08:30.600
of this right triangle.

00:08:30.600 --> 00:08:32.809
Well, what the hypotenuse
of this right triangle?

00:08:32.809 --> 00:08:34.308
Well, it's going
to be the magnitude

00:08:34.308 --> 00:08:40.309
of the total
gravitational force.

00:08:40.309 --> 00:08:41.690
I guess you could say that.

00:08:41.690 --> 00:08:46.870
And so you could say that is mg.

00:08:46.870 --> 00:08:48.770
We could write it like this.

00:08:48.770 --> 00:08:52.250
But that's really-- well,
I could write it like that.

00:08:52.250 --> 00:08:54.010
And so this is going
to be equal to what?

00:08:54.010 --> 00:08:56.350
We have the, if we're looking
at this angle right here,

00:08:56.350 --> 00:08:59.950
we have the adjacent
over the hypotenuse.

00:08:59.950 --> 00:09:01.240
Remember.

00:09:01.240 --> 00:09:02.796
We can do this in a new color.

00:09:02.796 --> 00:09:05.130
We can do this in a new color.

00:09:05.130 --> 00:09:08.490
SOH CAH TOA.

00:09:08.490 --> 00:09:11.230
Cosine is adjacent
over hypotenuse.

00:09:11.230 --> 00:09:14.580
So this is equal to
cosine of the angle.

00:09:14.580 --> 00:09:18.240
So cosine of theta is
equal to the adjacent

00:09:18.240 --> 00:09:19.230
over the hypotenuse.

00:09:19.230 --> 00:09:22.350
So if you multiply both
sides by the magnitude

00:09:22.350 --> 00:09:26.630
of the hypotenuse, you get the
component of our vector that

00:09:26.630 --> 00:09:30.240
is perpendicular to the
surface of the plane

00:09:30.240 --> 00:09:38.870
is equal to the magnitude
of the force due to gravity

00:09:38.870 --> 00:09:41.020
times the cosine of theta.

00:09:41.020 --> 00:09:42.720
Times the cosine of theta.

00:09:42.720 --> 00:09:44.272
We'll apply this
in the next video

00:09:44.272 --> 00:09:46.480
just so you can make the
numbers a lot more concrete.

00:09:46.480 --> 00:09:48.438
Sometimes just the notation
makes it confusing.

00:09:48.438 --> 00:09:50.970
You'll see it's really actually
pretty straightforward.

00:09:50.970 --> 00:09:54.520
And then this second thing,
we can use the same logic.

00:09:54.520 --> 00:09:57.120
If we think about the parallel
vector right over here,

00:09:57.120 --> 00:10:02.490
the magnitude of the
component of the force

00:10:02.490 --> 00:10:05.320
due to gravity that is
parallel to the plane

00:10:05.320 --> 00:10:08.550
over the magnitude of the
force due to gravity--

00:10:08.550 --> 00:10:13.230
which is the
magnitude of mg-- that

00:10:13.230 --> 00:10:16.910
is going to be equal to what?

00:10:16.910 --> 00:10:20.160
This is the opposite
side to the angle.

00:10:20.160 --> 00:10:22.720
So the blue stuff is the
opposite side, or at least

00:10:22.720 --> 00:10:25.270
its length, is the
opposite side of the angle.

00:10:25.270 --> 00:10:27.670
And then right over
here this magnitude

00:10:27.670 --> 00:10:29.200
of mg, that is the hypotenuse.

00:10:29.200 --> 00:10:31.260
So you have the opposite
over the hypotenuse.

00:10:31.260 --> 00:10:32.790
Opposite over hypotenuse.

00:10:32.790 --> 00:10:35.090
Sine of an angle is
opposite over hypotenuse.

00:10:35.090 --> 00:10:39.090
So this is going to be
equal to the sine of theta.

00:10:39.090 --> 00:10:41.490
This is equal to
the sine of theta.

00:10:41.490 --> 00:10:46.600
Or you multiply both sides
times the magnitude of the force

00:10:46.600 --> 00:10:49.590
due to gravity and
you get the component

00:10:49.590 --> 00:10:57.550
of the force due to gravity
that is parallel to the ramp

00:10:57.550 --> 00:11:07.260
is going to be the force
due to gravity total times

00:11:07.260 --> 00:11:08.810
sine of theta.

00:11:08.810 --> 00:11:11.480
Times sine of theta.

00:11:11.480 --> 00:11:13.950
And hopefully you should
see where this came from.

00:11:13.950 --> 00:11:17.640
Because if you ever have to
derive this again 30 years

00:11:17.640 --> 00:11:21.880
after you took a physics class,
you should be able to do it.

00:11:21.880 --> 00:11:27.830
But if you know this right
here, and this right here,

00:11:27.830 --> 00:11:30.220
we can all of a sudden start
breaking down the forces

00:11:30.220 --> 00:11:32.129
into things that
are useful to us.

00:11:32.129 --> 00:11:33.670
Because we could
say, hey, look, this

00:11:33.670 --> 00:11:35.410
isn't moving down
into this plane.

00:11:35.410 --> 00:11:36.930
So maybe there's
some normal force

00:11:36.930 --> 00:11:39.750
that's completely netting
it out in this example.

00:11:39.750 --> 00:11:41.882
And maybe if there's
nothing to keep it up,

00:11:41.882 --> 00:11:43.590
and there's no friction,
maybe this thing

00:11:43.590 --> 00:11:45.850
will start accelerating
due to the parallel force.

00:11:45.850 --> 00:11:47.510
And we'll think a
lot more about that.

00:11:47.510 --> 00:11:50.112
And if you ever forget these,
think about them intuitively.

00:11:50.112 --> 00:11:52.320
You don't have to go through
this whole parallel line

00:11:52.320 --> 00:11:53.700
and transversal and all of that.

00:11:57.860 --> 00:11:59.650
If this angle went
down to 0, then we'll

00:11:59.650 --> 00:12:01.680
be talking about
essentially a flat surface.

00:12:01.680 --> 00:12:03.720
There is no inclination there.

00:12:03.720 --> 00:12:06.830
And if this angle goes down
to 0, then all of the force

00:12:06.830 --> 00:12:11.250
should be acting perpendicular
to the surface of the plane.

00:12:11.250 --> 00:12:14.690
So if this going to 0, if
the perpendicular force

00:12:14.690 --> 00:12:17.810
should be the same thing as
the total gravitational force.

00:12:17.810 --> 00:12:19.310
And that's why it's
cosine of theta.

00:12:19.310 --> 00:12:21.680
Because cosine of
0 right now is 1.

00:12:21.680 --> 00:12:23.590
And so these would
equal each other.

00:12:23.590 --> 00:12:26.450
And if this is equal to 0,
then the parallel component

00:12:26.450 --> 00:12:27.960
of gravity should go to 0.

00:12:27.960 --> 00:12:29.540
Because gravity
will only be acting

00:12:29.540 --> 00:12:32.890
downwards, and once again,
if sine of theta is 0.

00:12:32.890 --> 00:12:35.430
So the force of gravity that
is parallel will go to 0.

00:12:35.430 --> 00:12:38.060
So if you ever forget, just do
that little intuitive thought

00:12:38.060 --> 00:12:40.050
process and you'll
remember which one is sine

00:12:40.050 --> 00:12:42.260
and which one is cosine.


Normal force in an elevator

WEBVTT
Kind: captions
Language: en

00:00:00.484 --> 00:00:01.900
What I want to do
in this video is

00:00:01.900 --> 00:00:03.940
think about how the
normal force might

00:00:03.940 --> 00:00:06.060
be different in
different scenarios.

00:00:06.060 --> 00:00:09.120
And since my 2 and 1/2-year-old
son is obsessed with elevators,

00:00:09.120 --> 00:00:11.790
I thought I would
focus on those.

00:00:11.790 --> 00:00:13.670
So here I've drawn
four scenarios.

00:00:13.670 --> 00:00:15.420
And we could imagine
them almost happening

00:00:15.420 --> 00:00:16.840
in some type of a sequence.

00:00:16.840 --> 00:00:20.500
So in this first
picture right over here,

00:00:20.500 --> 00:00:24.650
I'm going to assume that
the velocity is equal to 0.

00:00:24.650 --> 00:00:27.599
Or another way to think about it
is this elevator is stationary.

00:00:27.599 --> 00:00:30.140
And everything we're going to
be talking about in this video,

00:00:30.140 --> 00:00:31.850
I'm talking about in
the vertical direction.

00:00:31.850 --> 00:00:34.224
That's the only dimension
we're going to be dealing with.

00:00:34.224 --> 00:00:39.320
So this is 0 meters per second
in the vertical direction.

00:00:39.320 --> 00:00:42.290
Or another way to think about
it, this thing is not moving.

00:00:42.290 --> 00:00:44.850
Now also it is
also-- and this may

00:00:44.850 --> 00:00:47.000
be somewhat obvious to
you-- but its acceleration

00:00:47.000 --> 00:00:50.350
is also 0 meters
per second squared

00:00:50.350 --> 00:00:52.710
in this picture right over here.

00:00:52.710 --> 00:00:55.810
Then let's say that I'm sitting
in this transparent elevator.

00:00:55.810 --> 00:00:56.910
And I press the button.

00:00:56.910 --> 00:01:01.090
So the elevator begins
to accelerate upwards.

00:01:01.090 --> 00:01:03.160
So in this video
right over here,

00:01:03.160 --> 00:01:05.040
or in this screen
right over here,

00:01:05.040 --> 00:01:10.472
let's say that the acceleration
is 2 meters per second.

00:01:10.472 --> 00:01:12.430
And I'll use the convention
that positive means

00:01:12.430 --> 00:01:14.690
upwards or negative
means downwards.

00:01:14.690 --> 00:01:17.391
We're only going to be operating
in this one dimension right

00:01:17.391 --> 00:01:17.890
here.

00:01:17.890 --> 00:01:21.975
I could write 2 meters per
second times the j unit vector

00:01:21.975 --> 00:01:23.850
because that tells us
that we are now moving.

00:01:23.850 --> 00:01:25.391
Why don't we just
leave it like that.

00:01:25.391 --> 00:01:29.490
That tells us that we are
moving in the upward direction.

00:01:29.490 --> 00:01:32.242
And let's say we do
that for 1 second.

00:01:32.242 --> 00:01:34.200
And then we get to this
screen right over here.

00:01:34.200 --> 00:01:36.130
So we had no velocity.

00:01:36.130 --> 00:01:37.130
We move.

00:01:37.130 --> 00:01:38.390
We accelerate.

00:01:38.390 --> 00:01:40.800
Let me-- oh, this is 2
meters per second squared.

00:01:40.800 --> 00:01:43.820
Let me make sure I--
It's 2 meters per second.

00:01:43.820 --> 00:01:45.470
This is acceleration here.

00:01:45.470 --> 00:01:47.330
So we do that for 1 second.

00:01:47.330 --> 00:01:49.790
And then at the end of 1
second, we stop accelerating.

00:01:49.790 --> 00:01:53.290
So here, once we get to this
little screen over here,

00:01:53.290 --> 00:01:55.920
our acceleration
goes back to 0 meters

00:01:55.920 --> 00:01:57.504
per second squared
in the j direction,

00:01:57.504 --> 00:01:59.711
only you don't have to write
that because it's really

00:01:59.711 --> 00:02:00.230
just 0.

00:02:00.230 --> 00:02:03.640
But now we have some velocity.

00:02:03.640 --> 00:02:05.750
We did that just for
the sake of simplicity.

00:02:05.750 --> 00:02:08.979
Let's say this screen
lasted for 1 second.

00:02:08.979 --> 00:02:13.770
So now our velocity is going
to be 2 meters per second

00:02:13.770 --> 00:02:20.280
in the j direction, or
in the upwards direction.

00:02:20.280 --> 00:02:23.230
And then let's say we
do that for 10 seconds.

00:02:23.230 --> 00:02:24.990
So at least at the
constant velocity,

00:02:24.990 --> 00:02:26.140
we travel for 20 meters.

00:02:26.140 --> 00:02:28.600
We travel a little bit while
we're accelerating, too.

00:02:28.600 --> 00:02:31.160
But we're getting
close to our floor.

00:02:31.160 --> 00:02:32.893
And so the elevator
needs to decelerate.

00:02:37.630 --> 00:02:38.890
So then it decelerates.

00:02:38.890 --> 00:02:42.060
The acceleration here
is negative 2 meters

00:02:42.060 --> 00:02:46.810
per second squared times--
in the j direction.

00:02:46.810 --> 00:02:48.890
So it's actually
accelerating downwards now.

00:02:48.890 --> 00:02:51.974
It has to slow it down to
get it back to stationary.

00:02:51.974 --> 00:02:53.640
So what I want to do
is think about what

00:02:53.640 --> 00:02:56.310
would be the normal
force, the force

00:02:56.310 --> 00:02:58.610
that the floor of the
elevator is exerting

00:02:58.610 --> 00:03:01.620
on me in each of
these situations.

00:03:01.620 --> 00:03:03.990
And we're going to assume
that we are operating

00:03:03.990 --> 00:03:06.650
near the surface of the Earth.

00:03:06.650 --> 00:03:08.510
So in every one of
these situations,

00:03:08.510 --> 00:03:10.870
if we're operating near
the surface of the Earth,

00:03:10.870 --> 00:03:13.474
I have some type of
gravitational attraction

00:03:13.474 --> 00:03:15.140
to the Earth and the
Earth has some type

00:03:15.140 --> 00:03:18.030
of gravitational
attraction to me.

00:03:18.030 --> 00:03:20.400
And so let's say that
I'm-- I don't know.

00:03:20.400 --> 00:03:23.070
Let's just make the math simple.

00:03:23.070 --> 00:03:24.860
Let's say that I'm
some type of a toddler.

00:03:24.860 --> 00:03:25.880
And I'm 10 kilograms.

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So maybe this is
my son, although I

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think he's 12 kilograms.

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But we'll keep it simple.

00:03:37.730 --> 00:03:38.865
Oh, let me be clear.

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He doesn't weigh 10 kilograms.

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That's wrong.

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He has a mass of 10 kilograms.

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Weight is the force
due to gravity.

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Mass of the amount of stuff,
the amount of matter there is.

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Although I that's not
a rigorous definition.

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So the mass of the individual,
of this toddler sitting

00:03:54.090 --> 00:03:56.610
in the elevator,
is 10 kilograms.

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So what is the force of gravity.

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Or another way to
think about it,

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what is this person's weight?

00:04:02.640 --> 00:04:05.040
Well, in this vignette
right over here,

00:04:05.040 --> 00:04:07.570
in this picture right
over here, its mass

00:04:07.570 --> 00:04:10.450
times the gravitational
field near the surface

00:04:10.450 --> 00:04:13.300
of the Earth, the 9.8
meters per second squared.

00:04:13.300 --> 00:04:14.760
Let me write that over here.

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The gravitational field near
the surface of the Earth

00:04:18.610 --> 00:04:22.469
is 9.8 meters per
second squared.

00:04:22.469 --> 00:04:24.510
And the negative tells
you it is going downwards.

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So you multiply this
times 10 kilograms.

00:04:29.310 --> 00:04:33.550
The downward force,
the force of gravity,

00:04:33.550 --> 00:04:38.220
is going to be 10 times negative
9.8 meters per second squared.

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So negative 98 newtons.

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And I could say that that's
going to be in the j direction.

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Well, what's going to be the
downward force of gravity here?

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Well, it's going to
be the same thing.

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We're still near the
surface of the Earth.

00:04:51.610 --> 00:04:54.151
We're going to assume that the
gravitational field is roughly

00:04:54.151 --> 00:04:55.990
constant, although
we know it slightly

00:04:55.990 --> 00:04:58.330
changes with the distance
from the center of the Earth.

00:04:58.330 --> 00:04:59.913
But when we're dealing
on the surface,

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we assume that it's
roughly constant.

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And so what we'll assume we have
the exact same force of gravity

00:05:06.830 --> 00:05:07.420
there.

00:05:07.420 --> 00:05:10.500
And of course, this person's
mass, this toddler's mass,

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does not change, depending
on going up a few floors.

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So it's going to have the same
force of gravity downwards

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in every one of
these situations.

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In this first
situation right here,

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this person has no acceleration.

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If they have no acceleration
in any direction,

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and we're only
concerning ourselves

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with the vertical
direction right here,

00:05:32.950 --> 00:05:36.970
that means that there must
be no net force on them.

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This is from Newton's
first law of motion.

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But if there's no
net force on them,

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there must be some force that's
counteracting this force.

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Because if there
was nothing else,

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there would be a net force of
gravity and this poor toddler

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would be plummeting to
the center of the Earth.

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So that net force
in this situation

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is the force of the floor
of the elevator supporting

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the toddler.

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So that force would
be an equal force

00:06:05.640 --> 00:06:06.980
but in the opposite direction.

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And in this case, that
would be the normal force.

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So in this case,
the normal force

00:06:11.230 --> 00:06:17.600
is 98 newtons in
the j direction.

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So it just completely
bounces off.

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There's no net force
on this person.

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They get to hold their
constant velocity of 0.

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And they don't plummet to
the center of the Earth.

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Now, what is the net force
on this individual right

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over here?

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Well, this individual
is accelerating.

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There is acceleration
going on over here.

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So there must be some
type of net force.

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Well, let's think about
what the net force must

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be on this person, or on
this toddler, I should say.

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The net force is going to
be the mass of this toddler.

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It's going to be 10 kilograms
times the acceleration

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of this toddler, times 2 meters
per second squared, which

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is equal to 20 kilogram meters
per second squared, which

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is the same thing as
20 newtons upwards.

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20 newtons upwards
is the net force.

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So if we already have
the force due to gravity

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at 98 newtons downwards--
that's the same thing here;

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that's that one
right over there,

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98 newtons downwards-- we need a
force that not only bounces off

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that 98 newtons downwards to
not only keep it stationary,

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but is also doing
another 20 newtons

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in the upwards direction.

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So here we need a force
in order for the elevator

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to accelerate the toddler
upwards at 2 meters per second,

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you have a net force
is positive 20 newtons,

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or 20 newtons in the
upward direction.

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Or another way to think about
it, if you have negative 98

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newtons here, you're going
to need 20 more than that

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in the positive direction.

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So you're going to need 118
newtons now in the j direction.

00:08:00.040 --> 00:08:03.110
So here, where the elevator
is accelerating upward,

00:08:03.110 --> 00:08:07.110
the normal force is
now 20 newtons higher

00:08:07.110 --> 00:08:08.220
than it was there.

00:08:08.220 --> 00:08:12.140
And that's what's allowing
this toddler to accelerate.

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Now let's think
about this situation.

00:08:14.340 --> 00:08:18.980
No acceleration, but
we do have velocity.

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So here we were stationary.

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Here we do have velocity.

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And you might be
tempted to think,

00:08:24.560 --> 00:08:27.770
oh, maybe I still
have some higher force

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here because I'm moving upwards.

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I have some upwards velocity.

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But remember Newton's
first law of motion.

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If you're at a
constant velocity,

00:08:36.450 --> 00:08:38.120
including a constant
velocity of 0,

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you have no net force on you.

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So this toddler right over
here, once the toddler

00:08:42.740 --> 00:08:44.780
gets to this stage,
the net forces

00:08:44.780 --> 00:08:46.549
are going to look
identical over here.

00:08:46.549 --> 00:08:48.840
And actually, if you're
sitting in either this elevator

00:08:48.840 --> 00:08:52.560
or this elevator, assuming it's
not being bumped around it all,

00:08:52.560 --> 00:08:55.350
you would not be able
to tell the difference

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because your body is
sensitive to acceleration.

00:08:59.330 --> 00:09:02.270
Your body cannot sense its
velocity if it has no air,

00:09:02.270 --> 00:09:05.670
if it has no frame of reference
or nothing to see passing by.

00:09:05.670 --> 00:09:07.310
So to the toddler
there, it doesn't

00:09:07.310 --> 00:09:09.470
know whether it is
stationary or whether it

00:09:09.470 --> 00:09:10.870
has constant velocity.

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It would be able to
tell this-- it would

00:09:12.910 --> 00:09:15.970
feel that kind of
compression on its body.

00:09:15.970 --> 00:09:18.570
And that's what its nerves
are sensitive towards,

00:09:18.570 --> 00:09:20.280
perception is sensitive to.

00:09:20.280 --> 00:09:22.665
But here it's identical
to the first situation.

00:09:22.665 --> 00:09:26.200
And Newton's first law tells
there's no net force on this.

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So it's just like
the first situation.

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The normal force, the force of
the elevator on this toddler's

00:09:31.770 --> 00:09:35.650
shoes, is going to be
identical to the downward force

00:09:35.650 --> 00:09:37.000
due to gravity.

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So the normal force here
is going to be 98 newtons.

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Completely nets out the
downward, the negative 98

00:09:43.140 --> 00:09:43.640
newtons.

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So once again, this
is in the j direction,

00:09:45.440 --> 00:09:47.710
in the positive j direction.

00:09:47.710 --> 00:09:51.370
And then when we
are about to get

00:09:51.370 --> 00:09:54.040
to our floor, what is happening?

00:09:54.040 --> 00:09:58.350
Well, once again we
have a net acceleration

00:09:58.350 --> 00:10:00.840
of negative 2 meters per second.

00:10:00.840 --> 00:10:03.060
So if you have a negative
acceleration, so once again

00:10:03.060 --> 00:10:04.740
what is the net force here?

00:10:04.740 --> 00:10:06.530
The net force over
here is going to be

00:10:06.530 --> 00:10:09.450
the mass of the
toddler, 10 kilograms,

00:10:09.450 --> 00:10:14.110
times negative 2
meters per second.

00:10:14.110 --> 00:10:16.760
And this was right here
in the j direction.

00:10:16.760 --> 00:10:18.390
That's the vertical direction.

00:10:18.390 --> 00:10:20.600
Remember j is just
the unit vector

00:10:20.600 --> 00:10:22.860
in the vertical
direction facing upwards.

00:10:22.860 --> 00:10:29.270
So negative 2 meters per second
squared in the j direction.

00:10:29.270 --> 00:10:34.120
And this is equal to negative
20 kilogram meters per second

00:10:34.120 --> 00:10:37.910
squared in the j direction, or
negative 20 newtons in the j

00:10:37.910 --> 00:10:38.900
direction.

00:10:38.900 --> 00:10:42.220
So the net force on this
is negative 20 newtons.

00:10:42.220 --> 00:10:46.240
So we have the force of
gravity at negative 98 newtons

00:10:46.240 --> 00:10:47.840
in the j direction.

00:10:47.840 --> 00:10:50.950
So we're fully
compensating for that

00:10:50.950 --> 00:10:53.450
because we're still going
to have a net negative force

00:10:53.450 --> 00:10:56.580
while this child
is decelerating.

00:10:56.580 --> 00:10:59.410
And that negative net force
is a negative net force

00:10:59.410 --> 00:11:02.150
of-- I keep repeating
it-- negative 20.

00:11:02.150 --> 00:11:08.240
So we're only going to have
a 78 newton normal force here

00:11:08.240 --> 00:11:11.250
that counteracts all but
20 newtons of the force

00:11:11.250 --> 00:11:13.220
due to gravity.

00:11:13.220 --> 00:11:15.280
So this right over
here is going to be

00:11:15.280 --> 00:11:18.295
78 newtons in the j direction.

00:11:18.295 --> 00:11:20.170
And so I really want
you to think about this.

00:11:20.170 --> 00:11:22.700
And I actually really want you
to think about this next time

00:11:22.700 --> 00:11:24.180
you're sitting in the elevator.

00:11:24.180 --> 00:11:27.770
The only time that you realize
that something is going on

00:11:27.770 --> 00:11:30.380
is when that elevator is
really just accelerating

00:11:30.380 --> 00:11:32.090
or when it's just decelerating.

00:11:32.090 --> 00:11:35.220
When it's just accelerating,
you feel a little bit heavier.

00:11:35.220 --> 00:11:38.289
And when it's just decelerating,
you feel a little bit lighter.

00:11:38.289 --> 00:11:40.580
And I want you to think a
little bit about why that is.

00:11:40.580 --> 00:11:42.420
But while it's moving
at a constant velocity

00:11:42.420 --> 00:11:44.750
or is stationary, you
feel like you're just

00:11:44.750 --> 00:11:48.910
sitting on the surface
of the planet someplace.