- 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
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