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Let's consider the
following problem.

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Suppose we have
an inclined plane.

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And on this inclined
plane, we have a block.

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And down here at the bottom
of the inclined plane

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is a little bumper.

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It can be a spring of some type.

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And what we have is the block
is sliding down the inclined

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plane, bouncing off the
bumper, and sliding up

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the inclined plane.

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And what we'd like to do
is find the acceleration

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of the block for both cases.

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So when it's sliding down, let's
do an analysis on that motion.

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Well, the first
thing we have to do

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is, again, identify our system.

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In this case, just the
block will be our system.

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So we'll put in the
block as our system.

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And now we have to analyze
the forces on the block

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as it's sliding down.

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There is a friction
with a coefficient mu k.

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And so we have the
gravitational force, mg.

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We have a normal
force of the surface

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on the block pointing up.

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And because the block
is sliding down,

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remember that our friction
force is opposing that motion.

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Now when you have a
free body force diagram,

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we have to decide how should we
choose our coordinate systems.

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What we want to
encourage you is not

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to choose a coordinate
system, unit vectors,

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horizontal and vertical,
just because that's the way

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one always does it, but to use
the constraints of the system

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to help you choose
the coordinate system.

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Now a technique for doing
that is to draw what you think

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will be the acceleration
on your sketch.

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So I expect that a is going
to go down the inclined plane.

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That's taking into
account that there

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is no motion perpendicular
to the inclined plane.

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Drawing the
acceleration-- remember,

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acceleration is not a
force-- but this gives me

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a clue for how I should
choose my unit vectors.

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I'll choose them, so
that I have points down.

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And so I essentially have
a one dimensional motion.

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And I'll choose j
hat pointing out.

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Now I'll introduce
an angle theta.

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That's the same
angle theta here.

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And one has to be a little
bit careful in doing

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these types of sketches.

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But this is also
the angle theta.

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And now I've chosen a
coordinate system, unit vectors.

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I've indicated relevant angles.

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And the rest of
the problem is just

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vector decomposition in
applying Newton's second law.

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So we'll write
down F equals m a.

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And I'm going to
indicate a d for we're

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talking about the
acceleration downwards.

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And we have two
directions to analyze,

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the i hat and the
j hat directions.

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So in the direction
tangent to the surface,

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we have a gravitational force,
downward, mg sine theta.

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We have the kinetic friction
force opposing that motion.

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And that's equal to m a d,
where a d is the x component

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of the acceleration.

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In the vertical direction,
we have the normal force

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minus m g cosine theta.

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And here's where our choice
of coordinate system helps us.

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The constraint of the
system is that a y

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is 0 when we choose y to be
the perpendicular direction.

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So this side is 0.

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Now I look at this,
and I see a f and N.

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And that's only two
equations and three unknowns.

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But now I have a for slope
for my friction, which

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is that the kinetic friction is
equal to a coefficient mu k N.

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And I can see from this
equation, that's mu k mg cosine

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theta.

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And now I'm in position
in which to substitute

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my friction into this equation
and solve for the acceleration.

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Notice the mass will cancel.

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And what I get is that the
acceleration going down

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is just equal to g times
sine theta minus mu k

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cosine of theta.

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And so that's how we can
analyze the acceleration

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when the block is sliding down.

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Next, we want to
analyze the acceleration

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when the block is going up.

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Now when you have
a case like that,

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it's sometimes easier,
again, just to draw.

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We'll still choose N.

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Now, the interesting
thing is, which way

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is the acceleration pointing?

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Just because the
block is going up,

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the acceleration is
still pointing down.

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How do you know that?

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Well, we'll see that when
we analyze the forces.

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So we still have N. We have mg.

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But now, the crucial
point is-- this

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is going up-- the
crucial point here

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is that the kinetic friction
is always opposing the motion.

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So when the block of sliding up,
the kinetic friction is down.

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And all that we have
is a sign change here.

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This is now plus.

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If I choose my i hat and j
hat in the same direction,

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notice both of these forces
have a positive component

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in the i hat direction.

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So the acceleration
will be down.

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And rather than do all
the algebra, all I'm doing

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is changing the sign.

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And I get that a up is equal to
g sign theta plus mu k cosine

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theta.

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And although it may seem
slightly counter-intuitive

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that the acceleration as the
block is going up in magnitude

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is bigger than the acceleration
as the block is going down.

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And that's because
of the direction

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of the kinetic friction force.