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0:00:01 --> 00:00:07
Okay, you did have some problems
with physical pendulums,

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and I want to talk a little bit
more about physical pendulums.

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Let's first look at the picture
in very general terms.

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I have here a solid object,
which is rotating about point P

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about an axis vertical
to the blackboard,

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and here at C
is the center of mass.

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The object has a mass M,
and so there is here a force Mg,

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and let the separation
be here b.

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I'm going to offset it
over an angle theta,

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and I'm going to oscillate it.

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Clearly, there has to be
a force at the pin.

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If there were no force
at the pin,

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this object would be accelerated
down with acceleration g,

14
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and that's not
what's going to happen.

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But I don't care
about that force

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because I'm going to take
the torque about point P.

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Remember when we had a spring,
just a one-dimensional case,

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we had F equals ma, and that,
for the spring, became minus kx,

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and the minus sign indicates
that it's a restoring force.

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So we now get
something very similar.

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In rotation,
force becomes torque,

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mass becomes moment of inertia,

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and acceleration becomes
angular acceleration.

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So now we have minus r cross F,

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and the minus sign indicates
that it is restoring.

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So if I take the torque relative
to point P, then I have...

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This is the position vector,
which has magnitude b.

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The force is Mg, and I have to
multiply by the sine of theta.

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So I have b times Mg times
the sine of theta

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and that now equals minus...
I can bring the minus here--

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minus the moment of inertia
about point P times alpha,

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and alpha is
the angular acceleration,

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which is theta double dot.

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I bring them together,

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and I use the small-angle
approximation, small angles.

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Then the sine of theta
is approximately theta

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if theta is in radians.

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And so I bring this all
on one side,

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so I get theta double dot
plus bMg

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divided by the moment of inertia
about that point P

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times theta-- now this is
my small-angle approximation--

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equals zero.

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And this is
a well-known equation.

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It is clearly a simple
harmonic oscillation in theta

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because this is a constant.

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And so we're going to get
as a solution

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that theta equals
theta maximum--

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you can call that
the angular amplitude--

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times cosine omega t plus phi.

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This omega is
the angular frequency

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It is a constant.

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Omega here, theta dot,
is the angular velocity,

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which isnot a constant.

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The two are
completely different.

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That is the angular frequency.

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So we know that the solution
to this differential equation

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gives me omega is the square
root of this constant.

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So it is bMg divided
by the moment of inertia

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about that point P.

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And so the period of oscillation
is two pi divided by omega,

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and so that is two pi times
the square root of I relative
to point P divided by bMg.

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And let's hang on to this
for a large part of this lecture

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because I'm going to apply this
to various geometries.

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Make sure that I have it
correct-- yes, I do.

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This is independent
of the mass of the object.

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Even though you will say
there is an M here,

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you will see that in all cases

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when we calculate the moment
of inertia about point P

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that there is always
a mass up here.

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So the mass will disappear,
as you will see very shortly.

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I have four objects here,

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and they all have different
moments of inertia.

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They're all going to rotate

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about an axis perpendicular
to the blackboard, so to speak,

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and we're going to massage
each one of them

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to predict their periods.

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Let's first go to the rod.

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So we first do the rod.

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We have the rod here.

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This is point P, and
here is the center of the rod.

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The rod has mass M
and it has length L.

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So we have here Mg.

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I don't have to worry
about this anymore.

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I simply go to this equation,
and I want to know

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what the period is of this rod,
of this oscillating rod.

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All I have to know now is

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what is the moment
of inertia about P.

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And I know already that b in
that equation equals one-half L.

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So, what is
the moment of inertia

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of oscillation about point P?

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I have to apply now
the parallel axis theorem--

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which you also had to do
during the exam--

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which says it is
the moment of inertia

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of rotation
about the center of mass,

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which, in our case, is C--
the axes have to be parallel,

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so there is this axis
perpendicular to the blackboard,

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and this axis perpendicular
to the blackboard--

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plus the mass of the rod

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times the distance
between P and c squared--

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plus M times
this distance squared--

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so that is b squared,

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and b squared is
one-quarter L squared.

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What is the moment of inertia

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for rotation of a rod
about this axis?

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I looked it up in a table.

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I happen to remember it now,
because I am lecturing 801.

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Two months from now,
I will have forgotten.

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So I remember now that
it is 1/12 ML squared

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plus one-quarter ML squared.

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That becomes
one-third ML squared.

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And so the period T
becomes two pi

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times the square root
of this moment of inertia,

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which is the one-third ML
squared, divided by bMg,

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and b is one-half L
for this geometry.

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So one-half LMg, and notice,
indeed, as I anticipated,

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you always lose your M,
you also lose one L here,

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and so you get two pi times

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the square root of two-thirds L
divided by g.

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So that is the period
that we predict for the rod.

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So let's write that under here,

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because we are going
to compare them shortly.

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So this is two pi times the
square root of two-thirds L

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divided by g.

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The rod that we have here
is designed in such a way

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that the period is very close
to one second.

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That was our goal.

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So T is as close as we can get
it to 1.00 seconds.

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And so if you substitute
in this equation T equals one,

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you will find
that the length of this rod,

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if it is really pivoting
at the very end,

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should be
about 37.2 centimeters.

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So we did the best we can
when we made this rod.

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There is always
an uncertainty, of course--

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how you drill the holes and
where you drill the holes--

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so I would say the value that
we actually achieved is 37.2,

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probably with an uncertainty
of about three millimeters,

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so 0.3 centimeters.

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That's what we have.

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That is an error
of one part in 370.

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Let's make that...
round that off.

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That's a one-percent error
in the length.

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Since the length is
under the square root,

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the one-percent error becomes
half a percent error,

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so the period
that I then would predict

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is about 1
plus half a percent,

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so that is 0.005 seconds.

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So that, then, has
a one-half-percent error.

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So this is my predicted period.

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And so we're going to make
ten oscillations

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of the observed oscillations.

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We're going to get a number.

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My reaction time is not much
better than a tenth of a second.

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So we're going to get
a number there.

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We divide this number by ten.

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And so we can always calculate
the period, then,

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and then we get
a much improved error

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of a hundredth of a second,

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because we will divide
this number by ten,

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so this also is going to be
divided by ten.

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And let's see how close
we were able to get this

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to the 1.0 seconds.

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So, here is the rod.

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Turn this on, the timer.

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We'll offset the rod,

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and I will start it
when it stops somewhere.

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Now-- one, two, three, four,
five, six, seven, eight, nine,

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ten... not bad.

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9.92-- well within
the prediction.

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9.92, and so this becomes 0.992
plus or minus .01,

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and you see that is well within
the prediction that I made.

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Now, all these four objects
were designed in such a way

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that they have exactly
the same period of one second.

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And now comes the question,

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how do the dimensions relate
to each other

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in order to get them
a period of one second?

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So let's now calculate
the period

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for the other three objects:

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for the ring, for the disk
and for the pendulum.

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Let's start with the pendulum.

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For the pendulum,
center of mass is here.

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Here is point P.

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I give the pendulum
length little l--

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you see that on
the right blackboard there.

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So b equals l-- that's the
separation between P and C.

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The moment of inertia about
point P is very easy now.

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This has no mass-- the mass is
all here, mass capital M,

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so that must be
M times l squared.

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So I go to this equation,

195
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and so I ask, what is
the period of a pendulum?

196
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And you're not surprised
that you find

197
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two pi times the square root
of l over g.

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We've seen that before, but,
of course, it also comes out

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if you do it in this
more complicated way.

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So here we get two pi times
the square root of l over g.

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00:11:55 --> 00:12:01
Let's now do the ring.

202
00:12:00 --> 00:12:06
 

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This is the ring.

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Pivot about point P here.

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Here is the center of mass

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right in the middle,
in the middle of nowhere.

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Right here is
the center of mass.

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And so the distance b is
the radius of the ring,

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so we have to calculate the
moment of inertia about point P.

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00:12:30 --> 00:12:36
Again, we have to use now
the parallel axis theorem.

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It is the moment of inertia
for rotation about this axis

213
00:12:37 --> 00:12:43
through point C,
through a center of mass.

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That is MR squared, if R
is the radius of the ring.

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All the mass is
at the circumference,

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all at a distance R
from the center of mass.

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And then we have to add
to that--

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according to
the parallel axis theorem--

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the mass times this distance
squared, and this distance is R,

220
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so you get MR squared.

221
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So it is 2MR squared.

222
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So what is the period
of an oscillation?

223
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I of P, that is 2MR squared,
divided by bMg-- b is R-- RMg.

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225
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I lose one R, I lose an M,
and we have seen this before.

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I derived this
in lecture number 21.

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I remember everything by lecture
number, believe it or not.

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So the period, now,

229
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is the same as a pendulum
with length two R.

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This is the ring.

231
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And so here we have

232
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two pi times the square root
of 2R divided by g.

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I'll make a comparison
very shortly.

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I just want to finish them all.

235
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And I now would like to do
the disk, last but not least.

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00:14:07 --> 00:14:13
 

237
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For the disk,
all we have to do now

238
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is calculate
the moment of inertia.

239
00:14:13 --> 00:14:19
This was very close to your...
the problem you had on the exam.

240
00:14:18 --> 00:14:24
So, here you have the disk,
a solid disk.

241
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This is point P,
this is the center of mass,

242
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but now it's solid,
so again, b is R,

243
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the separation between c and b.

244
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And the moment of inertia
for rotation about point P

245
00:14:36 --> 00:14:42
is now the moment of inertia

246
00:14:38 --> 00:14:44
for rotation
about the center of mass,

247
00:14:40 --> 00:14:46
which you look up in a table--

248
00:14:41 --> 00:14:47
in the case of your exam, it
was given on the front cover--

249
00:14:44 --> 00:14:50
one-half MR squared.

250
00:14:48 --> 00:14:54
That's the moment of inertia

251
00:14:50 --> 00:14:56
for rotation
about the center of mass.

252
00:14:52 --> 00:14:58
And now we have to add
Mb squared, and b equals R,

253
00:14:58 --> 00:15:04
so we have to add MR squared.

254
00:15:02 --> 00:15:08
So we find
three-halves MR squared.

255
00:15:08 --> 00:15:14
So what is the period
of oscillation?

256
00:15:11 --> 00:15:17
Two pi times
three-halves MR squared

257
00:15:18 --> 00:15:24
divided by bMg-- b is R--

258
00:15:22 --> 00:15:28
RMg, and that equals two pi
times the square root...

259
00:15:32 --> 00:15:38
I lose my M as always,
I lose an R.

260
00:15:36 --> 00:15:42
I get here
three-halves R divided by g.

261
00:15:40 --> 00:15:46
And so let's write that
down here, so we have

262
00:15:45 --> 00:15:51
two pi times the square root
of three-halves R

263
00:15:54 --> 00:16:00
divided by g.

264
00:15:56 --> 00:16:02
So we have them all four there,

265
00:15:58 --> 00:16:04
and so we can now make
a meaningful comparison.

266
00:16:04 --> 00:16:10
We want the periods
to be the same,

267
00:16:07 --> 00:16:13
so we can hang on
to those numbers,

268
00:16:09 --> 00:16:15
so we don't need this anymore.

269
00:16:11 --> 00:16:17
We want the periods
to be the same.

270
00:16:14 --> 00:16:20
We already have established

271
00:16:16 --> 00:16:22
that the period of the rod
is close to the one second,

272
00:16:19 --> 00:16:25
so we're not going
to measure them anymore.

273
00:16:22 --> 00:16:28
We just want to compare them--

274
00:16:24 --> 00:16:30
whether, indeed, they have
the same period of oscillation--

275
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by making them oscillate
in unison.

276
00:16:31 --> 00:16:37
But we want to know

277
00:16:32 --> 00:16:38
what the relative ratios are
of these dimensions

278
00:16:34 --> 00:16:40
the way we designed it.

279
00:16:35 --> 00:16:41
So let's first go to the rod,
then, and make a comparison

280
00:16:39 --> 00:16:45
between the rod
and the pendulum.

281
00:16:41 --> 00:16:47
So if you look at the rod--

282
00:16:43 --> 00:16:49
and we use the pendulum as our
standard, with length little l--

283
00:16:48 --> 00:16:54
then you see the period will be
the same if 1½ R is little l...

284
00:16:53 --> 00:16:59
Oh, sorry, the rod.

285
00:16:55 --> 00:17:01
If two-thirds capital L
is little l.

286
00:16:58 --> 00:17:04
So, for the rod,
two-thirds L equals little l,

287
00:17:06 --> 00:17:12
so capital L is 1½ l.

288
00:17:10 --> 00:17:16
It has to be exactly 1½ times
the length of the pendulum,

289
00:17:15 --> 00:17:21
so this length has to be
exactly 1½ times this length

290
00:17:18 --> 00:17:24
to the center of mass,

291
00:17:20 --> 00:17:26
which is the center
of that billiard ball.

292
00:17:22 --> 00:17:28
And that's what we tried,
to the best that we could.

293
00:17:24 --> 00:17:30
So let's now go to the ring.

294
00:17:27 --> 00:17:33
For the ring to have
the same period, one second,

295
00:17:32 --> 00:17:38
2R has to be the same
as the length of the pendulum.

296
00:17:37 --> 00:17:43
So 2R equals l,
so we can put in here for L,

297
00:17:47 --> 00:17:53
this has to be 1½ times l,

298
00:17:51 --> 00:17:57
and here, this now,
which is 2R, has to be l.

299
00:18:00 --> 00:18:06
Very nonintuitive,
that this length here

300
00:18:03 --> 00:18:09
is the same as
the diameter of the ring.

301
00:18:06 --> 00:18:12
Not at all obvious.

302
00:18:09 --> 00:18:15
And so now we go for the disk.

303
00:18:13 --> 00:18:19
So now we require that
three-halves R, 1½ R...

304
00:18:20 --> 00:18:26
We want that to be l, so we want
R to be 2l divided by three.

305
00:18:26 --> 00:18:32
So we want the diameter 2R
to be 4l divided by three.

306
00:18:31 --> 00:18:37
And when you look at the disk,

307
00:18:33 --> 00:18:39
it's hard to see that
it is exactly four-thirds,

308
00:18:36 --> 00:18:42
but you can see that it is
longer than the pendulum--

309
00:18:39 --> 00:18:45
it should be one-third longer--
but it is not as long as the rod

310
00:18:44 --> 00:18:50
because the rod is 1½ times
the length of the pendulum,

311
00:18:48 --> 00:18:54
and so we can now complete
that picture.

312
00:18:51 --> 00:18:57
And we have now
that the diameter here--

313
00:18:58 --> 00:19:04
2R-- is now
four-thirds times l.

314
00:19:04 --> 00:19:10
And so what we can do now,
we can play with them.

315
00:19:07 --> 00:19:13
We can oscillate them
simultaneously

316
00:19:09 --> 00:19:15
and just see how well
they track each other.

317
00:19:12 --> 00:19:18
There's no sense in giving
you the dimensions.

318
00:19:16 --> 00:19:22
The rod was 37.2 centimeters.

319
00:19:19 --> 00:19:25
Let me write that down,
because we calculated that.

320
00:19:23 --> 00:19:29
So this was 37.2 centimeters,
and I think that translates

321
00:19:30 --> 00:19:36
into-- for the pendulum--
24.8 centimeters.

322
00:19:36 --> 00:19:42
But for the others

323
00:19:38 --> 00:19:44
I leave it up to you
to calculate the dimensions.

324
00:19:41 --> 00:19:47
Yes, 25... 24.8 is correct.

325
00:19:44 --> 00:19:50
So timing is not useful anymore.

326
00:19:47 --> 00:19:53
Let's just see
how these two go together.

327
00:19:51 --> 00:19:57
So we offset them,
and then we let them go.

328
00:19:54 --> 00:20:00
And they go
pretty much in unison.

329
00:19:56 --> 00:20:02
If you wait long enough,
of course,

330
00:19:57 --> 00:20:03
you will see
there is a difference.

331
00:19:59 --> 00:20:05
You can never make them
exactly the same,

332
00:20:02 --> 00:20:08
but they track
each other nicely.

333
00:20:05 --> 00:20:11
We can now also use
the rod and the disk.

334
00:20:07 --> 00:20:13
 

335
00:20:12 --> 00:20:18
They track each other
beautifully.

336
00:20:13 --> 00:20:19
They're both very close
to 1.00 seconds.

337
00:20:20 --> 00:20:26
And we can have
the disk versus the pendulum.

338
00:20:25 --> 00:20:31
 

339
00:20:27 --> 00:20:33
And you see they track
each other very nicely.

340
00:20:31 --> 00:20:37
But wait long enough
and you will see

341
00:20:33 --> 00:20:39
that, of course, the periods
will be different.

342
00:20:36 --> 00:20:42
 

343
00:20:39 --> 00:20:45
So, this is my last word
on physical pendulums,

344
00:20:41 --> 00:20:47
but you may see it
again on the final.

345
00:20:43 --> 00:20:49
Not maybe-- you can almost
count on that, I'm telling you.

346
00:20:48 --> 00:20:54
Okay, I want to discuss now some
other interesting oscillation--

347
00:20:52 --> 00:20:58
again, simple harmonic--

348
00:20:54 --> 00:21:00
and that is liquid in a u-tube,
which you see there.

349
00:21:01 --> 00:21:07
If I have here a tube,
which has everywhere...

350
00:21:07 --> 00:21:13
it's open on both sides

351
00:21:10 --> 00:21:16
and everywhere
the same cross-section,

352
00:21:16 --> 00:21:22
and I put a liquid in here,
in equilibrium, just like that,

353
00:21:23 --> 00:21:29
and the liquid has mass M.

354
00:21:27 --> 00:21:33
It has density rho.

355
00:21:29 --> 00:21:35
The area of the tube is A, and
the length of the liquid is l.

356
00:21:39 --> 00:21:45
So this is l.

357
00:21:44 --> 00:21:50
 

358
00:21:46 --> 00:21:52
I'm going to offset it,
the liquid,

359
00:21:49 --> 00:21:55
and I wanted to see it
oscillate, and I wanted to see

360
00:21:53 --> 00:21:59
whether I could calculate
the period of the oscillation.

361
00:21:58 --> 00:22:04
The total mass of the liquid
that I have...

362
00:22:03 --> 00:22:09
the total mass is the volume,

363
00:22:06 --> 00:22:12
which is the area times
the length times rho.

364
00:22:11 --> 00:22:17
I'm going to offset it

365
00:22:13 --> 00:22:19
so that this is higher
over a distance y.

366
00:22:19 --> 00:22:25
So this, then, is lower
over a distance y.

367
00:22:24 --> 00:22:30
 

368
00:22:27 --> 00:22:33
So this distance is also y,
same as that.

369
00:22:31 --> 00:22:37
So the liquid now is here,
and then I release it,

370
00:22:34 --> 00:22:40
and it will start to oscillate.

371
00:22:36 --> 00:22:42
 

372
00:22:39 --> 00:22:45
Well, when it starts
to oscillate,

373
00:22:45 --> 00:22:51
there comes a time that
the liquid, the whole liquid

374
00:22:48 --> 00:22:54
is going to slosh back and forth
and so everywhere in the tube,

375
00:22:53 --> 00:22:59
the velocity at any moment
in time will be the same

376
00:22:57 --> 00:23:03
because the cross-section
is not changing--

377
00:22:59 --> 00:23:05
it's the same everywhere.

378
00:23:01 --> 00:23:07
See, if there is
a certain velocity here v,

379
00:23:03 --> 00:23:09
then it's the same
as the velocity here,

380
00:23:06 --> 00:23:12
as the velocity there,
as the velocity there.

381
00:23:10 --> 00:23:16
And that, of course, is y dot.

382
00:23:13 --> 00:23:19
That's the first derivative
of that position here.

383
00:23:19 --> 00:23:25
I'm going to write down

384
00:23:21 --> 00:23:27
the conservation of total
energy, mechanical energy.

385
00:23:24 --> 00:23:30
I assume that there is
no energy loss,

386
00:23:27 --> 00:23:33
although there probably is some.

387
00:23:29 --> 00:23:35
Friction inside the liquid
will probably generate some heat

388
00:23:32 --> 00:23:38
and that will cause
some damping.

389
00:23:34 --> 00:23:40
You will see that
when we do the experiment.

390
00:23:37 --> 00:23:43
For now, I will assume
that that's not the case.

391
00:23:40 --> 00:23:46
So what, now, is then the total
energy of the system-- that is,

392
00:23:43 --> 00:23:49
the sum of the kinetic energy
plus the potential energy?

393
00:23:47 --> 00:23:53
And if we assume
that that's constant,

394
00:23:49 --> 00:23:55
we will be able to find
the period of the oscillation

395
00:23:53 --> 00:23:59
very shortly, as you will see.

396
00:23:55 --> 00:24:01
The kinetic energy
of the liquid is easy.

397
00:23:59 --> 00:24:05
That is one-half M times
the velocity squared,

398
00:24:03 --> 00:24:09
and the velocity, we agreed,
is y dot squared.

399
00:24:09 --> 00:24:15
Now the potential energy.

400
00:24:11 --> 00:24:17
I call the potential energy
here, I call that u equals zero.

401
00:24:16 --> 00:24:22
When the liquid is standing here
and the liquid is standing here,

402
00:24:22 --> 00:24:28
I call that potential energy
zero.

403
00:24:23 --> 00:24:29
 

404
00:24:27 --> 00:24:33
The mass that is now above this
level here, I call that delta M,

405
00:24:32 --> 00:24:38
and delta M is the area
times y times rho.

406
00:24:40 --> 00:24:46
This is how much mass
there is here.

407
00:24:42 --> 00:24:48
It was taken away from here
and was put here.

408
00:24:46 --> 00:24:52
How much work do you have to do

409
00:24:48 --> 00:24:54
to take this liquid
and put it there?

410
00:24:50 --> 00:24:56
Well, that's the same
when you take this liquid

411
00:24:54 --> 00:25:00
and put it here.

412
00:24:55 --> 00:25:01
And when you bring this liquid
which was therehere,

413
00:24:59 --> 00:25:05
then you have moved it up
over a distance y,

414
00:25:03 --> 00:25:09
and so the gravitational
potential energy increases

415
00:25:08 --> 00:25:14
by delta M-- this is
the amount of mass here--

416
00:25:12 --> 00:25:18
times g times h, and h is y.

417
00:25:14 --> 00:25:20
Mgh, remember?

418
00:25:16 --> 00:25:22
That's the increase
in potential energy.

419
00:25:18 --> 00:25:24
And so I move an amount
of mass which is delta M...

420
00:25:23 --> 00:25:29
I move it over a distance y.

421
00:25:26 --> 00:25:32
I bring it here, but that makes
no difference, of course.

422
00:25:29 --> 00:25:35
And so this is the total energy,
and this is now a constant.

423
00:25:36 --> 00:25:42
So I'm going to substitute
in there the A, l and rho,

424
00:25:40 --> 00:25:46
so I get one-half Al rho
velocity squared plus A rho g.

425
00:25:52 --> 00:25:58
And I get
a y squared equals a constant,

426
00:25:56 --> 00:26:02
because I have a y here
and I have a y there.

427
00:26:00 --> 00:26:06
We've done this before-- this
is the conservation of energy,

428
00:26:03 --> 00:26:09
and in order to find

429
00:26:05 --> 00:26:11
the period of the simple
harmonic oscillation,

430
00:26:09 --> 00:26:15
we take the time derivative.

431
00:26:11 --> 00:26:17
By the way, before we do that,

432
00:26:14 --> 00:26:20
this is... this was delta M
and this is an A, right? Yeah.

433
00:26:19 --> 00:26:25
A, rho-- yeah, that's it.

434
00:26:22 --> 00:26:28
So we lose A, we lose rho,

435
00:26:25 --> 00:26:31
and we continue
with what we have.

436
00:26:29 --> 00:26:35
And so we're going to take

437
00:26:31 --> 00:26:37
the derivative versus time
of this equation.

438
00:26:34 --> 00:26:40
That gives me one-half l.

439
00:26:36 --> 00:26:42
The two pops out,
and the two becomes a one,

440
00:26:40 --> 00:26:46
so I get 2y dot,
then I apply the chain rule,

441
00:26:45 --> 00:26:51
so I get y double dot.

442
00:26:47 --> 00:26:53
Here I get plus g.

443
00:26:49 --> 00:26:55
The two pops out, becomes 2y,

444
00:26:52 --> 00:26:58
and then I get the chain rule,
y dot, and that equals zero.

445
00:26:57 --> 00:27:03
I lose my y dot, because
I have y dot in both terms.

446
00:27:02 --> 00:27:08
This two eats up this two.

447
00:27:05 --> 00:27:11
And so I find
that y double dot plus 2g

448
00:27:11 --> 00:27:17
divided by l times y equals
zero, and that was my goal,

449
00:27:18 --> 00:27:24
because this is clearly
a simple harmonic oscillation,

450
00:27:22 --> 00:27:28
because this is a constant.

451
00:27:23 --> 00:27:29
And it'll oscillate
in the following way:

452
00:27:26 --> 00:27:32
y equals y max times
the cosine of omega t plus phi,

453
00:27:30 --> 00:27:36
then this is
the angular frequency,

454
00:27:33 --> 00:27:39
which is directly related
to the period.

455
00:27:37 --> 00:27:43
Omega, angular frequency equals
the square root of 2g over l,

456
00:27:43 --> 00:27:49
and so the period will be

457
00:27:45 --> 00:27:51
two pi times the square root
of l over 2g.

458
00:27:49 --> 00:27:55
So this is the period
for an oscillating liquid.

459
00:27:55 --> 00:28:01
Notice that it is the period
that you would have had...

460
00:27:58 --> 00:28:04
would have obtained
from a pendulum

461
00:28:00 --> 00:28:06
if the length of the pendulum
were l... l over two--

462
00:28:03 --> 00:28:09
not at all obvious,
not at all intuitive.

463
00:28:08 --> 00:28:14
You see our setup here.

464
00:28:11 --> 00:28:17
I have to know what l is,
and that is not so easy.

465
00:28:16 --> 00:28:22
Because of this radius here,
if I measure l on the outside,

466
00:28:23 --> 00:28:29
it's substantially larger
than on the inside.

467
00:28:25 --> 00:28:31
You may not think
it's a big difference,

468
00:28:27 --> 00:28:33
but it's huge-- it's
a nine-centimeter difference

469
00:28:30 --> 00:28:36
between the outside
and the inside.

470
00:28:32 --> 00:28:38
If I take the average value
between the two...

471
00:28:36 --> 00:28:42
if I take the average,
I find 72 centimeters,

472
00:28:41 --> 00:28:47
and I could be off by one.

473
00:28:43 --> 00:28:49
 

474
00:28:47 --> 00:28:53
If I use this number for l and I
substitute it in this equation,

475
00:28:53 --> 00:28:59
then I find my predicted period,
which is 1.204,

476
00:29:02 --> 00:29:08
and because of this error that I
have of one, that would give me

477
00:29:05 --> 00:29:11
an uncertainty
of about 0.01 seconds.

478
00:29:11 --> 00:29:17
However, before
we start measuring it--

479
00:29:13 --> 00:29:19
and I will do ten oscillations

480
00:29:15 --> 00:29:21
to get a reasonable,
accurate result--

481
00:29:18 --> 00:29:24
I want to warn you.

482
00:29:21 --> 00:29:27
I make a prediction--

483
00:29:22 --> 00:29:28
that the period
that we will measure

484
00:29:25 --> 00:29:31
will probably be
larger than this,

485
00:29:27 --> 00:29:33
and I can think of two reasons.

486
00:29:29 --> 00:29:35
The first is that the damping
of this liquid will be huge.

487
00:29:33 --> 00:29:39
You will see
how quickly it damps.

488
00:29:35 --> 00:29:41
In the past, we have never
taken damping into account,

489
00:29:38 --> 00:29:44
and we won't do that in 801.

490
00:29:40 --> 00:29:46
But the damping has the effect
on making the period longer.

491
00:29:43 --> 00:29:49
We've always ignored that,

492
00:29:45 --> 00:29:51
and in most of the
demonstrations that we did--

493
00:29:47 --> 00:29:53
like just now--
that was acceptable.

494
00:29:49 --> 00:29:55
It may not be acceptable
for the liquid.

495
00:29:52 --> 00:29:58
But now there is a second point
that I want you to think about.

496
00:29:56 --> 00:30:02
Is it correct that I take
the average length,

497
00:30:00 --> 00:30:06
namely the average value

498
00:30:02 --> 00:30:08
between the outer length
and the inner length?

499
00:30:06 --> 00:30:12
I don't think it is.

500
00:30:08 --> 00:30:14
I want you to think about
why that is not correct.

501
00:30:11 --> 00:30:17
Look carefully

502
00:30:12 --> 00:30:18
where that l comes
into my differential equation

503
00:30:15 --> 00:30:21
and you will probably come up
with the right answer.

504
00:30:18 --> 00:30:24
And I claim that the actual l
that we should have taken

505
00:30:21 --> 00:30:27
is a little bit larger--

506
00:30:22 --> 00:30:28
I don't know how much larger,
but it's a little bit larger--

507
00:30:26 --> 00:30:32
and so that will also make
the observed period

508
00:30:28 --> 00:30:34
become larger
than the predicted one.

509
00:30:30 --> 00:30:36
So I'm not too optimistic

510
00:30:32 --> 00:30:38
that we will go and hit this
the way we want to hit it,

511
00:30:35 --> 00:30:41
but that's good, because
that's where the physics lies--

512
00:30:38 --> 00:30:44
that you see that
there are other factors

513
00:30:40 --> 00:30:46
that have to be taken
into account.

514
00:30:42 --> 00:30:48
I'll turn this one on.

515
00:30:43 --> 00:30:49
Is it on now? Is it zeroed?

516
00:30:44 --> 00:30:50
I'll make it completely dark
in the classroom,

517
00:30:47 --> 00:30:53
because you're going to see...

518
00:30:49 --> 00:30:55
otherwise,
you can't see the liquids.

519
00:30:51 --> 00:30:57
 

520
00:30:52 --> 00:30:58
So you see the liquid now.

521
00:30:56 --> 00:31:02
Oh, you see
these equations, too.

522
00:30:59 --> 00:31:05
Okay, it's zeroed.

523
00:31:03 --> 00:31:09
So, let me try to give this
a swing, a large swing.

524
00:31:07 --> 00:31:13
It damps so enormously

525
00:31:09 --> 00:31:15
that I really want to get
a very large swing.

526
00:31:12 --> 00:31:18
 

527
00:31:14 --> 00:31:20
That's nice.

528
00:31:16 --> 00:31:22
Now-- one... two... three...
four... five... six...

529
00:31:24 --> 00:31:30
seven... eight... nine...

530
00:31:27 --> 00:31:33
Ten!

531
00:31:30 --> 00:31:36
Ah, not bad.

532
00:31:33 --> 00:31:39
12.18-- not bad.

533
00:31:37 --> 00:31:43
We'll get a little bit of light.

534
00:31:41 --> 00:31:47
12.18.

535
00:31:43 --> 00:31:49
So observed...

536
00:31:46 --> 00:31:52
Ten T observed... is it 12.18?

537
00:31:52 --> 00:31:58
Okay, my reaction time is 0.1.

538
00:31:55 --> 00:32:01
So T observed is 1.2...

539
00:32:01 --> 00:32:07
let's make it two
plus or minus 0.01 seconds.

540
00:32:06 --> 00:32:12
Oh, that's not bad.

541
00:32:07 --> 00:32:13
It actually...
there is an overlap.

542
00:32:09 --> 00:32:15
If you add this one here,
you get one-to-one,

543
00:32:11 --> 00:32:17
and if you subtract it here,
you get also one-to-one.

544
00:32:14 --> 00:32:20
So it's not bad.

545
00:32:15 --> 00:32:21
I expected it
to be a little higher,

546
00:32:16 --> 00:32:22
but it's close enough
to be happy.

547
00:32:19 --> 00:32:25
Think about why l should have
been taken a little larger.

548
00:32:24 --> 00:32:30
 

549
00:32:27 --> 00:32:33
Now one more
very interesting oscillation--

550
00:32:31 --> 00:32:37
a torsional pendulum.

551
00:32:33 --> 00:32:39
There's a wire there,
2½-meter steel wire,

552
00:32:35 --> 00:32:41
and there's hanging something
on the bottom,

553
00:32:37 --> 00:32:43
which we're going to offset,

554
00:32:38 --> 00:32:44
and then it's going
to oscillate back and forth.

555
00:32:41 --> 00:32:47
That's called
a torsional pendulum.

556
00:32:45 --> 00:32:51
And we're going to calculate

557
00:32:46 --> 00:32:52
the period of oscillation
of the torsional pendulum,

558
00:32:49 --> 00:32:55
and they have
wonderful properties.

559
00:32:51 --> 00:32:57
They are in a way like a spring,
like a one-dimensional spring.

560
00:32:56 --> 00:33:02
Remember
the one-dimensional spring

561
00:32:58 --> 00:33:04
that we had a period which was
independent of the amplitude?

562
00:33:03 --> 00:33:09
Well, within reason, of course.

563
00:33:05 --> 00:33:11
If you make the amplitude
too large,

564
00:33:07 --> 00:33:13
then you get permanent
deformation of the spring.

565
00:33:10 --> 00:33:16
But you never had to make

566
00:33:11 --> 00:33:17
any small-angle approximation
with the spring

567
00:33:15 --> 00:33:21
as we had to do
with the pendulum.

568
00:33:17 --> 00:33:23
Here is the pendulum,
the torsional pendulum.

569
00:33:23 --> 00:33:29
Here's a bar,
and there is a weight here

570
00:33:26 --> 00:33:32
and there's a weight here--

571
00:33:27 --> 00:33:33
I'll tell you more
about that later.

572
00:33:29 --> 00:33:35
It's hanging here
from the ceiling.

573
00:33:31 --> 00:33:37
It has a certain length l.

574
00:33:33 --> 00:33:39
This is point P.

575
00:33:34 --> 00:33:40
And we're going to twist it

576
00:33:37 --> 00:33:43
and then we are going
to let it oscillate-- this bar--

577
00:33:41 --> 00:33:47
in a horizontal plane.

578
00:33:43 --> 00:33:49
So when you look from above,
you will see the bar here--

579
00:33:49 --> 00:33:55
see point P here--

580
00:33:50 --> 00:33:56
and then we can offset it
over an angle theta,

581
00:33:55 --> 00:34:01
and then it will oscillate
back and forth.

582
00:34:00 --> 00:34:06
The torque relative to point P
is now very similar

583
00:34:10 --> 00:34:16
to what we had with a spring.

584
00:34:15 --> 00:34:21
We have a minus sign;

585
00:34:19 --> 00:34:25
again, that illustrates
that it is restoring.

586
00:34:22 --> 00:34:28
Instead of a k now,
we have kappa,

587
00:34:25 --> 00:34:31
which is what we call
the torsional spring constant,

588
00:34:29 --> 00:34:35
and now we have an angle
which we call theta.

589
00:34:33 --> 00:34:39
So we generate a torque which is
proportional to the angle,

590
00:34:38 --> 00:34:44
very similar
to the linear spring

591
00:34:39 --> 00:34:45
whereby we generate a force

592
00:34:41 --> 00:34:47
which is proportional
to the linear displacement.

593
00:34:44 --> 00:34:50
Now you generate a torque

594
00:34:45 --> 00:34:51
which is linearly proportional
to the angle.

595
00:34:48 --> 00:34:54
And this is

596
00:34:50 --> 00:34:56
the moment of inertia
about point P times alpha,

597
00:34:53 --> 00:34:59
and alpha is theta double dot.

598
00:34:56 --> 00:35:02
So we're going to get

599
00:34:57 --> 00:35:03
that theta double dot
plus kappa times theta

600
00:35:03 --> 00:35:09
divided by I of P equals zero.

601
00:35:06 --> 00:35:12
Kappa, by the way,
is the torsional constant.

602
00:35:14 --> 00:35:20
 

603
00:35:16 --> 00:35:22
So we have
a differential equation.

604
00:35:18 --> 00:35:24
It's clear that
you're going to see

605
00:35:21 --> 00:35:27
a simple harmonic oscillation.

606
00:35:24 --> 00:35:30
This is a constant,

607
00:35:25 --> 00:35:31
and so you're going to get
theta equals theta maximum

608
00:35:30 --> 00:35:36
times the cosine
omega t plus phi.

609
00:35:32 --> 00:35:38
It's getting boring.

610
00:35:34 --> 00:35:40
This is the angular frequency,
angular frequency...

611
00:35:38 --> 00:35:44
And angular frequency is
the square root of kappa

612
00:35:42 --> 00:35:48
divided by the moment
of inertia about point P.

613
00:35:46 --> 00:35:52
And therefore the period-- which
is two pi divided by omega--

614
00:35:51 --> 00:35:57
equals two pi times
the square root

615
00:35:54 --> 00:36:00
moment of inertia about point P
divided by kappa.

616
00:35:58 --> 00:36:04
 

617
00:36:01 --> 00:36:07
Well... how about kappa?

618
00:36:06 --> 00:36:12
Kappa is a function

619
00:36:08 --> 00:36:14
of the cross-sectional area here
A and the length l.

620
00:36:14 --> 00:36:20
And it's also a function of
what kind of material you have.

621
00:36:17 --> 00:36:23
Whether you have steel or nylon
makes a big difference.

622
00:36:19 --> 00:36:25
That's very intuitive,
of course.

623
00:36:22 --> 00:36:28
Remember that
in an earlier lecture

624
00:36:25 --> 00:36:31
when we stressed a wire to
the point that it was breaking.

625
00:36:30 --> 00:36:36
We dealt with Young's modulus.

626
00:36:33 --> 00:36:39
We had a wire

627
00:36:34 --> 00:36:40
and we had a mass hanging
at the end of the wire.

628
00:36:38 --> 00:36:44
And we discussed
the vertical oscillation.

629
00:36:41 --> 00:36:47
We could stress it
and let it go,

630
00:36:43 --> 00:36:49
and then we would get
an oscillation, like a spring,

631
00:36:47 --> 00:36:53
and that spring constant that we
found, then, was Young's modulus

632
00:36:51 --> 00:36:57
times the cross-sectional area
here divided by the length.

633
00:36:55 --> 00:37:01
And that was kind of pleasing.

634
00:36:57 --> 00:37:03
The thicker the bar,
the stiffer it is;

635
00:37:00 --> 00:37:06
the longer the wire,
the less stiff it is.

636
00:37:03 --> 00:37:09
Well, there is something
very similar here,

637
00:37:05 --> 00:37:11
but I don't want to go
into the details

638
00:37:07 --> 00:37:13
of exactly
how you derive here kappa.

639
00:37:10 --> 00:37:16
It's a little bit
more complicated.

640
00:37:12 --> 00:37:18
But indeed, the same is true.

641
00:37:15 --> 00:37:21
If you make the wire thicker,
then kappa will go up,

642
00:37:20 --> 00:37:26
and if you make the wire longer,
kappa will go down.

643
00:37:23 --> 00:37:29
That's immediately obvious.

644
00:37:24 --> 00:37:30
If you have a very short rod and
you try to twist that rod...

645
00:37:28 --> 00:37:34
It's clamped at the top, and you
twist it and it's very short,

646
00:37:32 --> 00:37:38
you would need a tremendous
torque for ten degrees.

647
00:37:35 --> 00:37:41
If you make the wire
a hundred meters long

648
00:37:37 --> 00:37:43
and you want to twist it
ten degrees, it takes nothing.

649
00:37:40 --> 00:37:46
So you can immediately see that,
of course, the value for kappa--

650
00:37:44 --> 00:37:50
the torsional constant--
is a function of the length.

651
00:37:48 --> 00:37:54
It will go down
when the length goes up.

652
00:37:51 --> 00:37:57
We have a wire here
which is 2½ meters long,

653
00:37:54 --> 00:38:00
and the thickness of the wire,
the diameter--

654
00:37:57 --> 00:38:03
according to the manufacturer,
it's a piano string--

655
00:38:01 --> 00:38:07
the thickness is
25/1,000 of an inch.

656
00:38:04 --> 00:38:10
And if I calculate kappa
to the best of my ability,

657
00:38:08 --> 00:38:14
well, I find that kappa
should be very close

658
00:38:12 --> 00:38:18
to four times ten to the minus
four newton-meters per radian.

659
00:38:21 --> 00:38:27
And so all we have to do now

660
00:38:22 --> 00:38:28
is to calculate the moment
of inertia of the system,

661
00:38:25 --> 00:38:31
and then we can predict

662
00:38:27 --> 00:38:33
what the period of
this pendulum is going to be--

663
00:38:31 --> 00:38:37
which is not my goal.

664
00:38:32 --> 00:38:38
You will see, my goal
is going to be a different one.

665
00:38:36 --> 00:38:42
Look at the bar
and look at the wire.

666
00:38:40 --> 00:38:46
The wire is so thin

667
00:38:41 --> 00:38:47
that the moment of inertia
relative to point P of the wire

668
00:38:46 --> 00:38:52
is very close to zero.

669
00:38:48 --> 00:38:54
Remember, it is proportional
to our square.

670
00:38:51 --> 00:38:57
But you can forget about that.

671
00:38:53 --> 00:38:59
Almost all moment of inertia
is in this system.

672
00:38:59 --> 00:39:05
I'll blow up that system
for you-- here it is.

673
00:39:02 --> 00:39:08
You'll see it there,
and it has on both sides...

674
00:39:06 --> 00:39:12
it has 200 grams.

675
00:39:09 --> 00:39:15
It has 0.2 kilograms and
it has here 0.2 kilograms.

676
00:39:16 --> 00:39:22
And this mass is
almost negligible.

677
00:39:19 --> 00:39:25
And this distance here
is 30 centimeters

678
00:39:24 --> 00:39:30
and this distance is
30 centimeters.

679
00:39:29 --> 00:39:35
So to a very good approximation

680
00:39:32 --> 00:39:38
the moment of inertia for
rotation about that point P--

681
00:39:37 --> 00:39:43
this rotation-- will be
this mass times radius squared

682
00:39:41 --> 00:39:47
plus this mass times
radius squared.

683
00:39:45 --> 00:39:51
So that will be twice,
because I have two masses,

684
00:39:48 --> 00:39:54
times the mass times
the radius squared,

685
00:39:53 --> 00:39:59
and that is about 0.036
kilogram meters squared.

686
00:40:02 --> 00:40:08
And so when I use that
into our equation--

687
00:40:05 --> 00:40:11
so I know now what kappa is,

688
00:40:06 --> 00:40:12
at least I have a reasonable
idea what kappa is--

689
00:40:09 --> 00:40:15
and I know what I of P is--

690
00:40:11 --> 00:40:17
that's really almost exclusively
determined by that cross-bar--

691
00:40:14 --> 00:40:20
I will find, then,
using that equation

692
00:40:17 --> 00:40:23
that the period is
very close to 60 seconds.

693
00:40:21 --> 00:40:27
 

694
00:40:23 --> 00:40:29
My goal is not to prove to you
that it is close to 60 seconds.

695
00:40:27 --> 00:40:33
My goal is to show you

696
00:40:29 --> 00:40:35
that for this dimension, which
is very thin and very long,

697
00:40:33 --> 00:40:39
that we can make
that angle theta maximum--

698
00:40:37 --> 00:40:43
this angle-- amazingly large.

699
00:40:40 --> 00:40:46
You're not talking about
ten degrees or 30 degrees.

700
00:40:43 --> 00:40:49
We can go much further.

701
00:40:45 --> 00:40:51
And what I want to test with you
is how far we can go.

702
00:40:49 --> 00:40:55
This is one
of the great things in life

703
00:40:52 --> 00:40:58
for you and for me--
a challenge:

704
00:40:54 --> 00:41:00
How far can you go
and get away with it?

705
00:40:57 --> 00:41:03
There comes a time that
if we make the angle too large

706
00:41:01 --> 00:41:07
that we permanently deform
the wire--

707
00:41:04 --> 00:41:10
it will not come back
to its original position.

708
00:41:07 --> 00:41:13
The same happens with a spring.

709
00:41:10 --> 00:41:16
If you take a spring, it is true

710
00:41:12 --> 00:41:18
that the period of oscillation
is independent of the amplitude

711
00:41:16 --> 00:41:22
but only up to a point.

712
00:41:18 --> 00:41:24
If you go too far, that
Hooke's Law no longer holds--

713
00:41:21 --> 00:41:27
that you deform it permanently--
then, of course,

714
00:41:24 --> 00:41:30
the period will become
a function of the extension,

715
00:41:27 --> 00:41:33
and the same is true here.

716
00:41:29 --> 00:41:35
So if we twist it too much up,

717
00:41:31 --> 00:41:37
then, of course, we will
permanently deform it,

718
00:41:35 --> 00:41:41
and then the period will not
be independent of theta maximum.

719
00:41:40 --> 00:41:46
Having said that, I would like
to start asking you for advice.

720
00:41:46 --> 00:41:52
What kind of angle in this
direction shall we start with?

721
00:41:51 --> 00:41:57
What do you think is reasonable

722
00:41:54 --> 00:42:00
without total torture
for the wire?

723
00:41:57 --> 00:42:03
And then we'll write down
the times.

724
00:42:00 --> 00:42:06
So we're not really interested
in testing the 60 seconds,

725
00:42:04 --> 00:42:10
but we would rather
like to compare

726
00:42:07 --> 00:42:13
the various angles
that we give it.

727
00:42:09 --> 00:42:15
So what is the first one
we will try? Any idea?

728
00:42:12 --> 00:42:18
(echoing class ):
30 degrees? What?

729
00:42:14 --> 00:42:20
STUDENT:
Six pi.

730
00:42:16 --> 00:42:22
LEWIN:
The first try?! You are cruel!

731
00:42:21 --> 00:42:27
Man! The first one,
you want six pi?!

732
00:42:25 --> 00:42:31
You're out of your mind.

733
00:42:27 --> 00:42:33
(class laughs )

734
00:42:29 --> 00:42:35
LEWIN:
I'm willing to go
one rotation, okay?

735
00:42:32 --> 00:42:38
You think that's nothing--
it's peanuts for you, right?

736
00:42:35 --> 00:42:41
Okay, I would like to go
theta maximum of 360 degrees--

737
00:42:42 --> 00:42:48
so two pi--
and measure the period.

738
00:42:47 --> 00:42:53
In fact, to measure the period
takes a minute,

739
00:42:52 --> 00:42:58
and it's not necessary.

740
00:42:53 --> 00:42:59
We can measure half a period.

741
00:42:55 --> 00:43:01
Namely, we wait
until the pendulum stops,

742
00:42:57 --> 00:43:03
and we measure the time
until it stops again;

743
00:42:59 --> 00:43:05
that's half a period.

744
00:43:01 --> 00:43:07
Like with the spring--

745
00:43:02 --> 00:43:08
if it stops here and it stops
there, that's half a period.

746
00:43:06 --> 00:43:12
So we'll measure half a period.

747
00:43:09 --> 00:43:15
 

748
00:43:12 --> 00:43:18
Now, I don't know what
my reaction time is going to be,

749
00:43:14 --> 00:43:20
It may be
another tenth of a second,

750
00:43:16 --> 00:43:22
because the moment that it stops

751
00:43:18 --> 00:43:24
is not so well defined,
you will see.

752
00:43:20 --> 00:43:26
I'm just guessing--

753
00:43:22 --> 00:43:28
probably a little larger
than one-tenth of a second.

754
00:43:24 --> 00:43:30
Let's give it a shot.

755
00:43:27 --> 00:43:33
Let's try first 360 degrees.

756
00:43:31 --> 00:43:37
 

757
00:43:35 --> 00:43:41
You see, this is black
and this is a little red,

758
00:43:38 --> 00:43:44
so we will rotate it
one rotation.

759
00:43:43 --> 00:43:49
This is back here where it was.

760
00:43:45 --> 00:43:51
Yeah, have you seen that?
360 degrees.

761
00:43:47 --> 00:43:53
Okay, now, I will first let it
go and wait until it stops.

762
00:43:50 --> 00:43:56
I always do that,
and then I start the time.

763
00:43:53 --> 00:43:59
And then when it stops again,
I'll stop the time,

764
00:43:56 --> 00:44:02
and then we have half a period.

765
00:43:58 --> 00:44:04
So let it first do
its own thing...

766
00:44:01 --> 00:44:07
very slowly, very gently.

767
00:44:05 --> 00:44:11
It should take roughly
30 seconds for half a rotation.

768
00:44:09 --> 00:44:15
so you'll see now that it...
it's now at equilibrium again,

769
00:44:14 --> 00:44:20
because we wound it up
one rotation

770
00:44:17 --> 00:44:23
and so back to equilibrium.

771
00:44:19 --> 00:44:25
And now it's going to stop
very shortly,

772
00:44:22 --> 00:44:28
and when it stops,
I want to start the timer.

773
00:44:25 --> 00:44:31
Now!

774
00:44:27 --> 00:44:33
Okay, so now it goes back, and
we'll wait until it stops again.

775
00:44:33 --> 00:44:39
That gives us half a period.

776
00:44:35 --> 00:44:41
 

777
00:44:48 --> 00:44:54
Okay.

778
00:44:49 --> 00:44:55
 

779
00:44:54 --> 00:45:00
Now!

780
00:44:56 --> 00:45:02
28.75.

781
00:44:59 --> 00:45:05
 

782
00:45:05 --> 00:45:11
28.8.

783
00:45:08 --> 00:45:14
What are we going to do now?

784
00:45:10 --> 00:45:16
 

785
00:45:12 --> 00:45:18
Three rotations?
Five rotations? Three.

786
00:45:16 --> 00:45:22
Are we in favor of three?

787
00:45:18 --> 00:45:24
Who is responsible for
permanent damage to the wire?

788
00:45:22 --> 00:45:28
Do you accept
responsibility in life?

789
00:45:24 --> 00:45:30
Three is a lot.

790
00:45:26 --> 00:45:32
Three is six pi, man, six.

791
00:45:30 --> 00:45:36
9I can start it, because it will
take a while before it stops.

792
00:45:34 --> 00:45:40
Three rotations--
first we have to be sure

793
00:45:37 --> 00:45:43
that it is more or less
back at equilibrium,

794
00:45:40 --> 00:45:46
which is always a difficult
thing, because it's so slow.

795
00:45:43 --> 00:45:49
 

796
00:45:49 --> 00:45:55
Yeah, close enough.

797
00:45:51 --> 00:45:57
Okay, three-- shall we go
clockwise or counterclockwise?

798
00:45:54 --> 00:46:00
It should make no difference.

799
00:45:55 --> 00:46:01
I went this way first;
shall we go back?

800
00:45:57 --> 00:46:03
Yeah? You can sleep with that?

801
00:45:59 --> 00:46:05
Okay? Okay.

802
00:46:03 --> 00:46:09
One... two... three.

803
00:46:10 --> 00:46:16
 

804
00:46:12 --> 00:46:18
Six pi.

805
00:46:13 --> 00:46:19
 

806
00:46:16 --> 00:46:22
The piano wire.

807
00:46:18 --> 00:46:24
Okay, let's go.

808
00:46:20 --> 00:46:26
I will start the timer
when it stops.

809
00:46:24 --> 00:46:30
We have some time.

810
00:46:26 --> 00:46:32
Six pi.

811
00:46:27 --> 00:46:33
 

812
00:46:39 --> 00:46:45
If we rotate it six...
three times,

813
00:46:41 --> 00:46:47
then it will rotate six times
back before it stops.

814
00:46:44 --> 00:46:50
I hope you realize that.

815
00:46:45 --> 00:46:51
(student speaks up
in background )

816
00:46:48 --> 00:46:54
 

817
00:46:50 --> 00:46:56
LEWIN:
Was I too late?

818
00:46:52 --> 00:46:58
Thank you for pointing that out.

819
00:46:54 --> 00:47:00
So if you rotate it three times
and then let it go,

820
00:46:57 --> 00:47:03
it goes first three times back,
then it's at equilibrium,

821
00:47:01 --> 00:47:07
and then it winds three times up
again before it stops.

822
00:47:04 --> 00:47:10
Notice it's going
much faster now, but the time--

823
00:47:07 --> 00:47:13
that's the whole thing--

824
00:47:10 --> 00:47:16
should be very close
to that number again.

825
00:47:13 --> 00:47:19
 

826
00:47:18 --> 00:47:24
28.5.

827
00:47:19 --> 00:47:25
Oh! Not bad.

828
00:47:21 --> 00:47:27
 

829
00:47:24 --> 00:47:30
Shall we now go all the way?

830
00:47:27 --> 00:47:33
What do you want to do now?

831
00:47:29 --> 00:47:35
Break the wire or...

832
00:47:32 --> 00:47:38
 

833
00:47:34 --> 00:47:40
Ten rotations?

834
00:47:35 --> 00:47:41
 

835
00:47:37 --> 00:47:43
You'd love to see that, right?

836
00:47:40 --> 00:47:46
It will go like mad,
ten rotations.

837
00:47:44 --> 00:47:50
Isn't it amazing
how much faster it goes?

838
00:47:46 --> 00:47:52
(imitates whirring )

839
00:47:48 --> 00:47:54
It's still 30 seconds.

840
00:47:50 --> 00:47:56
I must make sure
that I have my equilibrium.

841
00:47:53 --> 00:47:59
This was not equilibrium.

842
00:47:55 --> 00:48:01
 

843
00:47:57 --> 00:48:03
I know it's somewhere here.

844
00:48:00 --> 00:48:06
 

845
00:48:02 --> 00:48:08
No, it wasn't equilibrium.

846
00:48:05 --> 00:48:11
 

847
00:48:08 --> 00:48:14
I think this is it.

848
00:48:10 --> 00:48:16
Okay-- ten, right?

849
00:48:13 --> 00:48:19
Ready?

850
00:48:15 --> 00:48:21
One... two... three... four...

851
00:48:23 --> 00:48:29
five... six... seven...
eight... nine...

852
00:48:33 --> 00:48:39
Ten-- poor wire.

853
00:48:35 --> 00:48:41
We'll let it go
and we'll see what happens.

854
00:48:39 --> 00:48:45
When it stops,
I'll start the time.

855
00:48:42 --> 00:48:48
 

856
00:48:44 --> 00:48:50
(student speaks up
in background )

857
00:48:46 --> 00:48:52
LEWIN:
Excuse me?

858
00:48:47 --> 00:48:53
(student repeats )

859
00:48:48 --> 00:48:54
LEWIN:
Thank you.

860
00:48:49 --> 00:48:55
Thank you for pointing that out.

861
00:48:51 --> 00:48:57
Look how fast it's going.

862
00:48:53 --> 00:48:59
I mean, it's really going wacko.

863
00:48:55 --> 00:49:01
It has to do all that
in 30 seconds.

864
00:48:57 --> 00:49:03
 

865
00:49:04 --> 00:49:10
Now!

866
00:49:05 --> 00:49:11
 

867
00:49:07 --> 00:49:13
So now it has to go back
to its stopping.

868
00:49:10 --> 00:49:16
It has to make 20 rotations now,
20 rotations in 30 seconds--

869
00:49:13 --> 00:49:19
ten back to equilibrium
and then ten to come to a halt.

870
00:49:19 --> 00:49:25
 

871
00:49:23 --> 00:49:29
This is going to be

872
00:49:24 --> 00:49:30
your Thanksgiving
farewell demonstration.

873
00:49:27 --> 00:49:33
 

874
00:49:32 --> 00:49:38
Now!

875
00:49:35 --> 00:49:41
29.2-- fantastic.

876
00:49:38 --> 00:49:44
Okay, have a good Thanksgiving.

877
00:49:40 --> 00:49:46