I made two bizzare objects, they were both made from two cups of, uh, esspresso
one of them glued like this and the other one glued like this
so this one I can balance very carefully like this
on a support
whereas the other one I have really no chance of balancing
it's very unstable
so you might have the impression that this one
is the stable one and this one is the unstable one
because this bulges out and
and hugs whatever supported standing on
with this is well, going to tip.
Let's do the dynamic
version of the experiment so this is an
inclined plane and we'll try to let this roll down the slope.
here we go. And it rolls down
and rolls all the way down very happily.
next, this one. let's try to make it roll down the slope and
when I do that, oops, it went off the rails
oops, however hard I tried to center it, it really
doesn't want to go all the way down it's
very very difficult
almost made it, but not quite it doesn't
why is this such a good roller all the way down
and why is this such an unstable roller?
the difference is very interesting and
it has to do with one of the most
fundamental if simplest ideas in science
and engineering which really is at the
heart of today's technology called
the stability and instability. let's first look at
this one.
well as long as I can center this thing
exactly in the middle
well thats fine goes straight down, rolls straight down, there
is no problem there. But, you know in nature
you always make mistakes so there is
some error initial error. let's see if it has
erred slightly to the left like this so
it's now deviated slightly to the left.
in that case you see
this thing is going to be supported on
the left the rail here and on the right
rail here that is on the left
it's rolling on effectively a wheel that
has a large radius over here it's
effectively rolling on the wheel that has
small radius so you have a large wheel
on the side and small wheel on this side
and you know what they're going to do
going to start deviating rightward in
other words in such a way as to cancel
the initial error to the left so this is
a restoring mechanism or an automatic
correction mechanism whenever there is a
slight deviation to the left, the system
automatically corrects itself by going
right and if it deviates a little bit to the right
into the same thing the other way around
you start automatically going to the
left so there's an automatic correction
mechanism and that's why it's so stable just wrong
(Brady) Why, professor? Why can't the big one roll along slowly
and what's the physical thing that happens?
Aha, because if you have a
small wheel on the side a large wheel on
the side and if they roll both at the
same time you see the per same amount of
angular roll the smaller one will advance a small
distance and large one advance a large
distance so you see this one goes a
little but it this one goes large so it starts curving in this
direction and if you have a small one on
this side and large one on this side, it's the other way and starts curving
on this side, so whichever side is smaller that side is
the side toward which the whole thing
starts reeling
(Brady) it's because they're connected, isn't it
Yeah they are connected. That's right.
ok so that's why this thing is such a
stable roller because whenever there is
an initial error which is inevitable
automatically is correcting its course
so why it's wobbling around the center
it can still go down in contrast this
is a disaster because again you know if
you can if you can at all center it in the
middle there is no problem to roll straight down but
that's only the ideal scenario in
practice there's always initial error let's
as before assume that there is a slight error to
the left but then you see the left part
of this object is supported here on the
rail so it's rolling on a small wheel
whereas the right part is supported here
which is rolling on a large wheel so
whenever it's deviating slightly to the
left you have a small wheel on the left
and a large wheel on the right and you know what those
wheels will do they'll start moving
towards left so if there is an initial left error
there will be a tendency to start going even
further left so the error grows and grows
and goes out of control
and eventually of course the whole thing
moves off the rails so there is a mechanism
which is opposite to what we had before
that is, whenever there is a small initial error which is
inevitable there is some mechanism
a devilish mechanism that makes the error grow and
grow and that is called an instability
and in general in abstract terms
whenever there is a system that we want
to control or something that nature
gives and you know there are some
initial errors in the positioning well if
the error has a tendency to diminish
there's an automatic correction mechanism
that system is stable and if the error
tends to grow and goes out of control
that's the unstable mechanism and that is why this can
hardly go down the slope whereas this is very very
stable thing that keeps going down
(Brady) Professor, anyone watching is immediately right now
is going to start thinking about trains maybe. How does a train wheel work?
So, if a train is going straight there is no problem
but suppose it wants to round the corner
and it has the wheel that say on this
side and the outside wheel on this side
and they are connected. As it goes around the corner
the distance that this outer wheel has to travel is
actually longer than the distance this inner wheel
has to travel so there's a
problem because they are rotating at the
same rate
one of them must skid and that's a very
bad the situation so they cannot both roll
because the distances are different
so how do you correct for this effect as
long as the wheels are connected it
seems to be completely insurmountable as a
difficulty but today is a really nice
mechanism here is what you should do: effectively
you should put this kind of design on
the pair of rails you see so when it wants
to go around the corner the centrifugal
force make this go slightly outside but
that's very good news because as long as
it's on the outside that part of the wheel
becomes effectively larger and this part effectively smaller so the
distances that they travel are ultimately
adjusted and indeed the outer wheel does travel
the longer distance and the inner wheel a short distance and
because the wheels radii are different
and of course in real train
wheels it doesn't look
like that but actually the designs kind
of look like this cross-section and this
is sitting on the rail like this and
this is sitting on the rail like this
so as the wobbles left and right the
effective wheel size varies and that
automatically adjust for the unstable
rounding of corners its radial on the
full thing. And by the way if you designed
the wheels in this fashion that would be
really a disaster train that
immediately come off the rails just as this
couldn't go down the slope.
I wonder if I can actually make it
go down all the way. AHAHA -- the first time.
[laughing]
You've just undone
all that math there
one way to do it, is from my point of view, the right hand and pull it
over and then do this which produces
a perpendicular knot.