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Today, we will talk exclusively
about work and energy.
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First, let's do
a one-dimensional case.
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The work that a force
is doing,
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when that force is moving
from point A to point B--
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one-dimensional, here's point A
and here is point B--
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and the force is along
that direction or...
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either in this direction
or in this direction
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but it's completely
one-dimensional,
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that work is the integral
in going from A to B
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of that force dx,
if I call that the x-axis.
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The unit of work,
you can see, is newton-meters.
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So work is newton-meters,
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for which we...
we call that "joule."
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If there's more than one force
in this direction,
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you have to add these forces
in this direction vectorially,
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and then this is the work
that the forces do together.
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Work is a scalar, so this
can be larger than zero,
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it can be zero,
or it can be smaller than zero.
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If the force and the direction
in which it moves
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are in opposite directions,
then it is smaller than zero.
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If they're
in the same direction,
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either this way or that way,
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then the work is larger
than zero.
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F = ma, so therefore, I can
also write with this m dv/dt.
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And I can write down for dx,
I can write down v dt.
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I substitute that in there, so
the work in going from A to B
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is the integral from A to B
times the force,
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which is m dv/dt,
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dx which is v dt.
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And look what I can do.
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I can eliminate time,
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and I can now go to a integral
over velocity--
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velocity A to the velocity B,
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and I get m times v times dv.
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That's a very easy integral.
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That is 1/2 m v squared,
which I have to evaluate
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between vA and vB,
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and that is 1/2 m vB squared,
minus 1/2 m vA squared.
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1/2 m v squared is what we call
in physics "kinetic energy."
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Sometimes we write just a K
for that.
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It's the energy of motion.
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And so the work that is done
when a force moves from A to B
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is the kinetic energy
in point B-- you see that here--
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minus the kinetic energy
in point A,
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and this is called
the work-energy theorem.
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If the work is positive, then
the kinetic energy increases
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when you go from A to B.
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If the work is smaller
than zero,
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then the kinetic energy
decreases.
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If the work is zero, then there
is no change in kinetic energy.
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Let's do a simple example.
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Applying this
work-energy theorem,
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I have an object that I want
to move from A to B.
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I let gravity do that.
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I give it a velocity.
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Here's the velocity v of A,
and let the separation be h,
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and this could be
my increasing y direction.
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The object has a mass m,
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and so there is a force,
gravitational force
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which is mg,
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and if I want to give it
a vector notation,
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it's mg y roof, because this
is my increasing value of Y.
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When it reaches point B,
it comes to a halt,
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and I'm going to ask you now
what is the value of h.
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We've done that in the past
in a different way.
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Now we will do it purely based
on the energy considerations.
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So I can write down that
the work that gravity is doing
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in going from A to B,
that work is clearly negative.
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The force is in this direction
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and the motion is
in this direction,
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so the work that gravity is
doing in going from A to B
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equals minus mgh.
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That must be the kinetic energy
at that point B,
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so that this kinetic energy
at point B
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minus the kinetic energy
at point A, this is zero,
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because it comes to a halt here,
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and so you find that
1/2 m vA squared equals mgh.
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m cancels, and so you'll find
that the height that you reach
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equals vA squared divided by 2g.
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And this is something
we've seen before.
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It was easy for us to derive it
in the past,
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but now we've done it purely
based on energy considerations.
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I'd like to do a second example.
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I lift an object from A to B--
I, Walter Lewin.
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I take it at A.
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It has no speed here;
vA is zero.
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It has no speed there.
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And I bring it
from here to here.
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There's a gravitational force mg
in this direction,
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so the force by Walter Lewin
must be in this direction,
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so the motion and my force
are in the same direction,
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so the work that I'm doing
is clearly plus mgh.
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So the work that Walter Lewin is
doing is plus mgh
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when the object goes
from A to B.
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The work that gravity was doing
was minus mgh--
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we just saw that.
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So the net work that is done
is zero,
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and you see there is indeed
no change in kinetic energy.
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There was no kinetic energy
here to start with,
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and there was
no kinetic energy there.
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If I take my briefcase
and I bring it up here,
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I've done positive work.
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If I bring it down,
I've done negative work.
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If I bring it up,
I do again positive work.
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When I do positive work,
gravity does negative work.
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When I do negative work,
like I do now,
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gravity does positive work.
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And I can do that a whole day,
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and the net amount of work
that I have done is zero--
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positive work, negative work,
positive work, negative work.
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I will get very tired.
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Don't confuse getting tired
with doing work.
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I would have done no work
and I would be very tired.
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I think we would all agree
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that if I stand here
24 hours like this
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that I would get very tired.
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I haven't done any work.
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I might as well put it here
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and let the table just
hold that briefcase for me.
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So it's clear that
you can get very tired
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without having done any work.
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So this is the way
we define work in physics.
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Now let's go from one
dimensions to three dimensions.
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It is not very much different,
as you will see.
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I go in three dimensions
from point A to point B,
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and I now have a force...
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which could be pointing not
just along the x direction,
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but in general,
in all directions.
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Now the work that the force is
doing in going from A to B
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is F dot dr.
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r is the position in
three-dimensional space
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where the force is
at that moment,
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and dr is a small displacement.
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So if this is from A to B,
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then dr here,
if you're going this direction,
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this would be
the little vector dr.
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And here, that would be
a little vector dr.
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And the force itself
could be like this here,
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and the force could be
like this there.
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The force can obviously
change along this path.
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So let the force be...
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F of x, x roof,
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plus F of y, y roof,
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plus F of z, z roof.
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I'll move this A
up a little, put it here.
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And let dr-- the general
notation for vector dr--
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equals dx, x roof,
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plus dy, y roof,
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plus dz, z roof.
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It cannot be any more general.
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So the work that
this force is doing
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when it moves from A to B
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is the integral of this F dr.
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Let's first take a small
displacement over dr,
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then I get dw.
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That is simply Fx times dx--
it's a scalar--
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because this is a dot product...
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plus Fy dy, plus Fz dz.
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That is little bit amount
of work
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if the force is displaced
over a distance dr.
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Now I have to do the integral
over the entire path to get W.
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From A to B, that's an integral
going from A to B,
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integral going from A to B.
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I don't need this anymore.
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Integral in going from A to B,
integral in going from A to B.
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Now we're home free,
because we already did this.
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This is a one-dimensional
problem,
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and a one-dimensional problem,
we already know the outcome.
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The integral F dx,
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we found that is
1/2 m vB squared
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minus m vA squared,
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which in this case
is obviously
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the velocity in the x direction,
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because this is
a one-dimensional problem.
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And the one-dimensional problem
indicates
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that the velocity
that I'm dealing with
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is the component
in this direction.
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So we have that
this is 1/2 m v B squared--
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and this is the x component--
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minus vA squared,
and that is the x component.
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This is also a one-dimensional
problem now,
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except that now I deal
with the component...
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with the y component
of the velocity,
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so I get 1/2 m
times vB y squared
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minus vA y squared,
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plus 1/2 m vB z squared
minus vA z squared.
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And now we're home free,
because what you see here
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is you see v squared
in the x direction,
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v squared y component,
v squared z component.
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And if you add those three up,
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you get exactly
the square of the velocity.
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You get the square of the speed.
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So if you add up
these three terms,
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you get vB squared...
I lost my m.
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Let me put my m in there.
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1/2 m times vB squared,
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and here you see Ax squared,
Ay squared, Az squared
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minus vA squared,
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and you get exactly the same
result that you had before,
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namely that the work done is the
difference in kinetic energy.
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You can always think
of these as speeds.
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Velocity squared is the speed.
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It's the magnitude squared
of the velocity.
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All right, I'd like
to return to gravity
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and work on
a three-dimensional situation.
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We have here, let this be x,
this be y and this be z.
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And there is here, this is
the increasing value of y.
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And there's here point A in
three dimensions like this.
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And there is here point B,
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so you get a rough idea
about the three dimensions.
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And y of B minus y of A
equals h.
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It's a given--
there is a height difference
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between A and between B.
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There is a gravitational force.
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The object moves from A to B.
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Suppose it moves
in some crazy way.
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Of course, gravity alone
could not do that.
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There has to be another force
if it goes in a strange way.
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But I'm only calculating now
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the work that's going
to be done by gravity.
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The other forces
I ignore for now.
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I only want to know the work
that gravity is doing.
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The object has a mass m,
and so there is a force mg,
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and I can write down the force
in vector notation.
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It's in this direction.
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So now I notice that there
is only a value for F of y,
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but there is no value
for F of x,
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and there is no value
for Fx; they are zero.
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And so F of y equals minus mg.
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And so if I calculate now
the work in going from A to B,
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this is the integral in going
from A to B of F dot dr,
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and the only term that I have
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is the one that deals
with the y direction.
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The other terms
have nothing in it,
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so it is the integral in
going from A to B of Fy dy.
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And that equals minus mg,
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because we have the minus mg,
times y of B minus y of h,
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so that is minus mg times h.
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And what you see here, that it
is completely independent
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of the path that I have chosen.
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It doesn't matter how I move.
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The only thing that matters
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is the difference in height
between point A and point B.
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h could be larger than zero,
if B is above A.
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It could be smaller than zero
if B is below A.
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It could be equal to zero
if B has the same height as A.
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Whenever the work
that is done by a force
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is independent of its path--
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it's only determined
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by the starting point
and the end point--
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that force is called
a "conservative force."
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It's a very important concept
in physics.
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I will repeat it.
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Whenever the work
that is done by a force
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in going from one point
to another
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is independent of the path--
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it's only determined
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by the starting point
and the end point--
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we call that
a conservative force.
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Gravity is a conservative force.
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It's very clear.
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Suppose that I do the work--
that I go from A to B
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in some very strange way.
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Then it is very clear that
the work that I would have done
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would be plus mgh,
because my force, of course,
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is exactly in the opposite
direction as gravity.
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So whenever gravity is
doing positive work,
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I would be doing negative work.
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If I hold it in my hand,
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when I'm doing positive work,
gravity is doing negative work.
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Again, I'm going
to concentrate now
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on a case where we deal
with gravity only.
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When there's only gravity,
then we have
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that minus mgh is the work done
in going from A to B
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equals minus mg, times y of B
minus y of A,
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and that now is the kinetic
energy at point B
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minus the kinetic energy
at point A.
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This is the work-energy theorem.
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Look closely here.
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00:17:43 --> 00:17:49
I can rearrange this, and I can
bring the Bs in one side,
282
00:17:45 --> 00:17:51
I can bring the As on one side.
283
00:17:48 --> 00:17:54
I then get mg times y of B plus
the kinetic energy at point B
284
00:17:56 --> 00:18:02
equals mg times y of A
285
00:18:00 --> 00:18:06
plus the kinetic energy
at point A.
286
00:18:04 --> 00:18:10
And this is truly
an amazing result.
287
00:18:09 --> 00:18:15
We call mgy...
we give that a name,
288
00:18:14 --> 00:18:20
and we call that "gravitational
potential energy."
289
00:18:21 --> 00:18:27
Often we write for that PE,
or we write for that a u.
290
00:18:26 --> 00:18:32
And what you're seeing here is
291
00:18:28 --> 00:18:34
that the sum of potential energy
at point B
292
00:18:33 --> 00:18:39
and the kinetic energy
at point B
293
00:18:36 --> 00:18:42
is the same as
the potential energy at A
294
00:18:39 --> 00:18:45
and the kinetic energy
at point A.
295
00:18:43 --> 00:18:49
One can be converted
into the other
296
00:18:44 --> 00:18:50
and it can be converted back.
297
00:18:46 --> 00:18:52
Kinetic energy can be converted
back to potential energy,
298
00:18:49 --> 00:18:55
and potential energy
can be converted back,
299
00:18:51 --> 00:18:57
but the sum of them-- which we
call "mechanical energy"--
300
00:18:55 --> 00:19:01
is conserved.
301
00:18:58 --> 00:19:04
And mechanical energy
is only conserved
302
00:19:01 --> 00:19:07
if the force is
a conservative force.
303
00:19:04 --> 00:19:10
It's extremely useful.
304
00:19:06 --> 00:19:12
We will use it many times, but
you have to be very careful.
305
00:19:09 --> 00:19:15
It's a dangerous tool
because it's only true
306
00:19:11 --> 00:19:17
when the force is conservative.
307
00:19:16 --> 00:19:22
Spring forces are
also conservative,
308
00:19:18 --> 00:19:24
but, for instance, friction
is not a conservative force.
309
00:19:23 --> 00:19:29
If I move an object
from here to here...
310
00:19:31 --> 00:19:37
Let's suppose
I move this object,
311
00:19:34 --> 00:19:40
and I go along a straight line,
312
00:19:36 --> 00:19:42
then the friction is
doing negative work,
313
00:19:38 --> 00:19:44
I am doing positive work.
314
00:19:40 --> 00:19:46
But now suppose I go from here
to here through this routing.
315
00:19:45 --> 00:19:51
You can see that the work
I have to do is much more.
316
00:19:51 --> 00:19:57
Friction is not
a conservative force.
317
00:19:54 --> 00:20:00
The frictional force
remains constant,
318
00:19:57 --> 00:20:03
dependent on the friction,
319
00:19:58 --> 00:20:04
the kinetic
friction coefficient,
320
00:20:00 --> 00:20:06
is always the same...
321
00:20:01 --> 00:20:07
the frictional force, which
I have to overcome as I move,
322
00:20:04 --> 00:20:10
and so if I go all the way here
323
00:20:06 --> 00:20:12
and then all the way back to
this point where I wanted to be,
324
00:20:10 --> 00:20:16
then I have done a lot more work
325
00:20:11 --> 00:20:17
than if I go along
the shortest distance.
326
00:20:13 --> 00:20:19
So friction is a classic example
327
00:20:15 --> 00:20:21
of a force that is
not conservative.
328
00:20:19 --> 00:20:25
If I look at this result--
329
00:20:24 --> 00:20:30
the sum of gravitational
potential energy
330
00:20:26 --> 00:20:32
and kinetic energy is conserved
for gravitational force--
331
00:20:29 --> 00:20:35
then it is immediately obvious
332
00:20:31 --> 00:20:37
where we put the zero
of kinetic energy.
333
00:20:35 --> 00:20:41
The zero of kinetic energy
334
00:20:36 --> 00:20:42
is when the object
has no velocity,
335
00:20:39 --> 00:20:45
because kinetic energy
equals 1/2 m v squared.
336
00:20:44 --> 00:20:50
So if the object has
no velocity,
337
00:20:46 --> 00:20:52
then there is no kinetic energy.
338
00:20:48 --> 00:20:54
How about potential energy?
339
00:20:51 --> 00:20:57
Well, you will say, sure,
340
00:20:53 --> 00:20:59
potential energy must be zero
when y is zero,
341
00:20:56 --> 00:21:02
because that's the way
that we defined it.
342
00:20:58 --> 00:21:04
You see?
343
00:20:59 --> 00:21:05
mgy is gravitational
potential energy.
344
00:21:03 --> 00:21:09
So you would think that
u is zero when y is zero.
345
00:21:07 --> 00:21:13
Not an unreasonable thing
to think.
346
00:21:09 --> 00:21:15
But where is y a zero?
347
00:21:12 --> 00:21:18
Is y zero at the surface
of the Earth?
348
00:21:14 --> 00:21:20
Or is y zero
at the floor of 26.100?
349
00:21:17 --> 00:21:23
Or is y zero here,
or is y zero at the roof?
350
00:21:21 --> 00:21:27
Well, you are completely free
to choose
351
00:21:25 --> 00:21:31
where you put u equals zero.
352
00:21:28 --> 00:21:34
It doesn't matter
353
00:21:29 --> 00:21:35
as long as point A and point B
are close enough together
354
00:21:34 --> 00:21:40
that the gravitational
acceleration, g,
355
00:21:36 --> 00:21:42
is very closely the same
for both points.
356
00:21:40 --> 00:21:46
The only thing that matters then
357
00:21:42 --> 00:21:48
is how far they are
separated vertically.
358
00:21:45 --> 00:21:51
The only thing that matters is
that uB minus uA...
359
00:21:50 --> 00:21:56
uB minus uA would be mgh.
360
00:21:55 --> 00:22:01
It is only the h that matters,
361
00:21:57 --> 00:22:03
and so you can then
simply choose your zero
362
00:22:01 --> 00:22:07
anywhere you want to.
363
00:22:04 --> 00:22:10
It's easy to see.
364
00:22:06 --> 00:22:12
Suppose I have here point A
and I have here point B.
365
00:22:12 --> 00:22:18
And suppose
this separation was h.
366
00:22:17 --> 00:22:23
Well, if you prefer to call
zero potential energy at A,
367
00:22:23 --> 00:22:29
I have no problem with that.
368
00:22:25 --> 00:22:31
So we can call this
u equals zero here.
369
00:22:29 --> 00:22:35
Then you would have
to call this u...
370
00:22:32 --> 00:22:38
you have to call it plus mgh.
371
00:22:35 --> 00:22:41
If you say,
372
00:22:36 --> 00:22:42
"No, I don't want to do that;
I want to call this zero"...
373
00:22:39 --> 00:22:45
that's fine.
374
00:22:40 --> 00:22:46
Then this becomes minus mgh.
375
00:22:43 --> 00:22:49
If you prefer to call
this zero, that's fine, too.
376
00:22:46 --> 00:22:52
Then this will have a positive
gravitational potential energy,
377
00:22:50 --> 00:22:56
and this will have one
that is higher than this one
378
00:22:52 --> 00:22:58
by this amount.
379
00:22:54 --> 00:23:00
If you say, "I'd like
to call this zero,"
380
00:22:57 --> 00:23:03
of course the same holds.
381
00:22:58 --> 00:23:04
What matters is
382
00:22:59 --> 00:23:05
what the difference between
potential energy is.
383
00:23:02 --> 00:23:08
That is what we need
384
00:23:03 --> 00:23:09
when we apply the conservation
of mechanical energy.
385
00:23:07 --> 00:23:13
That is what we need
in order to evaluate
386
00:23:10 --> 00:23:16
how the object changes
its kinetic energy.
387
00:23:12 --> 00:23:18
So where you choose your zero
is completely up to you.
388
00:23:18 --> 00:23:24
As long as A and B
are close enough
389
00:23:21 --> 00:23:27
so that there is
no noticeable difference
390
00:23:23 --> 00:23:29
in the gravitational
acceleration g.
391
00:23:27 --> 00:23:33
Before the end of this hour,
I will also evaluate
392
00:23:30 --> 00:23:36
the situation that g
is changing.
393
00:23:33 --> 00:23:39
When you go far way from
the Earth, g is changing.
394
00:23:37 --> 00:23:43
So let us first do...
look at a consequence
395
00:23:43 --> 00:23:49
of the conservation
of mechanical energy.
396
00:23:48 --> 00:23:54
Very powerful concept, and as
long as we deal with gravity,
397
00:23:52 --> 00:23:58
you can always use it.
398
00:23:55 --> 00:24:01
You see here on the desk
399
00:23:57 --> 00:24:03
something that looks
like a roller coaster,
400
00:24:00 --> 00:24:06
and I'm going to slide
an object from this direction.
401
00:24:09 --> 00:24:15
Let's clean it
a little bit better.
402
00:24:15 --> 00:24:21
Here is that roller coaster.
403
00:24:18 --> 00:24:24
This is a circle,
and then it goes up again.
404
00:24:24 --> 00:24:30
And let the circle have
a radius R.
405
00:24:27 --> 00:24:33
This point will be A.
406
00:24:30 --> 00:24:36
I release it with zero speed.
407
00:24:32 --> 00:24:38
I assume that there is
no friction for now.
408
00:24:36 --> 00:24:42
This point will be B.
409
00:24:40 --> 00:24:46
And I define here y equals zero,
410
00:24:45 --> 00:24:51
or what is even more important,
I define that u equals zero.
411
00:24:49 --> 00:24:55
And this is the direction,
positive direction, of y.
412
00:24:53 --> 00:24:59
At A, the object has
no velocity, no speed.
413
00:24:58 --> 00:25:04
At B, of course, it does.
414
00:25:00 --> 00:25:06
It has converted some potential
energy to kinetic energy.
415
00:25:04 --> 00:25:10
At this point C,
this has reached
416
00:25:07 --> 00:25:13
a maximum velocity
that it can ever have
417
00:25:11 --> 00:25:17
because all the potential energy
has been converted
418
00:25:13 --> 00:25:19
to kinetic energy.
419
00:25:15 --> 00:25:21
And at this point D,
if it ever reaches that point,
420
00:25:20 --> 00:25:26
that will be the velocity, say.
421
00:25:24 --> 00:25:30
Okay, I start off, point A is at
a distance h above this level,
422
00:25:33 --> 00:25:39
and so I apply now
423
00:25:34 --> 00:25:40
the conservation
of mechanical energy.
424
00:25:37 --> 00:25:43
So I know that u at A plus
the kinetic energy at A--
425
00:25:41 --> 00:25:47
which is zero--
426
00:25:44 --> 00:25:50
must be u at B
plus kinetic energy at B,
427
00:25:48 --> 00:25:54
must be u at C
plus kinetic energy at C,
428
00:25:52 --> 00:25:58
must be u at D
plus kinetic energy at D.
429
00:25:56 --> 00:26:02
If there is no friction,
430
00:25:57 --> 00:26:03
if there are no other forces,
only gravity.
431
00:25:59 --> 00:26:05
So we lose no... no energy goes
lost in terms of friction.
432
00:26:04 --> 00:26:10
We know that this
height difference is 2R.
433
00:26:14 --> 00:26:20
And so now I can write this
in general terms of y...
434
00:26:18 --> 00:26:24
Take this point B.
435
00:26:21 --> 00:26:27
Think of that being at a
location y above the zero line.
436
00:26:26 --> 00:26:32
Then I can write down now
that uA, which is mgh...
437
00:26:31 --> 00:26:37
That was a given when I started.
438
00:26:32 --> 00:26:38
That was all the energy I had.
439
00:26:33 --> 00:26:39
That was my total
mechanical energy.
440
00:26:36 --> 00:26:42
If I call this u zero,
which is free choice I have,
441
00:26:39 --> 00:26:45
equals u of B, which is
now mgy, plus 1/2 m v squared
442
00:26:46 --> 00:26:52
at that position y.
443
00:26:49 --> 00:26:55
This should hold...
what you see there should hold
444
00:26:51 --> 00:26:57
for every point
that I have here.
445
00:26:53 --> 00:26:59
It should for A, for B, for C,
for D, for any point.
446
00:26:57 --> 00:27:03
I lose my m,
and so you find here that...
447
00:27:03 --> 00:27:09
We summarize it at v squared
equals 2g, times h minus y.
448
00:27:15 --> 00:27:21
So this should hold
for all these points.
449
00:27:17 --> 00:27:23
Therefore, it should
also hold for point D.
450
00:27:19 --> 00:27:25
However, at point D, there
is something very important.
451
00:27:25 --> 00:27:31
There is a requirement.
452
00:27:27 --> 00:27:33
There is a requirement
453
00:27:28 --> 00:27:34
that there is a centripetal
acceleration,
454
00:27:30 --> 00:27:36
which is in this direction,
a centripetal.
455
00:27:35 --> 00:27:41
And that centripetal
acceleration is a must
456
00:27:39 --> 00:27:45
for this object
to reach that point D.
457
00:27:42 --> 00:27:48
And that centripetal
acceleration, as we remember
458
00:27:44 --> 00:27:50
from when we played
with the bucket of water,
459
00:27:47 --> 00:27:53
that is v squared divided by R.
460
00:27:51 --> 00:27:57
And this must be larger
or equal to
461
00:27:56 --> 00:28:02
the gravitational
acceleration g.
462
00:27:58 --> 00:28:04
If it is not larger, the bucket
of water would not have made it
463
00:28:02 --> 00:28:08
to that point D.
464
00:28:04 --> 00:28:10
So this is my second equation
that I'm going to use,
465
00:28:08 --> 00:28:14
so look very carefully.
466
00:28:09 --> 00:28:15
So v squared must be larger
or equal than gR,
467
00:28:14 --> 00:28:20
so I have here v squared,
which is 2g times h minus y.
468
00:28:20 --> 00:28:26
But y for that point D is 2R,
469
00:28:24 --> 00:28:30
so I put in a 2R,
must be larger or equal to gR.
470
00:28:31 --> 00:28:37
I lose my g, so 2h minus 4R
must be larger or equal to R,
471
00:28:40 --> 00:28:46
so h must be larger
or equal to 2½R.
472
00:28:45 --> 00:28:51
This is a classic result
473
00:28:47 --> 00:28:53
that almost every person who has
taken physics will remember.
474
00:28:51 --> 00:28:57
It is by no means intuitive.
475
00:28:53 --> 00:28:59
It means that if I have
this ball here--
476
00:28:56 --> 00:29:02
and I will show you
that shortly--
477
00:28:58 --> 00:29:04
and I let the ball go
into this roller coaster,
478
00:29:01 --> 00:29:07
that it will not make this point
unless I release it from a point
479
00:29:08 --> 00:29:14
that is at least 2½ times
the radius of this circle
480
00:29:12 --> 00:29:18
above the zero level.
481
00:29:14 --> 00:29:20
If I do it any lower,
it will not make it.
482
00:29:17 --> 00:29:23
So think about this.
483
00:29:19 --> 00:29:25
That is something that you could
not have just easily predicted.
484
00:29:22 --> 00:29:28
It's a very strong result,
but it is not something
485
00:29:25 --> 00:29:31
that you say intuitively,
"Oh, yes, of course."
486
00:29:28 --> 00:29:34
It follows immediately
487
00:29:30 --> 00:29:36
from the conservation
of mechanical energy.
488
00:29:32 --> 00:29:38
So if I release it...
489
00:29:34 --> 00:29:40
That 2½ radius point,
by the way, is somewhere here.
490
00:29:38 --> 00:29:44
So if I release this object
way below that,
491
00:29:42 --> 00:29:48
it will not make this point.
492
00:29:44 --> 00:29:50
Let's do that.
493
00:29:47 --> 00:29:53
You see, it didn't make it.
494
00:29:48 --> 00:29:54
I go a little higher,
didn't make it.
495
00:29:51 --> 00:29:57
Go a little higher,
didn't make it.
496
00:29:54 --> 00:30:00
Go a little higher,
still didn't make it.
497
00:29:57 --> 00:30:03
Now I go to the 2½ mark...
498
00:30:03 --> 00:30:09
and now it makes it.
499
00:30:05 --> 00:30:11
2½ times the radius,
500
00:30:07 --> 00:30:13
conservation of mechanical
energy tells you
501
00:30:09 --> 00:30:15
that that is the minimum
it takes
502
00:30:11 --> 00:30:17
to just go through that point.
503
00:30:13 --> 00:30:19
Of course, if there were
no loss of energy at all,
504
00:30:16 --> 00:30:22
if there were no mechanical
energy lost--
505
00:30:18 --> 00:30:24
that means if there
were no friction--
506
00:30:20 --> 00:30:26
then if I were to release it
at this point,
507
00:30:22 --> 00:30:28
it would have to make it back
to this point again,
508
00:30:25 --> 00:30:31
with zero kinetic energy.
509
00:30:27 --> 00:30:33
But that's not the case.
510
00:30:28 --> 00:30:34
There is always
a little bit of friction
511
00:30:30 --> 00:30:36
with the track, for one thing,
and also, of course, with air.
512
00:30:33 --> 00:30:39
So if I release it
all the way here,
513
00:30:36 --> 00:30:42
you would not expect
514
00:30:38 --> 00:30:44
that it will bounce up
all the way to here.
515
00:30:40 --> 00:30:46
It will probably stop
somewhere there.
516
00:30:42 --> 00:30:48
It may not even
make it to the end.
517
00:30:43 --> 00:30:49
We can try that.
518
00:30:46 --> 00:30:52
Oh, it made it
somewhere to here--
519
00:30:47 --> 00:30:53
a little lower than that level.
520
00:30:49 --> 00:30:55
Of course there is some
friction, that is unavoidable.
521
00:30:55 --> 00:31:01
All right, this is
a classic one.
522
00:30:57 --> 00:31:03
There are many exams where
this problem has been given.
523
00:31:00 --> 00:31:06
I won't give it to you this
time, but it's a classic one.
524
00:31:03 --> 00:31:09
You see it on the general exams
for physics,
525
00:31:05 --> 00:31:11
and it's simply a matter
526
00:31:07 --> 00:31:13
of conservation
of mechanical energy.
527
00:31:11 --> 00:31:17
Let's now look at the situation
whereby A and B are so far apart
528
00:31:15 --> 00:31:21
that the gravitational
acceleration
529
00:31:17 --> 00:31:23
is no longer constant,
530
00:31:19 --> 00:31:25
and so you can
no longer simply say
531
00:31:21 --> 00:31:27
that the difference
in potential energy
532
00:31:23 --> 00:31:29
between point B and point A
is simply mgh.
533
00:31:29 --> 00:31:35
So now we are dealing with
a very important concept,
534
00:31:34 --> 00:31:40
and that is
the gravitational force.
535
00:31:37 --> 00:31:43
You can think of the Earth
acting on a mass
536
00:31:44 --> 00:31:50
or you can think of the sun
acting on a planet,
537
00:31:47 --> 00:31:53
whichever you prefer, but
that's what I want to deal with
538
00:31:52 --> 00:31:58
when the distances
are now very large.
539
00:31:55 --> 00:32:01
Let me first give you
the formal definition
540
00:31:58 --> 00:32:04
of gravitational
potential energy.
541
00:32:01 --> 00:32:07
The formal definition is
542
00:32:04 --> 00:32:10
that the gravitational
potential energy at a point P
543
00:32:08 --> 00:32:14
is the work that I,
Walter Lewin, have to do
544
00:32:11 --> 00:32:17
to bring that mass from
infinity to that point P.
545
00:32:17 --> 00:32:23
Now, you may say
that's very strange
546
00:32:19 --> 00:32:25
that in physics, there are
definitions which...
547
00:32:21 --> 00:32:27
where Walter Lewin comes in.
548
00:32:23 --> 00:32:29
Well, we can change it
to gravity,
549
00:32:24 --> 00:32:30
because my force is always
550
00:32:26 --> 00:32:32
the same force as gravity
with a minus sign,
551
00:32:28 --> 00:32:34
so it's also minus the work
that gravity does
552
00:32:31 --> 00:32:37
when the object moves from
infinity to that point P.
553
00:32:35 --> 00:32:41
I just like to think of it, it's
easier for me to think of it,
554
00:32:38 --> 00:32:44
as the work that I do.
555
00:32:41 --> 00:32:47
So if we apply that concept,
556
00:32:44 --> 00:32:50
then we first have to know what
is the gravitational force.
557
00:32:50 --> 00:32:56
If this is an object,
capital M--
558
00:32:52 --> 00:32:58
and you can think of this
559
00:32:53 --> 00:32:59
as being the Earth,
if you want to--
560
00:32:56 --> 00:33:02
and there is here an object
little m, then I have to know
561
00:33:00 --> 00:33:06
what the forces are
between the two.
562
00:33:02 --> 00:33:08
And this now is Newton's
Universal Law of Gravity,
563
00:33:07 --> 00:33:13
which he postulated...
564
00:33:10 --> 00:33:16
Universal Law of Gravity.
565
00:33:16 --> 00:33:22
566
00:33:19 --> 00:33:25
He says the force
that little m experiences,
567
00:33:25 --> 00:33:31
this force equals--
568
00:33:27 --> 00:33:33
I'll put a little m here
and a capital M here--
569
00:33:31 --> 00:33:37
so it is little m
experiences that force
570
00:33:33 --> 00:33:39
due to the presence
of capital M--
571
00:33:36 --> 00:33:42
equals little m times capital M
times a constant,
572
00:33:41 --> 00:33:47
which Newton, in his days,
didn't know yet
573
00:33:43 --> 00:33:49
what that value was,
574
00:33:44 --> 00:33:50
divided by r squared, if r
is the distance between the two.
575
00:33:50 --> 00:33:56
576
00:33:52 --> 00:33:58
This object, since Newton's
Third Law holds--
577
00:33:56 --> 00:34:02
action equals minus reaction--
578
00:33:58 --> 00:34:04
this force,
which I will indicate it
579
00:34:01 --> 00:34:07
as capital M, little m--
580
00:34:03 --> 00:34:09
it is the force that
this one experiences
581
00:34:06 --> 00:34:12
due to the presence
of this one--
582
00:34:08 --> 00:34:14
is exactly the same
in magnitude
583
00:34:09 --> 00:34:15
but opposite in direction,
584
00:34:12 --> 00:34:18
and that is the Universal Law
of Gravity.
585
00:34:16 --> 00:34:22
Gravity is always attractive.
586
00:34:20 --> 00:34:26
Gravity sucks--
that's the way to think of it.
587
00:34:22 --> 00:34:28
It always attracts.
588
00:34:23 --> 00:34:29
There is no such thing
as repelling forces.
589
00:34:26 --> 00:34:32
The gravitational constant G
is an extremely low number--
590
00:34:31 --> 00:34:37
6.67 times 10 to the minus 11--
591
00:34:35 --> 00:34:41
in our... as our units,
592
00:34:36 --> 00:34:42
which is newtons,
gram-meters per kilogram
593
00:34:39 --> 00:34:45
or something like that.
594
00:34:40 --> 00:34:46
That's an extremely low number.
595
00:34:43 --> 00:34:49
It means that
if I have two objects
596
00:34:46 --> 00:34:52
which are each one kilogram,
which are about one meter apart,
597
00:34:53 --> 00:34:59
which I have now here
about one meter,
598
00:34:55 --> 00:35:01
that the force which
they attract each other
599
00:34:59 --> 00:35:05
is only 6.67 times 10
to the minus 11 newtons.
600
00:35:03 --> 00:35:09
That is an extremely
small force.
601
00:35:09 --> 00:35:15
If this were the Earth, and I am
here and this is my mass,
602
00:35:16 --> 00:35:22
then I experience a force which
is given by this equation.
603
00:35:22 --> 00:35:28
This would be, then,
the mass of the Earth.
604
00:35:26 --> 00:35:32
Now, F equals ma.
605
00:35:30 --> 00:35:36
So if I'm here, I experience
a gravitational acceleration,
606
00:35:34 --> 00:35:40
and the gravitational
acceleration that I experience
607
00:35:37 --> 00:35:43
is therefore given
by MG divided by r squared.
608
00:35:42 --> 00:35:48
And so you see
609
00:35:43 --> 00:35:49
that the gravitational
acceleration that I experience
610
00:35:45 --> 00:35:51
at different distances
from the Earth,
611
00:35:47 --> 00:35:53
or, for that matter,
612
00:35:48 --> 00:35:54
at different distances
from the sun,
613
00:35:50 --> 00:35:56
is inversely proportional
with r squared.
614
00:35:52 --> 00:35:58
We have discussed that earlier
when we dealt with the planets,
615
00:35:56 --> 00:36:02
and we dealt with uniform
circular motions,
616
00:35:59 --> 00:36:05
and we evaluated
the centripetal acceleration.
617
00:36:01 --> 00:36:07
We came exactly
to that conclusion--
618
00:36:04 --> 00:36:10
that the gravitational
acceleration falls off
619
00:36:06 --> 00:36:12
as one over r squared.
620
00:36:09 --> 00:36:15
Ten times further away,
621
00:36:10 --> 00:36:16
the gravitational acceleration
is down by a factor of 100.
622
00:36:16 --> 00:36:22
If you are standing near
the surface of the Earth,
623
00:36:20 --> 00:36:26
then, of course, the force
that I will experience
624
00:36:23 --> 00:36:29
is my mass
times the mass of the Earth
625
00:36:28 --> 00:36:34
times the gravitational constant
626
00:36:30 --> 00:36:36
divided by the radius
of the Earth squared--
627
00:36:34 --> 00:36:40
just like we are
here in 26.100--
628
00:36:36 --> 00:36:42
and so this must be mg.
629
00:36:39 --> 00:36:45
That's the gravitational
acceleration
630
00:36:41 --> 00:36:47
if we drop an object here.
631
00:36:43 --> 00:36:49
And so you see that
this now is our famous g,
632
00:36:47 --> 00:36:53
and that is the famous 9.8.
633
00:36:50 --> 00:36:56
You substitute in there
the mass of the Earth,
634
00:36:53 --> 00:36:59
which is six times
10 to the 24 kilograms.
635
00:36:56 --> 00:37:02
You put in here the
gravitational constant,
636
00:36:59 --> 00:37:05
and you put in the radius
of the Earth,
637
00:37:01 --> 00:37:07
which is 6,400 kilometers,
638
00:37:03 --> 00:37:09
out pops our well-known number
639
00:37:05 --> 00:37:11
of 9.8 meters
per second squared.
640
00:37:09 --> 00:37:15
Okay, my goal was
to evaluate for you
641
00:37:14 --> 00:37:20
the gravitational
potential energy
642
00:37:18 --> 00:37:24
the way that it is defined
in general,
643
00:37:23 --> 00:37:29
not in a special case
when we are near the Earth.
644
00:37:28 --> 00:37:34
So we now have to move an object
from infinity to a point P,
645
00:37:36 --> 00:37:42
and we calculate the work
that I have to do.
646
00:37:40 --> 00:37:46
So here is capital M,
and here is that point P,
647
00:37:47 --> 00:37:53
and infinity is somewhere there.
648
00:37:51 --> 00:37:57
It's very, very far away,
and I come in from infinity
649
00:37:54 --> 00:38:00
with an object with mass m,
and I finally land at point P.
650
00:38:03 --> 00:38:09
Since gravity is
a conservative force,
651
00:38:06 --> 00:38:12
and since my force is always
the same in magnitude
652
00:38:09 --> 00:38:15
except in opposite direction,
653
00:38:11 --> 00:38:17
it doesn't matter how I move in;
654
00:38:13 --> 00:38:19
it will always come up
with the same answer.
655
00:38:16 --> 00:38:22
So we might as well do it
in a civilized way
656
00:38:20 --> 00:38:26
and simply move that object
in from infinity
657
00:38:24 --> 00:38:30
along a straight line.
658
00:38:26 --> 00:38:32
It should make no difference
659
00:38:28 --> 00:38:34
because gravity is
a conservative force.
660
00:38:32 --> 00:38:38
So infinity is somewhere there.
661
00:38:36 --> 00:38:42
The force that
I will experience,
662
00:38:40 --> 00:38:46
that I will have to produce,
is this force.
663
00:38:46 --> 00:38:52
664
00:38:48 --> 00:38:54
The force of gravity
is this one.
665
00:38:51 --> 00:38:57
The two are identical
666
00:38:52 --> 00:38:58
except that mine is
in this direction--
667
00:38:54 --> 00:39:00
this is increasing value of r--
668
00:38:56 --> 00:39:02
so mine would be plus m MG
divided by r squared
669
00:39:03 --> 00:39:09
if I'm here at location r.
670
00:39:06 --> 00:39:12
And let this be at a distance
capital R from this object.
671
00:39:12 --> 00:39:18
You can already see that the
gravitational potential energy,
672
00:39:14 --> 00:39:20
when I come from infinity
with a force in this direction
673
00:39:18 --> 00:39:24
and I move inward,
674
00:39:19 --> 00:39:25
you can already see that
gravitational potential energy
675
00:39:23 --> 00:39:29
will always be negative
for all points anywhere.
676
00:39:27 --> 00:39:33
It doesn't matter where I am,
it will always be negative.
677
00:39:31 --> 00:39:37
You may say, gee, that's
sort of a strange thing--
678
00:39:33 --> 00:39:39
negative potential energy.
679
00:39:36 --> 00:39:42
Well, that is not a problem.
680
00:39:38 --> 00:39:44
Remember that depending upon how
you define your zero level here,
681
00:39:43 --> 00:39:49
you also end up with negative
values for potential energy.
682
00:39:46 --> 00:39:52
So there's nothing sacred
about that.
683
00:39:48 --> 00:39:54
What is important, of course,
if we get the right answer
684
00:39:51 --> 00:39:57
for the gravitational
potential energy,
685
00:39:53 --> 00:39:59
that when we move away
from this object
686
00:39:56 --> 00:40:02
that the gravitational
potential energy increases.
687
00:39:59 --> 00:40:05
That's all that matters.
688
00:40:01 --> 00:40:07
But whether it is negative
or positive is irrelevant.
689
00:40:05 --> 00:40:11
So we already know
it's going to be negative,
690
00:40:08 --> 00:40:14
and so we can now evaluate
the work that I have to do
691
00:40:11 --> 00:40:17
when I go from infinity
to that position, capital R.
692
00:40:16 --> 00:40:22
So here comes the work
that Walter Lewin has to do
693
00:40:20 --> 00:40:26
when we go from infinity
to that point,
694
00:40:23 --> 00:40:29
which is capital R,
radius, from this object.
695
00:40:27 --> 00:40:33
Think of it as the sun or
the Earth; either one is fine.
696
00:40:30 --> 00:40:36
So that is the integral in going
from infinity to R of my force--
697
00:40:37 --> 00:40:43
which is plus, because it's
an increasing value of R--
698
00:40:41 --> 00:40:47
m MG divided by R squared dr.
699
00:40:48 --> 00:40:54
That's a very easy integral.
700
00:40:50 --> 00:40:56
This is minus one over r,
701
00:40:53 --> 00:40:59
so I get m MG
over r with a minus sign,
702
00:40:57 --> 00:41:03
and that has to be evaluated
between infinity and capital R.
703
00:41:03 --> 00:41:09
When I substitute for R,
infinity, I get a zero,
704
00:41:06 --> 00:41:12
and so the answer is
minus m MG over capital R.
705
00:41:14 --> 00:41:20
And this is the potential...
gravitational potential energy
706
00:41:18 --> 00:41:24
at any distance capital R
that you please
707
00:41:22 --> 00:41:28
away from this object.
708
00:41:25 --> 00:41:31
At infinity,
it's now always zero.
709
00:41:32 --> 00:41:38
Earlier, you had a choice
where you chose your zero.
710
00:41:35 --> 00:41:41
When you're near Earth
and when g doesn't change,
711
00:41:37 --> 00:41:43
you have a choice.
712
00:41:38 --> 00:41:44
Now you no longer have a choice.
713
00:41:40 --> 00:41:46
Now the gravitational potential
energy at infinity
714
00:41:44 --> 00:41:50
is fixed at zero.
715
00:41:48 --> 00:41:54
So let's look at this function,
716
00:41:51 --> 00:41:57
and let us make a plot
of this function
717
00:41:55 --> 00:42:01
as a function of distance.
718
00:41:59 --> 00:42:05
The one over r relationship
719
00:42:01 --> 00:42:07
of the gravitational
potential energy...
720
00:42:05 --> 00:42:11
the force, gravitational force,
falls off as one over r squared.
721
00:42:10 --> 00:42:16
722
00:42:15 --> 00:42:21
Here's zero.
723
00:42:17 --> 00:42:23
This is the gravitational
potential energy.
724
00:42:20 --> 00:42:26
All these values
here are negative,
725
00:42:22 --> 00:42:28
and here I plot it
as a function.
726
00:42:24 --> 00:42:30
I use the symbol little r now
instead of capital R.
727
00:42:27 --> 00:42:33
And so the curve would
be something like this.
728
00:42:33 --> 00:42:39
This is proportional
to one over r.
729
00:42:39 --> 00:42:45
If you move an object from A
to B and this separation is h,
730
00:42:51 --> 00:42:57
and if A and B are very apart,
731
00:42:54 --> 00:43:00
the difference in potential
energy is no longer mgh,
732
00:42:57 --> 00:43:03
but the difference
in potential energy
733
00:42:59 --> 00:43:05
is the difference between
this value and this value.
734
00:43:04 --> 00:43:10
And you have to use that
equation to evaluate that.
735
00:43:08 --> 00:43:14
But you can clearly see
that if I go from here to here--
736
00:43:11 --> 00:43:17
if I take an object
and go from here to here--
737
00:43:13 --> 00:43:19
that the potential energy
will increase,
738
00:43:15 --> 00:43:21
and that's all that matters.
739
00:43:17 --> 00:43:23
So it increases when you go
further away from the Earth
740
00:43:21 --> 00:43:27
if you look at the Earth,
741
00:43:23 --> 00:43:29
or from the sun
if you look at the sun.
742
00:43:26 --> 00:43:32
743
00:43:30 --> 00:43:36
Is there any disagreement
744
00:43:32 --> 00:43:38
between this result
that we have here
745
00:43:36 --> 00:43:42
and the result
that we found there?
746
00:43:39 --> 00:43:45
The answer is no.
747
00:43:41 --> 00:43:47
I invite you to go through
the following exercise.
748
00:43:44 --> 00:43:50
Take a point A in space,
749
00:43:47 --> 00:43:53
which is at a distance r of A
750
00:43:49 --> 00:43:55
from the center
of the Earth, say,
751
00:43:52 --> 00:43:58
and I do that... I start at
the surface of the Earth itself,
752
00:43:57 --> 00:44:03
so the radius is
the radius of the Earth.
753
00:44:00 --> 00:44:06
And I go to point B,
754
00:44:03 --> 00:44:09
which is a little bit
further away
755
00:44:05 --> 00:44:11
from the center of the Earth,
only a distance h.
756
00:44:09 --> 00:44:15
And h is way, way, way smaller
than the radius of the Earth.
757
00:44:15 --> 00:44:21
So I can calculate now
758
00:44:16 --> 00:44:22
what the difference
in potential energy is
759
00:44:18 --> 00:44:24
between point B and point A,
760
00:44:23 --> 00:44:29
and I can use, and I should use,
this equation.
761
00:44:26 --> 00:44:32
And when I use that equation and
you use the Taylor's expansion,
762
00:44:30 --> 00:44:36
the first order
of Taylor's expansion,
763
00:44:32 --> 00:44:38
you will immediately see
that the result that you find
764
00:44:35 --> 00:44:41
collapses into this result
765
00:44:39 --> 00:44:45
because the g
at the two points is so close
766
00:44:42 --> 00:44:48
that you will see
that you will find then
767
00:44:45 --> 00:44:51
that it is approximately mgh,
even though it is the difference
768
00:44:49 --> 00:44:55
between these two
rather clumsy terms.
769
00:44:53 --> 00:44:59
We will,
many, many times in the future,
770
00:44:56 --> 00:45:02
use the one over r relationship
771
00:44:58 --> 00:45:04
for gravitational
potential energy.
772
00:45:00 --> 00:45:06
We will get
very used to the idea
773
00:45:03 --> 00:45:09
that gravitational potential
energy is negative everywhere
774
00:45:06 --> 00:45:12
the way it's defined,
775
00:45:07 --> 00:45:13
and we will get used
to the idea that at infinity,
776
00:45:10 --> 00:45:16
the gravitational
potential energy is zero.
777
00:45:13 --> 00:45:19
But whenever we deal with near-
Earth situations like in 26.100,
778
00:45:18 --> 00:45:24
then, of course,
it is way more convenient
779
00:45:21 --> 00:45:27
to deal with the simplification
780
00:45:24 --> 00:45:30
that the difference in
gravitational potential energy
781
00:45:27 --> 00:45:33
is given by mgh.
782
00:45:30 --> 00:45:36
I always remember that-- mgh,
Massachusetts General Hospital.
783
00:45:34 --> 00:45:40
That's the best way that you can
remember these simple things.
784
00:45:37 --> 00:45:43
Now I want to return
785
00:45:40 --> 00:45:46
to the conservation
of mechanical energy.
786
00:45:44 --> 00:45:50
I have here a pendulum.
787
00:45:47 --> 00:45:53
I have an object
that weighs 15 kilograms,
788
00:45:50 --> 00:45:56
and I can lift it up one meter,
which I have done now.
789
00:45:53 --> 00:45:59
That means I've done work--
mgh is the work I have done.
790
00:45:57 --> 00:46:03
Believe me, I've increased the
potential energy of this object
791
00:46:00 --> 00:46:06
15 times 10,
so about 150 joules.
792
00:46:04 --> 00:46:10
If I let it fall, then that will
be converted to kinetic energy.
793
00:46:09 --> 00:46:15
If I would let it swing
from one meter height,
794
00:46:15 --> 00:46:21
and you would be there and it
would hit you, you'd be dead.
795
00:46:18 --> 00:46:24
150 joules is
enough to kill you.
796
00:46:21 --> 00:46:27
They use these devices--
it's called a wrecker ball--
797
00:46:25 --> 00:46:31
they use them
to demolish buildings.
798
00:46:28 --> 00:46:34
You lift up a very heavy object,
even heavier than this,
799
00:46:32 --> 00:46:38
and then you let it go,
you swing it,
800
00:46:35 --> 00:46:41
thereby converting gravitational
potential energy
801
00:46:37 --> 00:46:43
into kinetic energy,
802
00:46:40 --> 00:46:46
and that way,
you can demolish a building.
803
00:46:43 --> 00:46:49
You just let it hit...
804
00:46:46 --> 00:46:52
(glass shattering )
805
00:46:48 --> 00:46:54
and it breaks a building.
806
00:46:49 --> 00:46:55
And that's the whole idea
of wrecking.
807
00:46:52 --> 00:46:58
(laughter )
808
00:46:53 --> 00:46:59
So you're using, then,
809
00:46:55 --> 00:47:01
the conversion of gravitational
potential energy
810
00:46:59 --> 00:47:05
to kinetic energy.
811
00:47:01 --> 00:47:07
Now, I am such a strong believer
812
00:47:05 --> 00:47:11
of the conservation
of mechanical energy
813
00:47:10 --> 00:47:16
that I am willing to put
my life on the line.
814
00:47:16 --> 00:47:22
If I release that bob
from a certain height,
815
00:47:21 --> 00:47:27
then that bob
can never come back
816
00:47:25 --> 00:47:31
to a point where the height
is any larger.
817
00:47:30 --> 00:47:36
If I release it from this height
and it swings,
818
00:47:33 --> 00:47:39
then when it reaches here,
it could not be higher.
819
00:47:37 --> 00:47:43
There is a conversion
820
00:47:38 --> 00:47:44
from gravitational potential
energy to kinetic energy
821
00:47:40 --> 00:47:46
back to gravitational
potential energy,
822
00:47:42 --> 00:47:48
and it will come to a stop here.
823
00:47:44 --> 00:47:50
And when it swings back,
824
00:47:46 --> 00:47:52
it should not be able
to reach any higher,
825
00:47:49 --> 00:47:55
provided that I do not give
this object an initial speed
826
00:47:54 --> 00:48:00
when I stand here.
827
00:47:57 --> 00:48:03
I trust the conservation
of mechanical energy 100%.
828
00:48:03 --> 00:48:09
I may not trust myself.
829
00:48:07 --> 00:48:13
I'm going to release
this object,
830
00:48:10 --> 00:48:16
and I hope I will be able
to do it at zero speed
831
00:48:14 --> 00:48:20
so that when it comes back
it may touch my chin,
832
00:48:18 --> 00:48:24
but it may not crush my chin.
833
00:48:20 --> 00:48:26
I want you to be extremely
quiet, because this is no joke.
834
00:48:24 --> 00:48:30
If I don't succeed
in giving it zero speed,
835
00:48:27 --> 00:48:33
then this will be
my last lecture.
836
00:48:30 --> 00:48:36
(laughter )
837
00:48:32 --> 00:48:38
I will close my eyes.
838
00:48:33 --> 00:48:39
I don't want to see this.
839
00:48:35 --> 00:48:41
So please be very quiet.
840
00:48:40 --> 00:48:46
I almost didn't sleep all night.
841
00:48:42 --> 00:48:48
Three, two, one, zero.
842
00:48:45 --> 00:48:51
843
00:48:50 --> 00:48:56
(class laughs with relief )
844
00:48:51 --> 00:48:57
845
00:48:58 --> 00:49:04
Physics works
and I'm still alive!
846
00:49:01 --> 00:49:07
(applause )
847
00:49:02 --> 00:49:08
See you Wednesday.
848
00:49:04 --> 00:49:10
(applause continues )
849
00:49:06 --> 00:49:12
850
00:49:11 --> 00:49:17.000