- When I tug the end of this thread, a short distance,
the spool zips towards me,
and because it's on a slight incline.
When I let go, it rolls away again.
So just with the slightest movement,
I can cause the spool to zip back and forth.
How weird is that?
I stumbled upon this after watching a six year old
Physics Girl video, link in the description for that.
In the video, Diana poses the question,
what will happen when I pull on this cord?
Will the spool roll away from me or towards me?
Actually, I think the question
with the more counterintuitive answer is
will the cord get longer or shorter?
Because you know, if you tog on a cord that's wrapped
around a spool, you expect the cord to unspool a bit
so that you have more cord.
But based on the shot that I showed you
at the very beginning, you may already have figured
out that the spool will roll
towards you and the cord will get shorter.
The reason I started playing around with it for myself is
because I wanted to answer two follow on questions.
The first one, what happens in the limit
like you've got this distance here, which is a measure
of how much room there is left in the spool.
What happens as that distance approaches zero?
As the spool gets full?
As a teaser, you end up with infinite mechanical advantage.
And secondly, I wanted to know what happens if you go
beyond that point, if you go past the spool being full,
if the loose end of the cord is below the ground.
But before we get into that,
let's see if we can explain the counterintuitive answer
to the original question.
So when you pull on the rope
it gets shorter as the spool rolls towards you
and there are actually two ways
of looking at why that happens.
The first way which is less intuitive I would argue
is just to think about forces.
So forgetting all the detail, you've got some object
it has a pivot point here
and you are pulling it with a force from here.
So of course the object is going to move
around that pivot point in the same direction as the force.
But then of course the pivot point moves
and so you have a rolling motion.
The other way to think about it
which is a bit more intuitive for me
is to think about what happens
to the rope when you roll the spool away from you.
Let's roll it a quarter turn.
So this is the distance the spool has traveled
and it's equal to a quarter of the circumference.
Now, in doing so, we have unspooled some of the rope.
How much rope have we unspooled?
Well, just this amount here
which is less than the distance the spool has traveled.
In other words, we haven't unspooled enough
of the rope to compensate
for the distance the spool has traveled.
The result is that the rope is pulled from our hands.
The rope is pulled away from us.
That means if we do the opposite
and pull the rope towards us,
the spool should do the opposite as well
and roll towards us.
By the way, I'm assuming that there is always
enough friction between the spool and the table
so the spool always rolls instead of slipping.
Interestingly, if I pull the rope upwards
then the opposite happens.
The rope gets longer as the spool rolls away from me.
That means there must be a point somewhere in
between vertical and horizontal where the spool
neither rolls away, nor rolls towards the pulling force.
Interestingly, the angle of the rope doesn't just
affect the angle that the force is applied.
It also affects the point on the object.
The force is applied to
like when I'm pulling vertically upwards
the force is being applied from here.
But when I'm pulling horizontally,
the force is being applied down here.
Look, as I change the angle of the rope,
there comes a point where the direction
of the force passes through the pivot point.
And if the force is passing through the pivot point,
then it can't be applying a turning force
around that pivot point.
So that's the magic angle.
I won't go through the mathematics
but here it is on the screen.
That's the angle at which the spool will
neither roll towards or away
and then it's impossible to have enough friction.
The spool simply slides.
But what happens in the limit as the spool fills
up with rope, when you've got only a tiny gap
between the rope and the edge of the spool,
like the spool is almost full, you really do need to worry
about if there's enough friction to avoid slipping.
So I'm using this high friction mat here
and I've switched to a less stretchy rope.
And just like with the sewing thread for a really short tug
on the rope, the spool travels a really long way.
I have some serious mechanical advantage here, and actually
it's really easy to calculate the mechanical advantage.
It's just the distance traveled
by the force divided by the distance traveled by the load.
In this case, it's this distance divided by this distance
which works out to be this length divided by this length
which works out to be this radius divided
by itself minus this radius.
So if this radius is one
and this radius is very nearly one like 0.9,
then you have a mechanical advantage of 10.
Actually, it's the spool that has the mechanical advantage
over me, right?
Because the spool has a low force over a large distance.
Yeah, that's right.
My hand has a mechanical disadvantage
but it means I can get something to move a long distance
by applying a large force over a short distance.
And brilliantly, when you fill the spool up completely
you get a singularity
in the equation for mechanical advantage.
Not a singularity in that sense
but a singularity in the sense
of like in this simplified mathematical description
we get an infinity.
We have infinite mechanical advantage.
Obviously, you can't get that in the real world.
Something has to fail
and the first thing to fail is friction.
In this case, we just end up dragging the thing along.
But also when the spool is nearly full,
you hardly have to tilt the rope up at all
for the solution to flip and the spool to roll away.
And it explains our experience with toilet roll.
The spool of toilet roll is by definition always full.
It's always being pulled from the edge
of the spool because the loose end
of the toilet roll is the edge of the spool.
So what happens if you carry on past the spool being full?
Well, in reality,
the rope itself would become the outermost part
of the spool and you'd have the toilet roll situation.
But what if you could somehow keep the frame
of the spool on the ground so
that the rope was being pulled from under the ground?
Basically, I just wanna see what happens
to the maths as this radius gets bigger than this radius
without having to worry about real world constraints
like the fact that rope can't pass through a table.
Well, I can achieve that with this setup.
So this is now the radius of the spool.
The radius of the rope is greater
than the radius of the spool, which is what we wanted
and it's able to pass through the table.
This clear acrylic is now just there to hold
the rope in place.
It's not part of the spool geometry.
So will the spool roll towards me
or away from me when I pull on the rope?
Remember the two ways of thinking about it.
You can consider this to be the pivot point
and the force is being applied here.
Or you can think about what will happen
to the rope if I roll the spool one way or the other.
So this time the spool rolls away
from me when I pull on the rope and the rope gets longer.
Again, that makes sense from a pivot point, point of view.
And if you think about what happens when the spool rolls
towards you, well, this is how far the spool travels
and this is how much rope you are adding back to the spool.
In other words, the rope is being taken away
from you more quickly than the spool is coming towards you.
And again, the rope is pulled away from your hands.
So if you do the opposite and pull the rope towards you,
you should expect the spool to do the opposite as well.
It's interesting to note that in this setup,
the spool will always roll to the right regardless
of the angle I choose, whereas with this setup
there's an angle at which it flips.
And that makes sense because in this setup,
the force vector never passes through the pivot point.
There are two other limits to consider.
There's the limit as the spool tends towards being empty.
And if we assume that there is no central hole in the spool
then it's just being pulled from the very center.
And if we assume that the cord has no thickness,
then the infinitely thin central axis
of the spool will gather no cord.
In that scenario, the spool would travel
at exactly the same speed as the hand pulling on the cord.
But what about taking this setup to the extreme?
Just keep adding rope.
What happens in the limit of infinite rope?
Well, that's a slightly tricky thing to build
but what we're really saying is what happens in the limit
as this radius becomes infinitely larger than this radius
and we can achieve that
by seeing what happens as this radius tends towards zero.
I 3D printed this modification to give us a very
small spool radius, and what we notice is
that the spool hardly moves at all, and that makes sense.
Every rotation of the spool gives us loads of rope
but causes hardly any travel of the spool.
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