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When solving physics problems,
we have fundamental principles.
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For instance, we have
Newton's second law.
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And we also have an
energy principle,
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work non-conservative equals
the change in kinetic energy
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plus the change in
potential energy.
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And many times
students are challenged
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by-- do I use this law?
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Do I use that law?
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Why do I use one or the other?
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Do I use both at the same time?
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Can I get the same
answers, et cetera.
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Now I want to illustrate a
particular category of problems
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where we have some type
of circular motion,
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and why we need to use
both of these laws.
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So let's consider, let's go
back to our dome example at MIT.
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And let's say we have an
object that's on the dome.
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And now let's write out a free
body diagram for this object.
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So we have our
gravitational force.
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We have a normal force.
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And here, let's just assume
that our dome is frictionless,
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for simplicity.
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So what you see here is that
the object is displacing
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in a direction-- now here if
I choose r hat and theta hat,
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and if I choose a
coordinate system where
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I take my positive
angle theta this way,
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that the normal force and the
displacement are perpendicular.
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Now what that means is that the
work done by the normal force,
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F normal dot ds is 0,
because well, we're
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calling F normal equal
to the normal force
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and that the normal force is
perpendicular to displacement.
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So what that means
is all information
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about the normal force is
not included in the energy
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principle.
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In fact, the energy
principle is-- the work
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that's done is just the
amount-- this is angle theta,
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so this is angle theta--
so the actual work done
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by the gravitational force g
is the component of g mg sine
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theta times the displacement ds.
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So the only work
that's appearing here,
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and this is conservative.
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And so when we
integrated this work
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and got that the work done
by the gravitational force
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is minus the change in potential
energy, that's showing up here.
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So the part of the force that's
in the direction of the motion
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is giving us the
energy condition.
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But we're losing all
information about the forces
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in the radial direction.
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So because also what we
need is Newton's second law
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in the direction perpendicular
to the displacement.
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Displacement is in the
theta hat direction,
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so we need Newton's second
law in the radial direction.
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And that we have r hat,
is we have a normal force
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minus mg co-sine theta.
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And the object is
undergoing circular motion,
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so it's equal to minus
mv squared over R,
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where R is the
radius of the dome.
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This equation is
not at all included
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in the energy condition.
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You may say, well what about the
tangential Newton's second law,
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which is mg sine theta equals
mR d squared theta dt squared.
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But it's precisely
this equation that's
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integrated with respect
to the displacement
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and that gives us
our energy principle.
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So in summary, the
energy principle
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is the integration of
Newton's second law
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in the direction of motion.
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And we're completely
missing the application
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of Newton's second law in
the direction perpendicular
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to the motion.
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Energy does not
account for that.
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And that's why we
needed separately
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to apply both of the
principles of energy
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and Newton's second law
in the radial direction
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in order to figure out
how to solve this problem.
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Now this idea is true in general
when you have circular motion,
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you need the energy
equation, which
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is the tangentially
integrated Newton's second
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on the tangential direction.
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And you need Newton's second law
in the direction perpendicular
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to that circular motion.