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All right...
long weekend ahead of us.
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One more lecture to go.
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If I have an object,
mass m, in gravitational field,
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gravitational force
is in this direction,
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if this is
my increasing value of y,
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then this force,
vectorially written,
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equals minus mg y roof.
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Since this is
a one-dimensional problem,
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we will often simply write
F equals minus mg.
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This minus sign is important
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because that's
the increasing value of y.
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If this level here
is y equals zero,
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then I could call this
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gravitational
potential energy zero.
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And this is y...
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Then the gravitational potential
energy here equals plus mg y.
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This is u.
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So if I make a plot of the
gravitational potential energy
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as a function of y, then
I would get a straight line.
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This is zero.
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So this equals u...
equals mg y, plus sign.
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If I'm here at point A and
I move that object to point B,
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I, Walter Lewin, move it,
I have to do positive work.
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Notice that the gravitational
potential energy increases.
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If I do positive work, the
gravity is doing negative work.
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If I go from A to some
other point-- call it B prime--
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then I do negative work.
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Notice the gravitational
potential energy goes down.
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If I do negative work, then
gravity is doing positive work.
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I could have chosen my zero
point of potential energy
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anywhere I please.
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I could have chosen it
right here
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and nothing would change
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other than that I offset
the zero point
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of my potential energy.
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But again, if I go from A to B,
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the gravitational potential
energy increases
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by exactly the same amount--
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I have to do exactly
the same work.
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So you are free to choose,
when you are near Earth,
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where you choose your zero.
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Now we take the situation
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whereby we are not
so close to the Earth.
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Here is the Earth itself.
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Of course you can also
replace that by the sun
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if you want to.
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And this is
increasing value of r.
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The distance between here
and this object m equals r.
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I now know that there is
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a gravitational force
on this object--
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Newton's Universal
Law of Gravity--
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and that gravitational force
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equals minus m M-Earth G
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divided by this r squared,
r roof
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so this is a vectorial notation.
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Since it is really
one-dimensional, we would...
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Just like we did there,
we would delete the arrow
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and we would delete
the unit vector
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in the positive r direction
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and so we would
simply write it this way.
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The gravitational potential
energy we derived last time
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equals minus m M-Earth G
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divided by r--
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and notice, here is r
and here is r squared--
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and if you plot that, then
the plot goes sort of like this.
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This is r, this is
increasing potential energy--
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all these values here
are negative--
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and you get a curve
which is sort of like this.
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This is proportional
to one over r.
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Now, of course, if the Earth
had a radius which is this big,
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then, of course, this curve does
not exist, it stops right here.
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If I move from point A to point
B, with a mass m in my hand,
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notice that the gravitational
potential energy increases.
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I have to do positive work,
there is no difference.
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If I go from A
to another point, B prime,
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which is closer to the Earth,
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notice that the gravitational
potential energy decreases.
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I do negative work.
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If I do positive work,
gravity is doing negative work.
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If I do negative work,
gravity is doing positive work.
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Right here near Earth,
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where this one-over-r curve
hits the Earth,
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that is, of course,
exactly that line.
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That dependence on y is exactly
the same as the dependence on r
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and then you can simplify
matters when you are near Earth.
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When the gravitational
acceleration doesn't change
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you get a linear relation.
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But that's only
an exceptional case
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when you don't move very far.
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The gravitational force is
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in the direction opposite
the increasing potential energy.
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Notice that when I'm here,
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the gravitational force is
in this direction.
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Increasing potential energy is
this way.
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The force is in this way.
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When I'm here,
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gravitational potential energy
increases in this way;
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the gravitational force
is in this direction.
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When I'm here,
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the gravitational
potential energy increases
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in this direction.
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The gravitational force is
in this direction.
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When I'm here,
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the gravitational
potential energy increases
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in this direction.
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The gravitational force is
in this direction.
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The force is always
in the opposite direction
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than the increasing value
of the potential energy.
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If I release an object
at zero speed,
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it therefore will always move
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towards a lower
potential energy
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because the force will drive it
to lower potential energy.
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Now I change from gravity
to a spring.
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I have a spring
which is relaxed length l--
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I call this x equals zero--
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and I extend it
over a distance x
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and there's a mass m at the end
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and there will be
a spring force.
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And that spring force...
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F equals minus kx.
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It's a one-dimensional situation
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so I can write it without having
to worry about the arrows.
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There is no friction here.
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It's clear
that if I hold this in my hand,
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that the force Walter Lewin
equals plus kx.
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It's in the direction
of increasing x.
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If I call this point A
at x equals zero
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and I call this point B
at x equals x,
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then I can calculate the work
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that I have to do
to bring it from A to B.
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So the work
that Walter Lewin has to do
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to bring it from A to B
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is the integral
in going from A to B
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of my force, dx.
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It's a dot product,
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but since the angle
between the two is zero degrees,
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the cosine of the angle is one,
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so I can forget about the fact
that there's a dot product.
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I move it in this direction.
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So that becomes the integral
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in going from zero
to a position x
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of plus x plus kx dx,
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and that is one-half k
x squared.
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And this is what we call
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the potential energy
of the spring.
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This is potential energy.
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It means, then, that we...
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At x equals zero, we define
potential energy to be zero.
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You don't have to do that,
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but it would be ridiculous
to do it any other way.
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So in the case that we have the
near-Earth situation of gravity,
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we had a choice where we put
our zero potential energy.
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In the case that we deal
with very large distances,
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we do not have a choice anymore.
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We defined it in such a way
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that the potential energy
at infinity is zero.
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As a result of that, all
potential energies are negative.
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And here, with the spring,
you don't have a choice, either.
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You choose...
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at x equals zero, you choose
potential energy to be zero.
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So if you now make a plot
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of the potential energy
as a function of x,
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you get a parabola,
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and if you are here,
if the object is here,
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then the force is always
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in the direction opposing
the increasing potential energy.
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If you go this way,
potential energy increases.
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So it's clear that the force is
going to be in this direction.
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If you are here, the force is
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always in the direction opposing
increasing potential energy.
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Increasing potential energy
is in this direction,
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so the force is
in this direction.
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You see, it's a restoring force.
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The force is always
in the direction
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opposite to increasing
potential energy.
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If you release an object here
at zero speed,
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it will therefore go
towards lower potential energy.
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The force will drive it
to lower potential energy.
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If we know the force--
in this case, the spring force
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or in those cases,
the gravitational force--
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we were able to calculate
the potential energy.
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Now, the question is,
can we also go back?
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Suppose we knew
the potential energy.
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Can we then find
the force again?
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And the answer is yes, we can.
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Let's take the situation
of the spring first.
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We have that the potential
energy u of the spring
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equals plus one-half k x squared
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and if I take the derivative
of that versus x
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then I get plus kx,
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but the force, the spring force
itself, is minus kx,
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so this equals
minus the spring force.
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So we have
that du/dx equals minus F,
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and I put the x there because
it's a one-dimensional problem--
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it is only in the x direction.
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The minus sign is telling you
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that the force is always
pointing in the direction
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which is opposite to increasing
values of the potential energy.
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That is what
the minus sign is telling you.
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It's staring you in the face
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what I have been telling you
for the past five minutes.
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If we have
a three-dimensional situation
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that we know
the potential energy
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as a function of x, y and z,
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then we can go back
and find the forces
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as a function of x, y and z.
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It doesn't matter
whether these are springs
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or whether it is gravity
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or whether it's electric forces
or nuclear forces,
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you then find
that du/dx equals minus F of x,
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du/dy equals minus Fy
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and du/dz equals minus Fz.
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What does this mean?
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It means that if you're
in three-dimensional space,
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you move only
in the x direction.
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You keep y and z constant,
and the change equals minus Fx.
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That gives you the component
of the force in the x direction.
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You move only
in the y direction,
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you keep x and z constant,
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and then you find
the component of the force
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in the y direction.
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We call these
partial derivatives,
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so we don't give them a "d"
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but we give them
a little curled delta.
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If we go back to the situation
where we had gravity,
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we had there the situation
near Earth.
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We had u... was plus mg y,
so what is du/dy?
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This is
a one-dimensional situation
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so I don't have to use
the partial derivatives.
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I can simply say du/dy.
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That is plus mg, and notice
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that the gravitational force
was minus mg.
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Remember? The minus sign is
still there.
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It's still there.
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And so you see
that here, indeed,
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du/dy is minus
the gravitational force.
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Now we take the situation
that we are not near Earth--
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we have there--
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so we have u equals
minus m M-Earth G
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divided by r--
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there's only an r here--
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so du/dr...
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The derivative of one over r
isminus one over r squared.
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The minus sign eats up
this minus sign,
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so I get plus m M-Earth G
divided by r squared
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so the gravitational force...
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the gravitational force
equals minus that.
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It isminus du/dr
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and, indeed, that's
exactly what we have there--
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minus that value.
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So whenever you know
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the potential
as a function of space,
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you can always find the
three components of the forces
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in the three
orthogonal directions.
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Suppose I have
a curved surface--
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literally,
a surface here in 26.100,
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which sort of looks like this...
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something like this.
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I call this, arbitrarily,
y equals zero
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and I could call this
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u gravitational potential
energy zero, for that matter.
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So this is a function y
as a function of x
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and the curve itself
represents effectively
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the gravitational
potential energy.
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This is y and this is x.
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So the gravitational potential
energy u equals mg y,
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but y is a function of x,
so that is also u times m...
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excuse me, that is m times g
times that function of x.
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There are points here
where du/dx equals zero.
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I'll get a nice mg in here...
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Where's zero...
and where are those points?
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Those points are here, here,
here, here and here.
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If du/dx is zero,
it means that the force--
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the component of the force
in the x direction-- is zero,
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because du/dx is minus
the force in the x direction.
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So if we visit those points,
for instance here,
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then there is, of course,
gravity, mg,
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if there is an object
there in the y direction...
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in the minus y direction
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and there is a normal force
in the plus y direction
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and these two exactly cancel
each other.
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So the net result is that here,
here, here, here and there
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there is no force
on the object at all
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so the object is not going to
move, it's going to stay put.
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Well, yes,
it's going to stay put.
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However, there is
a huge difference
293
00:17:19 --> 00:17:25
between this point here
and that point there
294
00:17:22 --> 00:17:28
and you sense immediately
that difference.
295
00:17:25 --> 00:17:31
If I put a marble here,
296
00:17:27 --> 00:17:33
I will have a hell of a time
to keep the marble in place,
297
00:17:30 --> 00:17:36
because if there is a fly there
in the corner of 26.100
298
00:17:36 --> 00:17:42
which does something
299
00:17:37 --> 00:17:43
then the slightest amount
of force on this one
300
00:17:40 --> 00:17:46
and it will start to roll off.
301
00:17:42 --> 00:17:48
In fact, what will happen is
302
00:17:44 --> 00:17:50
it will go
to a lower potential energy.
303
00:17:47 --> 00:17:53
Here, however,
if this one is offset,
304
00:17:50 --> 00:17:56
then it will want to go
to a lower potential energy.
305
00:17:54 --> 00:18:00
The force is always opposing
306
00:17:56 --> 00:18:02
the direction
of increasing potential energy,
307
00:17:59 --> 00:18:05
so the force will drive it back
308
00:18:00 --> 00:18:06
and so that's why we call this
a stable equilibrium.
309
00:18:03 --> 00:18:09
It will always go back
to that point.
310
00:18:05 --> 00:18:11
And this is
an unstable equilibrium.
311
00:18:11 --> 00:18:17
We have a setup here,
312
00:18:13 --> 00:18:19
and I would like to show you
how that will work.
313
00:18:19 --> 00:18:25
So we do have something
that is a curved object.
314
00:18:24 --> 00:18:30
It's a track.
315
00:18:26 --> 00:18:32
Let me give you a little
bit better light condition.
316
00:18:30 --> 00:18:36
So you see there, there is
that object, a little ball.
317
00:18:36 --> 00:18:42
And no surprise, if I offset it
from the lowest point
318
00:18:40 --> 00:18:46
that it will be driven back
to that point-- that's trivial.
319
00:18:43 --> 00:18:49
What is less trivial is
that there is a point here
320
00:18:47 --> 00:18:53
whereby, indeed,
the net forces are zero
321
00:18:50 --> 00:18:56
and it is not easy
to achieve that,
322
00:18:53 --> 00:18:59
but I will try to put it there
so that it, indeed, stays put.
323
00:18:59 --> 00:19:05
I'm not too fortunate,
it is very difficult.
324
00:19:04 --> 00:19:10
I'm trying... no...
325
00:19:07 --> 00:19:13
Yeah! Did it.
326
00:19:09 --> 00:19:15
It's there, it's very unstable.
327
00:19:13 --> 00:19:19
I blow...
oh, it's not so unstable.
328
00:19:16 --> 00:19:22
And there it goes.
329
00:19:17 --> 00:19:23
So you see,
that's the difference
330
00:19:19 --> 00:19:25
between stable equilibrium
and unstable equilibrium.
331
00:19:21 --> 00:19:27
At the stable point,
332
00:19:23 --> 00:19:29
the second derivative of the
potential energy versus x
333
00:19:27 --> 00:19:33
is positive.
334
00:19:29 --> 00:19:35
At the unstable point, the
second derivative is negative.
335
00:19:33 --> 00:19:39
336
00:19:39 --> 00:19:45
I'm going to return now
to my spring
337
00:19:44 --> 00:19:50
and I'm going to show you
338
00:19:49 --> 00:19:55
that if you use the potential
energy of the spring alone
339
00:19:55 --> 00:20:01
that you can show that an object
that oscillates on a spring
340
00:20:00 --> 00:20:06
follows
a simple harmonic motion.
341
00:20:04 --> 00:20:10
So here...
is u as a function of x,
342
00:20:13 --> 00:20:19
and this is the parabola
that we already had
343
00:20:16 --> 00:20:22
which equals
one-half kx squared.
344
00:20:22 --> 00:20:28
Let the object be
at a position x maximum here.
345
00:20:27 --> 00:20:33
It's going to oscillate between
plus x max and minus x max.
346
00:20:37 --> 00:20:43
When it is at a random position
x, there is a force on it
347
00:20:43 --> 00:20:49
and the force is always
348
00:20:45 --> 00:20:51
in the direction opposing
the increasing potential energy
349
00:20:48 --> 00:20:54
so the force is clearly
in this direction.
350
00:20:50 --> 00:20:56
It's being driven back
to equilibrium.
351
00:20:53 --> 00:20:59
When it is there, it will have
a certain velocity.
352
00:20:56 --> 00:21:02
The velocity could either be
in this direction
353
00:20:58 --> 00:21:04
or it could be
in that direction.
354
00:21:00 --> 00:21:06
It has a certain speed
355
00:21:03 --> 00:21:09
and since spring forces
are conservative forces,
356
00:21:07 --> 00:21:13
I can now apply the conservation
of mechanical energy.
357
00:21:13 --> 00:21:19
We call this a potential well.
358
00:21:15 --> 00:21:21
The object is going to oscillate
in a potential well.
359
00:21:18 --> 00:21:24
Of course, it doesn't oscillate
like that.
360
00:21:20 --> 00:21:26
It really oscillates like this,
of course.
361
00:21:22 --> 00:21:28
It's a one-dimensional problem.
362
00:21:25 --> 00:21:31
The total energy
that I started with
363
00:21:27 --> 00:21:33
if I release it
here at zero speed
364
00:21:29 --> 00:21:35
equals one-half k x max squared.
365
00:21:34 --> 00:21:40
That is e total.
366
00:21:35 --> 00:21:41
That will always be the same
367
00:21:37 --> 00:21:43
if there is no friction
of any kind
368
00:21:40 --> 00:21:46
and I have to assume
that there is no friction.
369
00:21:43 --> 00:21:49
That must be, now,
one-half m v squared
370
00:21:48 --> 00:21:54
at a random position x,
371
00:21:50 --> 00:21:56
plus one-half k x squared,
372
00:21:53 --> 00:21:59
which is the potential energy
at position x.
373
00:21:56 --> 00:22:02
So this is the kinetic energy
374
00:21:58 --> 00:22:04
and this is
the potential energy.
375
00:22:02 --> 00:22:08
v is the first derivative
of position versus time,
376
00:22:08 --> 00:22:14
so I can write for this
an x dot.
377
00:22:11 --> 00:22:17
And now what I'm going to do,
378
00:22:14 --> 00:22:20
I'm going to rewrite
it slightly differently.
379
00:22:16 --> 00:22:22
I'll bring the x dot squared
to one side, my halfs go away
380
00:22:24 --> 00:22:30
and I divide by m,
381
00:22:27 --> 00:22:33
so I get plus k over m
times x squared,
382
00:22:33 --> 00:22:39
and then I get minus k x max
squared equals zero.
383
00:22:41 --> 00:22:47
Did I do that right?
384
00:22:42 --> 00:22:48
Yes, I divide...
385
00:22:44 --> 00:22:50
Oh, there's an m here,
and the m has to be here.
386
00:22:46 --> 00:22:52
And now what I'm going to do,
387
00:22:48 --> 00:22:54
I'm going to take the time
derivative of this equation.
388
00:22:52 --> 00:22:58
Now you will see something
remarkable falling in place.
389
00:22:55 --> 00:23:01
Just for free.
390
00:22:58 --> 00:23:04
I take the derivative
versus time.
391
00:22:59 --> 00:23:05
That gives me a two x dot, but
I have to apply the chain rule
392
00:23:04 --> 00:23:10
so I also get x double dot,
the second derivative--
393
00:23:07 --> 00:23:13
it's the acceleration--
394
00:23:09 --> 00:23:15
plus I get a two k over m
times x, with the chain rule
395
00:23:16 --> 00:23:22
gives me an x dot.
396
00:23:18 --> 00:23:24
This is a constant, that's
the total energy when I started,
397
00:23:21 --> 00:23:27
so the whole thing equals zero.
398
00:23:24 --> 00:23:30
I lose my two, I lose my x dot
because it's zero
399
00:23:29 --> 00:23:35
and what do I find?
400
00:23:30 --> 00:23:36
x double dot plus k over m
equals zero.
401
00:23:35 --> 00:23:41
And this makes my day
402
00:23:38 --> 00:23:44
because I know this is
a simple harmonic oscillation.
403
00:23:41 --> 00:23:47
You've seen
this equation before.
404
00:23:43 --> 00:23:49
We derived it
in a different way.
405
00:23:45 --> 00:23:51
We didn't use forces today.
406
00:23:47 --> 00:23:53
We only used the concept
of mechanical energy,
407
00:23:51 --> 00:23:57
which is conserved.
408
00:23:53 --> 00:23:59
We know the solution
to this equation...
409
00:23:57 --> 00:24:03
There is an x here.
410
00:23:59 --> 00:24:05
I heard someone mention the x--
thank you very much.
411
00:24:02 --> 00:24:08
The solution is:
412
00:24:04 --> 00:24:10
x equals x max times
the cosine omega t plus phi.
413
00:24:13 --> 00:24:19
This is the amplitude.
414
00:24:15 --> 00:24:21
And omega equals
the square root of k over m,
415
00:24:19 --> 00:24:25
and the period
for one oscillation
416
00:24:21 --> 00:24:27
equals two pi divided by omega.
417
00:24:26 --> 00:24:32
We were able to do this.
418
00:24:30 --> 00:24:36
We were able to apply
419
00:24:31 --> 00:24:37
the conservation
of mechanical energy
420
00:24:33 --> 00:24:39
because spring forces are
conservative forces.
421
00:24:36 --> 00:24:42
So you've seen,
in a completely different way,
422
00:24:40 --> 00:24:46
how you arrive
at the same result.
423
00:24:43 --> 00:24:49
Now I'm going to try
424
00:24:45 --> 00:24:51
something similar
to another potential well
425
00:24:50 --> 00:24:56
and that potential well
is a track
426
00:24:53 --> 00:24:59
and the track is
a perfect circle.
427
00:24:56 --> 00:25:02
And I'm going to slide down
that track an object mass m,
428
00:25:01 --> 00:25:07
and I'm going to evaluate
429
00:25:03 --> 00:25:09
the oscillation along
a perfect circular track.
430
00:25:09 --> 00:25:15
And to make it
as perfect as I can,
431
00:25:12 --> 00:25:18
I even have here
a pair of compasses...
432
00:25:17 --> 00:25:23
make it a...
433
00:25:20 --> 00:25:26
that's the track.
434
00:25:22 --> 00:25:28
And the track has a radius R,
435
00:25:28 --> 00:25:34
and at this moment in time
436
00:25:31 --> 00:25:37
the angle equals theta,
and here is the object.
437
00:25:36 --> 00:25:42
I call this x equals zero.
438
00:25:39 --> 00:25:45
That is also, of course,
where theta equals zero.
439
00:25:46 --> 00:25:52
This is increasing value of y,
and I choose this y equals zero.
440
00:25:54 --> 00:26:00
And so the gravitational
potential energy of this object
441
00:25:59 --> 00:26:05
is its own mg y, so I have
to know what this y is,
442
00:26:04 --> 00:26:10
and therefore I have to know
what this distance is.
443
00:26:09 --> 00:26:15
That's very easy.
444
00:26:12 --> 00:26:18
This one here
equals R cosine theta
445
00:26:17 --> 00:26:23
and so this one is
R minus R cosine theta.
446
00:26:23 --> 00:26:29
So the potential energy equals
447
00:26:25 --> 00:26:31
mg times R one minus cosine
theta, if I choose zero there.
448
00:26:33 --> 00:26:39
I'm free to change that
449
00:26:35 --> 00:26:41
but that's, of course,
a logical thing to do here.
450
00:26:39 --> 00:26:45
Notice if theta equals zero
and the cosine theta equals one,
451
00:26:42 --> 00:26:48
then you find u equals zero.
452
00:26:44 --> 00:26:50
That's, of course...
I have defined it.
453
00:26:47 --> 00:26:53
That's the way I defined
my y equals zero.
454
00:26:50 --> 00:26:56
So u equals zero.
455
00:26:51 --> 00:26:57
Notice that
when theta equals pi over two--
456
00:26:55 --> 00:27:01
if the object were here--
457
00:26:57 --> 00:27:03
that you find that the
potential energy u equals mg R.
458
00:27:02 --> 00:27:08
That's exactly right,
because then the distance
459
00:27:05 --> 00:27:11
between here
and the y zero is R,
460
00:27:08 --> 00:27:14
so this is the potential energy
as a function of angle theta.
461
00:27:14 --> 00:27:20
The velocity of that object
as a function of theta
462
00:27:22 --> 00:27:28
is given by R d theta/dt.
463
00:27:28 --> 00:27:34
And I can make you
see that very easily.
464
00:27:32 --> 00:27:38
Let this be the angle d theta
465
00:27:37 --> 00:27:43
so it moves in a short amount
of time over an angle d theta
466
00:27:41 --> 00:27:47
and the arc here is dS,
and the radius is R.
467
00:27:46 --> 00:27:52
The definition of theta--
468
00:27:48 --> 00:27:54
that's the definition of theta
which is in radians--
469
00:27:51 --> 00:27:57
is that dS divided by R
equals d theta.
470
00:27:57 --> 00:28:03
That's our definition
of radians.
471
00:27:59 --> 00:28:05
So I take the derivative, the
time derivative, left and right,
472
00:28:03 --> 00:28:09
so I get the dS/dt--
473
00:28:05 --> 00:28:11
which, of course, is
474
00:28:06 --> 00:28:12
the tangential velocity
along that arc--
475
00:28:10 --> 00:28:16
equals R times d theta/dt,
476
00:28:17 --> 00:28:23
for which you can write
R theta dot.
477
00:28:20 --> 00:28:26
d theta/dt... d theta/dt
is sometimes called omega,
478
00:28:24 --> 00:28:30
which is the angular velocity,
479
00:28:27 --> 00:28:33
but keep in mind
that in this case
480
00:28:30 --> 00:28:36
the angular velocity omega--
481
00:28:33 --> 00:28:39
if you want to call this omega,
which is the angular velocity--
482
00:28:36 --> 00:28:42
is changing with time.
483
00:28:38 --> 00:28:44
The angular velocity is zero
when you release it
484
00:28:40 --> 00:28:46
and is a maximum when it goes
through the lowest point.
485
00:28:46 --> 00:28:52
So I can now apply
486
00:28:48 --> 00:28:54
the conservation of mechanical
energy, because I know
487
00:28:53 --> 00:28:59
what the velocity is
at any angle of theta
488
00:28:56 --> 00:29:02
and I know what the kinetic...
what the potential energy is.
489
00:29:01 --> 00:29:07
So, let the total energy be
just the mechanical energy
490
00:29:04 --> 00:29:10
which depends on my initial
conditions wherever I start.
491
00:29:07 --> 00:29:13
Maybe it's just that I release
it here with zero speed;
492
00:29:09 --> 00:29:15
maybe I give it a little speed.
493
00:29:11 --> 00:29:17
It is a number,
it is a constant.
494
00:29:14 --> 00:29:20
So that is going to be
495
00:29:16 --> 00:29:22
one-half mv squared
at a random angle of theta.
496
00:29:20 --> 00:29:26
And that means this is v,
497
00:29:24 --> 00:29:30
so that is R squared
times theta dot squared.
498
00:29:29 --> 00:29:35
This is simply one-half
mv squared, nothing else,
499
00:29:34 --> 00:29:40
so this is the kinetic energy.
500
00:29:37 --> 00:29:43
Plus the potential energy,
501
00:29:41 --> 00:29:47
which is mg times R times
one minus cosine theta.
502
00:29:48 --> 00:29:54
And this is always the same,
503
00:29:52 --> 00:29:58
independent
of the angle of theta,
504
00:29:53 --> 00:29:59
because gravity is
a conservative force.
505
00:29:57 --> 00:30:03
So this is the conservation
of mechanical energy.
506
00:30:04 --> 00:30:10
This angle cosine theta is
really a pain in the neck,
507
00:30:09 --> 00:30:15
and therefore
what we're going to do
508
00:30:10 --> 00:30:16
is something
we have seen before--
509
00:30:13 --> 00:30:19
we are going to make a small
angle approximation...
510
00:30:18 --> 00:30:24
small angle approximation.
511
00:30:20 --> 00:30:26
And we're going to write
for cosine theta
512
00:30:24 --> 00:30:30
one minus theta squared
divided by two.
513
00:30:28 --> 00:30:34
That is a very,
very good approximation.
514
00:30:31 --> 00:30:37
That approximation isway better
than the one we did before
515
00:30:36 --> 00:30:42
when we simply said
the cosine of theta equals one.
516
00:30:39 --> 00:30:45
Remember we did that once?
517
00:30:41 --> 00:30:47
We said,
"Oh... for theta is very small.
518
00:30:44 --> 00:30:50
The cosine of theta is
about one."
519
00:30:46 --> 00:30:52
If we did that now,
we would be dead in the waters,
520
00:30:48 --> 00:30:54
because if we said
the cosine of theta is one,
521
00:30:50 --> 00:30:56
this becomes zero
and you end up with nonsense,
522
00:30:53 --> 00:30:59
because it would say
523
00:30:54 --> 00:31:00
that the mechanical energy
is changing all the time
524
00:30:57 --> 00:31:03
because this velocity is
changing all the time.
525
00:30:59 --> 00:31:05
So we cannot do that.
526
00:31:02 --> 00:31:08
We would kill ourselves
if we did that.
527
00:31:04 --> 00:31:10
The approximation is
really amazingly good.
528
00:31:08 --> 00:31:14
If I give you here
theta in radians
529
00:31:13 --> 00:31:19
and I give you here
the cosine of theta
530
00:31:15 --> 00:31:21
and here I give you one minus
theta squared over two,
531
00:31:20 --> 00:31:26
then if I take 1/60
of a radian--
532
00:31:24 --> 00:31:30
and I pick 1/60 since that is
approximately one degree.
533
00:31:30 --> 00:31:36
But I pick exactly 1/60--
534
00:31:32 --> 00:31:38
and I ask what the cosine is,
that is 0.999.
535
00:31:38 --> 00:31:44
And then I have an 861114.
536
00:31:43 --> 00:31:49
I just used my calculator.
537
00:31:45 --> 00:31:51
Then I calculate what one minus
theta squared over two is
538
00:31:49 --> 00:31:55
and I find 0.999861111.
539
00:31:56 --> 00:32:02
That isvery, very close.
540
00:31:59 --> 00:32:05
That is only... it only differs
by three parts in abillion.
541
00:32:03 --> 00:32:09
That is very close.
542
00:32:05 --> 00:32:11
That means the difference
between the two
543
00:32:07 --> 00:32:13
is only one-third
of a millionth of a percent.
544
00:32:12 --> 00:32:18
Suppose now I go
a little rougher
545
00:32:14 --> 00:32:20
and I go to one-fifth
of a radian,
546
00:32:17 --> 00:32:23
which is about 12 degrees, so
this is very roughly 12 degrees.
547
00:32:22 --> 00:32:28
Then the cosine of theta
equals 0.98007,
548
00:32:29 --> 00:32:35
and one minus theta squared
over two equals 0.98000.
549
00:32:35 --> 00:32:41
So that still is
amazingly close--
550
00:32:38 --> 00:32:44
that is, only differs
by seven parts in 100 -->
0:32:42 --> 00:32:48
so the difference is less
than 1/100 of a percent.
551
00:32:46 --> 00:32:52
So with this in mind,
I feel comfortable to pursue
552
00:32:50 --> 00:32:56
my conservation
of mechanical energy.
553
00:32:55 --> 00:33:01
And I'm going to replace
this cosine theta
554
00:32:57 --> 00:33:03
by one minus theta squared
divided by two.
555
00:33:02 --> 00:33:08
556
00:33:05 --> 00:33:11
So I will continue here--
557
00:33:07 --> 00:33:13
the center blackboard is always
nice, you can see it best--
558
00:33:13 --> 00:33:19
and I will massage
that equation a little further
559
00:33:19 --> 00:33:25
and, of course,
you can already guess
560
00:33:22 --> 00:33:28
what I am going to do when
I massage it a little further.
561
00:33:25 --> 00:33:31
I'm going to take
the time derivative
562
00:33:27 --> 00:33:33
just as I did in the case
of the spring.
563
00:33:33 --> 00:33:39
So we are going to get
that the mechanical energy--
564
00:33:38 --> 00:33:44
which is not changing--
565
00:33:40 --> 00:33:46
equals one-half m R squared
theta dot squared plus mg R.
566
00:33:52 --> 00:33:58
Cosine squared becomes one
minus theta squared over two
567
00:33:55 --> 00:34:01
so we have a minus
times minus becomes plus
568
00:33:58 --> 00:34:04
so I get simply
theta squared over two.
569
00:34:01 --> 00:34:07
And now I take
the time derivative...
570
00:34:06 --> 00:34:12
for this becomes zero...
equals...
571
00:34:10 --> 00:34:16
Now, I get a two out of here,
which eats up this one-half,
572
00:34:14 --> 00:34:20
so I get m R squared,
then I get theta dot,
573
00:34:20 --> 00:34:26
but the chain rule gives me
theta double dot.
574
00:34:24 --> 00:34:30
Excuse me?
575
00:34:25 --> 00:34:31
Anything wrong?
576
00:34:27 --> 00:34:33
I don't think so, thank you.
577
00:34:29 --> 00:34:35
So I have to take
the derivative of this one.
578
00:34:34 --> 00:34:40
The two flips out, which eats up
this two, so I get mg R
579
00:34:39 --> 00:34:45
and then I get a theta.
580
00:34:41 --> 00:34:47
With the chain rule,
gives me a theta dot.
581
00:34:45 --> 00:34:51
I lose my m, I lose one R,
I lose my theta dot--
582
00:34:52 --> 00:34:58
I picked the wrong one; I lose
my theta dot, not the theta--
583
00:34:59 --> 00:35:05
and what do I find?
584
00:35:01 --> 00:35:07
That theta double dot plus g
over R times theta equals zero.
585
00:35:11 --> 00:35:17
And I couldn't be happier,
586
00:35:13 --> 00:35:19
because this tells me
that the motion
587
00:35:16 --> 00:35:22
is that of a simple
harmonic oscillation.
588
00:35:20 --> 00:35:26
And the solution is x...
excuse me, not x.
589
00:35:24 --> 00:35:30
Theta equals
some maximum angle for theta.
590
00:35:28 --> 00:35:34
It's the amplitude in angle
591
00:35:31 --> 00:35:37
times the cosine
of omega t plus phi.
592
00:35:34 --> 00:35:40
This is the angle of frequency.
593
00:35:36 --> 00:35:42
It hasnothing, nothing to do
with that omega there,
594
00:35:39 --> 00:35:45
which is the angular velocity,
which is changing in time.
595
00:35:42 --> 00:35:48
This is a constant,
this is angular frequency
596
00:35:44 --> 00:35:50
and this omega equals
the square root of g over R,
597
00:35:49 --> 00:35:55
and so the period
of the oscillation is
598
00:35:51 --> 00:35:57
two pi times the square root
of R over g.
599
00:35:56 --> 00:36:02
And when you see that,
you say, "Hey!
600
00:36:02 --> 00:36:08
I have seen that before."
601
00:36:04 --> 00:36:10
Where have we seen this before?
602
00:36:06 --> 00:36:12
603
00:36:09 --> 00:36:15
Almost a carbon copy
604
00:36:11 --> 00:36:17
of something
that we have seen before.
605
00:36:13 --> 00:36:19
What is it?
606
00:36:14 --> 00:36:20
(class murmurs )
607
00:36:15 --> 00:36:21
LEWIN:
Excuse me, speak louder.
608
00:36:17 --> 00:36:23
(echoing class ):
Pendulum!
609
00:36:18 --> 00:36:24
We had a pendulum whereby we had
length l of a massless string.
610
00:36:24 --> 00:36:30
We had an object m
hanging on the end
611
00:36:26 --> 00:36:32
and what was it doing?
612
00:36:29 --> 00:36:35
It was going along a perfect arc
which is exactly identical.
613
00:36:34 --> 00:36:40
The problem is the same,
it's not a surprise,
614
00:36:36 --> 00:36:42
because now we have a surface
which is an exact, perfect arc.
615
00:36:41 --> 00:36:47
It's a circle,
we have no friction.
616
00:36:44 --> 00:36:50
We assumed
with the pendulum
617
00:36:46 --> 00:36:52
that there was
no friction either.
618
00:36:48 --> 00:36:54
So it shouldn't surprise us
619
00:36:49 --> 00:36:55
that you get exactly
the same period
620
00:36:51 --> 00:36:57
that you had
with the pendulum and...
621
00:36:53 --> 00:36:59
except that, of course
622
00:36:56 --> 00:37:02
with the pendulum,
what we called l is now R.
623
00:36:59 --> 00:37:05
Gravity is the only force
that does work
624
00:37:04 --> 00:37:10
and so it is justified
625
00:37:06 --> 00:37:12
to use the conservation
of mechanical energy
626
00:37:09 --> 00:37:15
because gravity is
a conservative force.
627
00:37:12 --> 00:37:18
We used the small angle
approximation to make it work.
628
00:37:18 --> 00:37:24
In the case of the spring,
629
00:37:20 --> 00:37:26
we had that the potential energy
was proportional to x squared
630
00:37:26 --> 00:37:32
and out came a perfect
simple harmonic oscillation,
631
00:37:29 --> 00:37:35
no approximation necessary.
632
00:37:32 --> 00:37:38
Now we forced
this potential energy...
633
00:37:37 --> 00:37:43
we forced it into being
dependent on theta squared.
634
00:37:39 --> 00:37:45
That's really what we did.
635
00:37:41 --> 00:37:47
You see, that is the term
of the potential energy
636
00:37:43 --> 00:37:49
that you have there.
637
00:37:44 --> 00:37:50
And by the approximation
of cosine theta being
638
00:37:48 --> 00:37:54
one minus theta squared
over two,
639
00:37:50 --> 00:37:56
we forced this term to become
quadratic in theta
640
00:37:53 --> 00:37:59
and therefore now,
with that approximation
641
00:37:56 --> 00:38:02
it becomes a perfect
simple harmonic oscillation.
642
00:38:02 --> 00:38:08
Now comes a key question.
643
00:38:04 --> 00:38:10
I said, "Gravity is really
the only force that does work."
644
00:38:09 --> 00:38:15
Is that true?
645
00:38:12 --> 00:38:18
There's no friction for now.
646
00:38:14 --> 00:38:20
Is that really true?
647
00:38:17 --> 00:38:23
When we had the pendulum,
it's true there is gravity.
648
00:38:22 --> 00:38:28
That's clear.
649
00:38:23 --> 00:38:29
There's a gravitational force,
which is mg,
650
00:38:27 --> 00:38:33
but there is also tension.
651
00:38:31 --> 00:38:37
We never mentioned that.
652
00:38:33 --> 00:38:39
We didn't even talk about it
653
00:38:34 --> 00:38:40
when we did the conservation
of mechanical energy.
654
00:38:37 --> 00:38:43
When the object is here,
sure, there is gravity
655
00:38:41 --> 00:38:47
and sure, there is no friction.
656
00:38:43 --> 00:38:49
So there is no force
along the arc,
657
00:38:45 --> 00:38:51
but there must be
a normal force.
658
00:38:49 --> 00:38:55
Is the tension
not doing any work?
659
00:38:52 --> 00:38:58
Is the normal force
not doing any work?
660
00:38:55 --> 00:39:01
Did we, perhaps,
forget something?
661
00:38:58 --> 00:39:04
Remember last week,
I put mylife on the line.
662
00:39:01 --> 00:39:07
I was so convinced
663
00:39:02 --> 00:39:08
that the conservation
of mechanical energy
664
00:39:04 --> 00:39:10
was going to work
665
00:39:05 --> 00:39:11
that I almost killed myself--
not quite--
666
00:39:09 --> 00:39:15
with this huge, 15½ kilogram
pendulum that I was swinging.
667
00:39:13 --> 00:39:19
I believed in the conservation
of mechanical energy
668
00:39:15 --> 00:39:21
and I overlooked the tension.
669
00:39:18 --> 00:39:24
Is it possible that the tension
does, perhaps, positive work?
670
00:39:21 --> 00:39:27
If that's the case,
I could havedied.
671
00:39:25 --> 00:39:31
What is the answer?
672
00:39:26 --> 00:39:32
Is the tension doing any work
673
00:39:28 --> 00:39:34
and in the case
of my circular track
674
00:39:32 --> 00:39:38
is, perhaps, the normal force
doing any work?
675
00:39:35 --> 00:39:41
What is the answer?
676
00:39:37 --> 00:39:43
(class murmurs )
677
00:39:38 --> 00:39:44
LEWIN:
I want to hear it loud
and clear!
678
00:39:40 --> 00:39:46
CLASS:
No.
679
00:39:41 --> 00:39:47
LEWIN:
No! Why is it no?
680
00:39:42 --> 00:39:48
Why is it not doing any work?
681
00:39:43 --> 00:39:49
Because what?
682
00:39:44 --> 00:39:50
(student answers )
683
00:39:46 --> 00:39:52
Exactly. You got it, man!
684
00:39:49 --> 00:39:55
That's it.
685
00:39:50 --> 00:39:56
The force is
always perpendicular
686
00:39:53 --> 00:39:59
to the direction of motion.
687
00:39:55 --> 00:40:01
And since work is a dot product
688
00:39:57 --> 00:40:03
between force and
the direction that it travels,
689
00:40:01 --> 00:40:07
neither the tension nor
the normal force does any work.
690
00:40:05 --> 00:40:11
So don't overlook the force,
691
00:40:07 --> 00:40:13
but do appreciate the fact
that they don't do any work.
692
00:40:11 --> 00:40:17
Great! So now I'm going to show
you a demonstration
693
00:40:17 --> 00:40:23
which I find one of the most
mind-boggling demonstrations
694
00:40:21 --> 00:40:27
that I have ever seen.
695
00:40:23 --> 00:40:29
We do have a circular track.
696
00:40:26 --> 00:40:32
You have it
right in front of you.
697
00:40:29 --> 00:40:35
That is a circle, although you
may not think it is, but it is.
698
00:40:33 --> 00:40:39
And that circle has a radius
699
00:40:37 --> 00:40:43
which, according to the
manufacturer, is 115 meters
700
00:40:43 --> 00:40:49
with an uncertainty of about...
I think it's about five meters.
701
00:40:48 --> 00:40:54
It is extremely difficult
to measure
702
00:40:50 --> 00:40:56
and even during transport,
you think it could change.
703
00:40:53 --> 00:40:59
Let me try to clean this
a little better.
704
00:40:57 --> 00:41:03
And so the radius of this... the
radius of curvature of our arc,
705
00:41:03 --> 00:41:09
which is also an air track,
706
00:41:05 --> 00:41:11
equals 115
plus or minus five meters.
707
00:41:11 --> 00:41:17
So we can calculate now
708
00:41:12 --> 00:41:18
what the period
of oscillations is.
709
00:41:15 --> 00:41:21
The whole track is
five meters long.
710
00:41:18 --> 00:41:24
So half the track is
about 2½ meters,
711
00:41:23 --> 00:41:29
so the angle theta maximum
712
00:41:26 --> 00:41:32
is approximately 2½ meters--
713
00:41:29 --> 00:41:35
which is half the length
of the track-- divided by 115
714
00:41:33 --> 00:41:39
and that is
an extremely small angle.
715
00:41:35 --> 00:41:41
That is about 1.2 degrees,
716
00:41:37 --> 00:41:43
because this is in radians
and this is in degrees.
717
00:41:40 --> 00:41:46
So the angle is very small,
so we should be able to make
718
00:41:44 --> 00:41:50
a perfect prediction
about the period.
719
00:41:47 --> 00:41:53
And I am going to do that.
720
00:41:49 --> 00:41:55
I take two pi times the square
root of R over G
721
00:41:52 --> 00:41:58
and R is 115, 115...
I divide it by G.
722
00:41:57 --> 00:42:03
I take the square root,
I multiply by two.
723
00:42:01 --> 00:42:07
I multiply by pi and I get 21.5.
724
00:42:07 --> 00:42:13
T-- and this is
a prediction...
725
00:42:13 --> 00:42:19
equals 21.5.
726
00:42:18 --> 00:42:24
The uncertainty in R
is about 4.3%.
727
00:42:23 --> 00:42:29
Since we have the square root
of R, that becomes 2.2%.
728
00:42:27 --> 00:42:33
So if I multiply that by .022,
729
00:42:30 --> 00:42:36
I get an uncertainty
of about 0.47.
730
00:42:33 --> 00:42:39
Let's call this 0.5 seconds.
731
00:42:36 --> 00:42:42
So this is a hard prediction
732
00:42:39 --> 00:42:45
what the period of an
oscillation should be--
733
00:42:43 --> 00:42:49
21.5 plus or minus
a half second.
734
00:42:47 --> 00:42:53
Now I'm going to observe it
and we're going to see
735
00:42:50 --> 00:42:56
what we're going to...
how this compares.
736
00:42:55 --> 00:43:01
I don't want to... I don't want
to oscillate it ten times.
737
00:42:58 --> 00:43:04
That will take three, four, five
minutes-- that's too long.
738
00:43:01 --> 00:43:07
It is not really necessary
739
00:43:02 --> 00:43:08
because my reaction time
is 0.1 second,
740
00:43:05 --> 00:43:11
so even if I did
only one oscillation,
741
00:43:08 --> 00:43:14
that would be enough to see
742
00:43:10 --> 00:43:16
whether it is coincident
with that...
743
00:43:12 --> 00:43:18
consistent with that number.
744
00:43:13 --> 00:43:19
However, it is such
a beautiful experiment.
745
00:43:16 --> 00:43:22
It's so much fun to see
that object go back and forth
746
00:43:19 --> 00:43:25
in 21 seconds, that I will go...
747
00:43:22 --> 00:43:28
For your pleasure
and for my own pleasure,
748
00:43:24 --> 00:43:30
I will go three oscillations.
749
00:43:25 --> 00:43:31
Not that it is necessary,
but I will do it.
750
00:43:28 --> 00:43:34
3T is going to be
something plus or minus...
751
00:43:31 --> 00:43:37
and this is my reaction time,
which is 0.1 second,
752
00:43:35 --> 00:43:41
and then we can all
divide that by three
753
00:43:38 --> 00:43:44
and then, of course,
754
00:43:39 --> 00:43:45
the error will go down
by a factor of three,
755
00:43:41 --> 00:43:47
and we will see whether this
number agrees with this one.
756
00:43:47 --> 00:43:53
All right, can you imagine
757
00:43:48 --> 00:43:54
someone making a track
like this...
758
00:43:50 --> 00:43:56
air track with a radius
of 115 meters?
759
00:43:53 --> 00:43:59
I mean, what is this?
760
00:43:54 --> 00:44:00
This may be eight meters.
761
00:43:55 --> 00:44:01
115 meters!
762
00:43:57 --> 00:44:03
That is something
like ten times higher...
763
00:44:00 --> 00:44:06
more-- 15 times higher
than this ceiling.
764
00:44:03 --> 00:44:09
Amazing that people
were able to do that.
765
00:44:05 --> 00:44:11
In fact, nowadays,
you can't even buy this anymore.
766
00:44:09 --> 00:44:15
This is probably some
50 years old, if not older.
767
00:44:14 --> 00:44:20
I have to get the air flowing
out of all these holes.
768
00:44:20 --> 00:44:26
There are many, many small holes
in here that you cannot see.
769
00:44:25 --> 00:44:31
The air is now blowing.
770
00:44:28 --> 00:44:34
And this object
is going to be put on here
771
00:44:33 --> 00:44:39
and just because of gravity,
it will go.
772
00:44:36 --> 00:44:42
That's all it is--
only gravity will do work.
773
00:44:39 --> 00:44:45
Here's the timer
and we're going to time it.
774
00:44:42 --> 00:44:48
I will start it off first
775
00:44:45 --> 00:44:51
and then when it comes back
to a stop, I will start to time
776
00:44:49 --> 00:44:55
because that's, for me,
a very sharp criterion.
777
00:44:51 --> 00:44:57
When the object comes back
and comes to a halt here,
778
00:44:55 --> 00:45:01
it's very easy for me
to start the timing.
779
00:44:59 --> 00:45:05
You may notice, as you watch,
780
00:45:01 --> 00:45:07
that some of the amplitude
will decrease
781
00:45:04 --> 00:45:10
because there is...
hold it, hold it, hold it!
782
00:45:08 --> 00:45:14
Because there is, of course,
a little bit of friction.
783
00:45:11 --> 00:45:17
It's very little,
but it is not zero.
784
00:45:15 --> 00:45:21
Enjoy this, just look at it.
785
00:45:17 --> 00:45:23
Isn't this incredible?
786
00:45:18 --> 00:45:24
It just goes simply by gravity.
787
00:45:21 --> 00:45:27
It's like a pendulum which has
a length of 115 meters.
788
00:45:26 --> 00:45:32
It's about to complete
its first oscillation.
789
00:45:29 --> 00:45:35
790
00:45:36 --> 00:45:42
It goes back...
791
00:45:39 --> 00:45:45
Actually, some of you may be
able to see the curvature.
792
00:45:42 --> 00:45:48
You can really see
that it is not straight.
793
00:45:45 --> 00:45:51
So we're coming up
to the second.
794
00:45:47 --> 00:45:53
795
00:45:53 --> 00:45:59
I better get back in position.
796
00:45:55 --> 00:46:01
797
00:46:05 --> 00:46:11
So when it stops here,
798
00:46:07 --> 00:46:13
it has made
three complete oscillations.
799
00:46:14 --> 00:46:20
Sixty-four point zero five.
800
00:46:21 --> 00:46:27
Let me turn this off.
801
00:46:27 --> 00:46:33
So 3T equals 64.05.
802
00:46:35 --> 00:46:41
I'm lazy-- 64.05,
I divide that by three.
803
00:46:40 --> 00:46:46
That is 21.35,
plus or minus .03.
804
00:46:47 --> 00:46:53
That's exactly in agreement
with the prediction,
805
00:46:49 --> 00:46:55
with the uncertainty
of the prediction.
806
00:46:55 --> 00:47:01
I have something very similar,
807
00:46:58 --> 00:47:04
and that is, again,
a curved track.
808
00:47:02 --> 00:47:08
It's not...
809
00:47:03 --> 00:47:09
Oop, I hope
I can retrieve that ball.
810
00:47:05 --> 00:47:11
It would be nice.
811
00:47:07 --> 00:47:13
812
00:47:08 --> 00:47:14
Hmm, what happened?
813
00:47:13 --> 00:47:19
Boy! You have to be...
814
00:47:17 --> 00:47:23
Gee, what's happening here?
815
00:47:19 --> 00:47:25
Oh, yeah, I got it,
got it, got it.
816
00:47:21 --> 00:47:27
Phew!
817
00:47:22 --> 00:47:28
Tricky to make a hole in here.
818
00:47:25 --> 00:47:31
This is an arc,
not unlike this one.
819
00:47:29 --> 00:47:35
There's more,
a little bit more friction
820
00:47:31 --> 00:47:37
and, in this case,
the radius is 85 centimeters.
821
00:47:36 --> 00:47:42
So we can calculate
what the maximum angle is.
822
00:47:41 --> 00:47:47
The radius is 85 centimeters
823
00:47:43 --> 00:47:49
and the arc to the edge
is about 20 centimeters.
824
00:47:48 --> 00:47:54
So we have now
a situation like this.
825
00:47:51 --> 00:47:57
R equals 85 centimeters
826
00:47:56 --> 00:48:02
and this here is
approximately 20 centimeters,
827
00:48:03 --> 00:48:09
so theta maximum is
roughly 20 divided by 85
828
00:48:09 --> 00:48:15
and that is something
like 13 degrees.
829
00:48:12 --> 00:48:18
13 degrees is
not a bad situation
830
00:48:15 --> 00:48:21
because the difference
between the cosine theta
831
00:48:18 --> 00:48:24
and one minus theta squared
over two
832
00:48:21 --> 00:48:27
is less than 1/100 of a percent,
it is that small.
833
00:48:25 --> 00:48:31
So I can make a prediction
834
00:48:27 --> 00:48:33
of the period
of this oscillation,
835
00:48:32 --> 00:48:38
predict and you can go through
exactly the same exercise.
836
00:48:36 --> 00:48:42
You take two pi times
the square root of R over d
837
00:48:42 --> 00:48:48
and you find 1.85.
838
00:48:47 --> 00:48:53
The uncertainty of this radius
is, of course, not very large
839
00:48:50 --> 00:48:56
but we are not certain
about the radius
840
00:48:52 --> 00:48:58
to about one centimeter,
841
00:48:54 --> 00:49:00
so it's 85 plus
or minus one centimeters.
842
00:48:56 --> 00:49:02
So that's about
a 1.2 percent error
843
00:49:00 --> 00:49:06
and so the error, then, in the
prediction will be 0.6 percent;
844
00:49:04 --> 00:49:10
it's about .01 seconds.
845
00:49:07 --> 00:49:13
So I expect...
this is my prediction.
846
00:49:12 --> 00:49:18
Now, Ireally want
to challenge this .01
847
00:49:16 --> 00:49:22
and so now I'm going
to make the observations
848
00:49:19 --> 00:49:25
and surely I'm going
to do it now 10 times,
849
00:49:22 --> 00:49:28
because then the uncertainty
will be 0.1 seconds--
850
00:49:25 --> 00:49:31
that's my reaction time--
851
00:49:27 --> 00:49:33
and so I have the final period
to an accuracy of .01 seconds
852
00:49:32 --> 00:49:38
and so we can compare
these numbers directly
853
00:49:35 --> 00:49:41
and that is what I will do now.
854
00:49:37 --> 00:49:43
I have here the timer
855
00:49:39 --> 00:49:45
and I'm going to oscillate
that back and forth--
856
00:49:42 --> 00:49:48
and that would only take
20 seconds--
857
00:49:45 --> 00:49:51
zero it, we started here.
858
00:49:48 --> 00:49:54
We have great confidence
in physics, right?
859
00:49:50 --> 00:49:56
We believe in physics.
860
00:49:51 --> 00:49:57
We believe in the conservation
of mechanical energy.
861
00:49:54 --> 00:50:00
Starts... are you counting?
862
00:49:58 --> 00:50:04
Is this two? Yeah?
863
00:50:00 --> 00:50:06
Is this three? Four?
864
00:50:03 --> 00:50:09
CLASS:
Four.
865
00:50:04 --> 00:50:10
LEWIN:
I don't believe you.
866
00:50:06 --> 00:50:12
Okay, we start again.
867
00:50:09 --> 00:50:15
Now!
868
00:50:11 --> 00:50:17
One, two, three,
869
00:50:17 --> 00:50:23
four, five,
870
00:50:22 --> 00:50:28
six, seven...
871
00:50:25 --> 00:50:31
I'm getting nervous.
872
00:50:27 --> 00:50:33
Eight, nine, ten.
873
00:50:33 --> 00:50:39
Holy smoke!
874
00:50:35 --> 00:50:41
22.7 seconds!
875
00:50:39 --> 00:50:45
It should have been 18!
876
00:50:41 --> 00:50:47
22.7 seconds.
877
00:50:44 --> 00:50:50
There must be something
fundamentally wrong
878
00:50:46 --> 00:50:52
with the conservation
of mechanical energy.
879
00:50:48 --> 00:50:54
Or is there something else?
880
00:50:51 --> 00:50:57
And what is the difference
between the two experiments?
881
00:50:54 --> 00:51:00
STUDENT:
Friction.
882
00:50:55 --> 00:51:01
LEWIN:
Excuse me?
883
00:50:56 --> 00:51:02
STUDENT:
Friction.
884
00:50:58 --> 00:51:04
LEWIN:
Oh, no, the friction is so low,
that is not the reason.
885
00:51:02 --> 00:51:08
There's a huge difference.
886
00:51:05 --> 00:51:11
Think about it when you take
your shower this weekend.
887
00:51:09 --> 00:51:15
There is a huge difference
888
00:51:10 --> 00:51:16
between this object moving
and that object moving
889
00:51:14 --> 00:51:20
and when you find out,
890
00:51:16 --> 00:51:22
that is the reason why that
is way slower, not friction.
891
00:51:20 --> 00:51:26
See you next Wednesday.
892
00:51:24 --> 00:51:30
893
00:51:29 --> 00:51:35.000