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Let's examine again
an object that's
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undergoing circular motion.
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And we'll choose our polar
coordinates r hat, theta hat.
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We'll make it
cylindrical with a k hat.
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Now that we've completed
our kinematic description
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of the motion, now let's see
how we apply Newton's second law
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to circular motion.
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Well, when we write
Newton's second law as F
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equals ma, that-- remember,
we can divide these two sides.
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This side, the how, is
a geometric description
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of the motion.
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And this side is
the why, and this
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is the dynamics of the motion.
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And the dynamics come from
analyzing the forces that
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are acting on this object.
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So when we're applying
this mathematically
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to circular motion, F equals
ma-- this is a vector equation.
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And so what we need to do is
think about each component
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separately.
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Sometimes I can just
distinguish the components
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that I'm talking
about over here.
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And so we have that the radial
component of the forces--
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and that comes from an
analysis of the dynamics,
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the physics of the problem--
and this side is mass times
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the radial component of
the acceleration, ar.
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Now these are very
different things.
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And it's by the
second law that we're
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equating in quantity
these two components.
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Now if we wrote that equation--
this side out in a little bit
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more detail-- we'll
save ourselves
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a little space when we handle
the tangential direction--
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the forces come from analysis
of free body force diagrams.
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And over here, we know the
acceleration is always inward.
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And I'll choose to write
this as r omega squared.
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And so this will be
our starting point
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for analyzing the radial
motion for an object that's
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undergoing circular motion.
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Now remember, there could
be a tangential motion, too.
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And in the tangential
direction, the tangential forces
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are equal to ma tangential.
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And as we saw, this is
again the second law,
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equating two different things.
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We have that we can write
the tangential force as r
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d squared theta dt squared.
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And sometimes we've been
writing that as r alpha z.
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But this equation
here is what we're
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going to apply for
the tangential forces.
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If the tangential forces
are 0, then there's
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no angular-- there's no
tangential acceleration.
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We know that for
circular motion,
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the radial force
can never be zero,
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because this term
is always non-zero
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and points radially inward.
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And now we'll look at
a variety of examples
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applying Newton's second
law to circular motion.