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Let's say I have some
type of a block here.

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And let's say this
block has a mass of m.

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So the mass of this
block is equal to m.

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And it's sitting on this-- you
could view this is an inclined

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plane, or a ramp, or
some type of wedge.

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And we want to think about what
might happen to this block.

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And we'll start thinking about
the different forces that

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might keep it in place
or not keep it in place

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and all of the rest.

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So the one thing we do know
is if this whole set up

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is near the surface
of the Earth--

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and we'll assume that it is
for the sake of this video--

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that there will be the
force of gravity trying

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to bring or attract this
mass towards the center

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of the Earth, and vice versa,
the center of the earth

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towards this mass.

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So we're going to have
some force of gravity.

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Let me start right at the center
of this mass right over here.

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And so you're going to
have the force of gravity.

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The force due to
gravity is going

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to be equal to the
gravitational field

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near the surface of the Earth.

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And so we'll call that g.

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We'll call that
g times the mass.

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Let me just write it.

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The mass times the
gravitational field

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near the surface of the Earth.

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And it's going to
be downwards, we

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know that, or at least towards
the surface of the Earth.

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Now, what else is going
to be happening here?

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Well, it gets a
little bit confusing,

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because you can't really say
that normal force is acting

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directly against this
force right over here.

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Because remember,
the normal force

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acts perpendicular to a surface.

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So over here, the surface is
not perpendicular to the force

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of gravity.

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So we have to think about it a
little bit differently than we

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do if this was sitting
on level ground.

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Well, the one thing we can do,
and frankly, that we should do,

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is maybe we can
break up this force,

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the force due to gravity.

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We can break it
up into components

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that are either
perpendicular to the surface

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or that are parallel
to the surface.

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And then we can use
those to figure out

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what's likely to happen.

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What are potentially the netting
forces, or balancing forces,

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over here?

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So let's see if we can do that.

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Let's see if we can
break this force vector,

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the force due to gravity,
into a component that

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is perpendicular to the
surface of this ramp.

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And also another
component that is parallel

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to the surface of this ramp.

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Let me do that in
a different color.

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That is parallel to the
surface of this ramp.

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And this is a little bit
unconventional notation,

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but I'll call this one over
here the force due to gravity

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that is perpendicular
to the ramp.

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That little upside down t, I'm
saying that's perpendicular.

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Because it shows a line
that's perpendicular to,

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I guess, this bottom line, this
horizontal line over there.

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And this blue thing
over here, I'm

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going to call this
the part of force

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due to gravity that is parallel.

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I'm just doing these
two upward vertical bars

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to show something that is
parallel to the surface.

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So this is the
component of force

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due to gravity that's
perpendicular, component

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of force that is parallel.

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So let's see if we
can use a little bit

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a geometry and
trigonometry, given

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that this wedge is at
a theta degree incline

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relative to the horizontal.

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If you were to measure
this angle right over here,

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you would get theta.

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So in future videos we'll
make it more concrete,

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like 30 degrees or 45
degrees or whatever.

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But let's just keep in general.

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If this is theta,
let's figure out

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what these components of
the gravitational force

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are going to be.

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Well, we can break out
our geometry over here.

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This, I'm assuming
is a right angle.

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And so if this is
a right angle, we

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know that the sum of the angles
in a triangle add up to 180.

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So if this angle, and this
90 degrees-- right angle

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says 90 degrees-- add
up to 180, then that

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means that this one and this one
need to add up to 90 degrees.

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Or, if this is theta,
this angle right over here

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is going to be 90 minus theta.

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Now, the other thing
that you may or may not

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remember from geometry
class is that if I

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have two parallel lines,
and I have a transversal.

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So I'm going to assume this
line is parallel to this line.

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And then I have a transversal.

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So let's say I have a
line that goes like this.

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We know from basic
geometry that this angle

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is going to be
equal to this angle.

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It comes from alternate
interior angles.

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And we prove it in
the geometry module,

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or in the geometry videos.

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But hopefully this makes a
little bit of intuitive sense,

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and you could even think
about how these angles would

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changes as the transversal
changes, and all of the rest.

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But the parallel
lines makes this angle

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similar to that angle, or
actually makes it identical,

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makes it congruent.

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This angle is going to be the
same measure as that angle.

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So can we apply that
anywhere over here?

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This line is perpendicular
to the surface of the Earth.

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Right over here that I'm
kind of shading in blue.

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And so is this force vector.

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It is also perpendicular to
the surface of the Earth.

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So this line over here and
this line over here in magenta

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are going to be parallel.

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I can even draw that.

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That line and that
line are both parallel.

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When you look at
it that way, you'll

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see that this big line over here
can be viewed as a transversal.

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Or you could have this
angle and this angle

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are going to be congruent.

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They're going to be
alternate interior angles.

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So this angle and this angle,
by the exact same idea here.

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It just looks a little
bit more confusing here

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because I have all
sorts of things.

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But this line and this
line are parallel.

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You can view this right
over here as a transversal.

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So this and this are
congruent angles.

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So this is 90 minus
theta degrees.

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This too will be 90
minus theta degrees.

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90 minus theta degrees.

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Now, given that, can we
figure out this angle?

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Well one thing, we're assuming
that this yellow force vector

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right here is perpendicular
to the surface of this plane

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or perpendicular to the
surface of this ramp.

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So that's perpendicular.

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This right here
is 90 minus theta.

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So what is this angle up
here going to be equal to?

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This angle, let
me do it in green.

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What is this angle up
here going to be equal to?

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So this angle plus 90
minus theta plus 90

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must be equal to 180, or this
angle plus 90 minus theta must

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be equal to-- let
me write this down.

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I don't want to do
too much in your head.

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So let's call it x.

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So x plus 90 minus theta.

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Plus this 90 degrees right over
here, plus this 90 degrees,

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needs to be equal
to 180 degrees.

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Let's see, we can subtract
180 degrees from both sides.

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So we subtract 90 twice,
you subtract 180 degrees

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and you get x minus
theta is equal to 0,

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or x is equal to theta.

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So whatever the inclination of
the plane is or of this ramp,

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that is also going to be
this angle right over here.

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And the value to
that is that now we

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can use our basic
trigonometry to figure out

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this component
and this component

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of the force of gravity.

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And to see that a
little bit clearer,

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let me shift this force
vector down over here.

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The parallel component,
let me shift it over here.

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And you can see the
perpendicular component

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plus the parallel component
is equal to the total force

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due to gravity.

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And you should also see that
this is a right triangle

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that I have set up over here.

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This is parallel to the plane.

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This is perpendicular
to the plane.

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And so we can use
basic trigonometry

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to figure out the magnitudes
of the perpendicular

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force due to gravity and the
parallel force due to gravity.

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Let's think about
it a little bit.

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I'll do it over here.

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The magnitude of
the perpendicular

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force due to gravity.

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Or I should say the
component of gravity

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that's perpendicular to
the ramp, the magnitude

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of that vector-- a
lot of fancy notation

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but it's really just the length
of this vector right over here.

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So the magnitude of
this over the hypotenuse

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of this right triangle.

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Well, what the hypotenuse
of this right triangle?

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Well, it's going
to be the magnitude

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of the total
gravitational force.

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I guess you could say that.

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And so you could say that is mg.

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We could write it like this.

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But that's really-- well,
I could write it like that.

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And so this is going
to be equal to what?

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We have the, if we're looking
at this angle right here,

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we have the adjacent
over the hypotenuse.

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Remember.

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We can do this in a new color.

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We can do this in a new color.

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SOH CAH TOA.

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Cosine is adjacent
over hypotenuse.

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So this is equal to
cosine of the angle.

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So cosine of theta is
equal to the adjacent

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over the hypotenuse.

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So if you multiply both
sides by the magnitude

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of the hypotenuse, you get the
component of our vector that

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is perpendicular to the
surface of the plane

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is equal to the magnitude
of the force due to gravity

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times the cosine of theta.

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Times the cosine of theta.

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We'll apply this
in the next video

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00:09:44,272 --> 00:09:46,480
just so you can make the
numbers a lot more concrete.

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Sometimes just the notation
makes it confusing.

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You'll see it's really actually
pretty straightforward.

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00:09:50,970 --> 00:09:54,520
And then this second thing,
we can use the same logic.

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If we think about the parallel
vector right over here,

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00:09:57,120 --> 00:10:02,490
the magnitude of the
component of the force

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00:10:02,490 --> 00:10:05,320
due to gravity that is
parallel to the plane

219
00:10:05,320 --> 00:10:08,550
over the magnitude of the
force due to gravity--

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00:10:08,550 --> 00:10:13,230
which is the
magnitude of mg-- that

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00:10:13,230 --> 00:10:16,910
is going to be equal to what?

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00:10:16,910 --> 00:10:20,160
This is the opposite
side to the angle.

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So the blue stuff is the
opposite side, or at least

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its length, is the
opposite side of the angle.

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And then right over
here this magnitude

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00:10:27,670 --> 00:10:29,200
of mg, that is the hypotenuse.

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So you have the opposite
over the hypotenuse.

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Opposite over hypotenuse.

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Sine of an angle is
opposite over hypotenuse.

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So this is going to be
equal to the sine of theta.

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This is equal to
the sine of theta.

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00:10:41,490 --> 00:10:46,600
Or you multiply both sides
times the magnitude of the force

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00:10:46,600 --> 00:10:49,590
due to gravity and
you get the component

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00:10:49,590 --> 00:10:57,550
of the force due to gravity
that is parallel to the ramp

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is going to be the force
due to gravity total times

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00:11:07,260 --> 00:11:08,810
sine of theta.

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Times sine of theta.

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And hopefully you should
see where this came from.

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Because if you ever have to
derive this again 30 years

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after you took a physics class,
you should be able to do it.

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But if you know this right
here, and this right here,

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we can all of a sudden start
breaking down the forces

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00:11:30,220 --> 00:11:32,129
into things that
are useful to us.

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00:11:32,129 --> 00:11:33,670
Because we could
say, hey, look, this

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00:11:33,670 --> 00:11:35,410
isn't moving down
into this plane.

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00:11:35,410 --> 00:11:36,930
So maybe there's
some normal force

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that's completely netting
it out in this example.

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And maybe if there's
nothing to keep it up,

249
00:11:41,882 --> 00:11:43,590
and there's no friction,
maybe this thing

250
00:11:43,590 --> 00:11:45,850
will start accelerating
due to the parallel force.

251
00:11:45,850 --> 00:11:47,510
And we'll think a
lot more about that.

252
00:11:47,510 --> 00:11:50,112
And if you ever forget these,
think about them intuitively.

253
00:11:50,112 --> 00:11:52,320
You don't have to go through
this whole parallel line

254
00:11:52,320 --> 00:11:53,700
and transversal and all of that.

255
00:11:53,700 --> 00:11:57,860


256
00:11:57,860 --> 00:11:59,650
If this angle went
down to 0, then we'll

257
00:11:59,650 --> 00:12:01,680
be talking about
essentially a flat surface.

258
00:12:01,680 --> 00:12:03,720
There is no inclination there.

259
00:12:03,720 --> 00:12:06,830
And if this angle goes down
to 0, then all of the force

260
00:12:06,830 --> 00:12:11,250
should be acting perpendicular
to the surface of the plane.

261
00:12:11,250 --> 00:12:14,690
So if this going to 0, if
the perpendicular force

262
00:12:14,690 --> 00:12:17,810
should be the same thing as
the total gravitational force.

263
00:12:17,810 --> 00:12:19,310
And that's why it's
cosine of theta.

264
00:12:19,310 --> 00:12:21,680
Because cosine of
0 right now is 1.

265
00:12:21,680 --> 00:12:23,590
And so these would
equal each other.

266
00:12:23,590 --> 00:12:26,450
And if this is equal to 0,
then the parallel component

267
00:12:26,450 --> 00:12:27,960
of gravity should go to 0.

268
00:12:27,960 --> 00:12:29,540
Because gravity
will only be acting

269
00:12:29,540 --> 00:12:32,890
downwards, and once again,
if sine of theta is 0.

270
00:12:32,890 --> 00:12:35,430
So the force of gravity that
is parallel will go to 0.

271
00:12:35,430 --> 00:12:38,060
So if you ever forget, just do
that little intuitive thought

272
00:12:38,060 --> 00:12:40,050
process and you'll
remember which one is sine

273
00:12:40,050 --> 00:00:00,000
and which one is cosine.