- This is a mock-up of a type of belt drive
called a crowned pulley.
And it behaves in a really counter-intuitive way.
It's worth disambiguating the word pulley at this point.
Like when you think of the word pulley,
you think of that device that redirects force
or multiplies a force, for example, a snatch block.
- Snatch block!
- This isn't the snatch block type of pulley.
- Snatch block.
(equipment clangs)
- It just so happens that in the world of belt drives
this part and this part are called pulleys.
The thing that distinguishes a crowned pulley
is this curved profile here.
what's surprising and counterintuitive
about the crowned pulley is that the band doesn't slip off.
Like if I place the band off center like this,
you would expect that when I start to turn it,
it will slip further and further
until it leaves the pulley altogether.
Like if I pull a rubber band against an inclined surface,
it slides down the incline.
But let's see what happens with the crowned pulley.
(drill whirring)
It's quite the opposite.
The band moves back towards the middle of the pulley.
How can that be?
This isn't just a curiosity.
It's actually really useful.
I asked Tom Lipton.
If he had any machines with a crowned pulley.
He very kindly sent me this video of his belt sander.
You might recognize Tom from a collaboration I did with him
on air bearings.
Link to that video and his channel in the description
and in the card.
It's worth just checking our intuition.
Why do we think the rubber band should slip off?
Well, rubber bands are stretchy.
When you pull them, they pull back.
You feel a restoring force.
If you'll permit me to anthropomorphize the rubber band,
when you extend it, it wants to be short again.
If you won't permit me to anthropomorphize the rubber band,
let's say that a shorter rubber band
is energetically favorable.
But either way you can see
that as the rubber band slips down the pulley,
it gets shorter and that's more energetically favorable.
So we should expect that to happen.
It's a bit like a ball on the top of a hill.
It's an unstable equilibrium.
If you push it a little in one direction,
it should roll all the way down the hill.
Except that doesn't happen in this case.
Our intuition is wrong.
I want to say at this point
that when you reach the end of this video
you should go and check out Matthias Wandel's channel.
He made a video about crowned pulleys 11 years ago.
I don't tend to make videos that have already been covered
by other people unless I can add something.
I feel like I can in this case.
But anyway, Matthias makes some really interesting builds
and you should check out his channel.
The link is in the card and the description.
To understand why the crowned pulley
behaves the way it does,
consider what happens to a rubber band when you stretch it
but only holding onto one edge.
You'll notice that the band arches.
This part ends up higher than it was before.
And you have a curve here and here.
You can see on the crowned pulley
that this edge is more stretched than this edge.
So we should expect the rubber band to curve upwards here.
It's very subtle because the difference in stretch
between the two edges is quite small in this case,
but it is there.
Because of that upward curve of the rubber band,
when you turn the crowned pulley,
it comes into contact with a piece of rubber band
that is higher than the rubber band
that's already on the pulley,
and so it slowly shifts upwards.
Just to prove that it is a belt effect,
here's a rubber band that is as wide as it is thick.
And you'll notice it does eventually slip off.
We should also expect the opposite counter-intuitive effect.
If the profile has a concave curve,
the rubber band should ride up one of the sides
when we intuitively expect it to be drawn into the middle.
And look, that's exactly what we find.
I actually think the bowing effect is easier to see
on this one.
And just for completeness, here's a thin elastic band
in the concave pulley not doing a thing.
One question we haven't answered yet is,
why does the rubber band bow
when it's stretched from one edge?
Well, you know me,
I like to build physical models of things,
so I've created this network of springs.
It's not a perfect analogy for a rubber band
because, well, it's a discretization of the rubber band.
Like a rubber band is a continuum of springiness
but we have discreet springs here.
But in any case, when you pull on
the top-most row of springs,
eventually these two springs here
begin to pull on the second row of springs.
And so now these seven springs are being pulled
into a straight line.
And as that line of springs straightens
it pushes up on these four springs,
and those springs push upwards on the top row of springs,
and so they bow upwards.
But actually, why are elastic bands stretchy
in the first place?
Why do they pull back?
Surprisingly, it's because of entropy.
You might know that elastic bands are made of rubber,
which is a polymer.
In other words, long chain molecules
like these long chains of beads here.
So this is a kind of model for the polymer.
You'll notice that all of the chains are jumbled up
just like in an elastic band when it's not stretched.
When you stretch an elastic band,
you line up those long chains like this.
So why isn't our model of an elastic band
contracting when we let go?
Well it's because our model doesn't have any heat yet.
Heat is just molecular jiggle.
So I can add heat to this model
just by jiggling the chains with my hands.
And you'll notice that when I do so,
the ends contract inwards just like a rubber band.
The reason atomic jiggle
causes the chains to become jumbled up
is because the jumbled-up state has higher entropy
and systems tend towards higher entropy.
The reason the jumbled state is higher entropy
and the reason systems tend towards higher entropy
would require a whole video to explain.
Fortunately, I've already made that video.
The link is in the card and the description.
I hope you check it out.
But isn't that cool?
The reason an elastic band stretches back
is because of an entropic force.
- Snatch block!
- It's not a snatch block.
Right?
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Thank you to Mitia Oven for the idea for this video
and thank you to Patrick Heard for the 3D-printed parts.
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(upbeat music)