- This video is sponsored by Wren.
I was working on this video where I needed a weight
to bounce up and down on a spring.
So I set that up, ready to record,
and then it just started misbehaving.
It wasn't bobbing up and down,
or at least it would for a while
and then it would switch to swinging like a pendulum.
And then after a while,
it would switch back to bobbing up and down.
And then it would go from bobbing up and down
to swinging like a pendulum.
Again, it would just go back and forth
between these two things.
It didn't do that if I changed the mass a little bit
or if I changed the length of this string a little bit.
So it must be that just by chance,
I chose a combination of length and mass
that made this thing happen.
And after a bit of research, I figured it out
it's something called autoparametric resonance.
What's autoparametric resonance?
Well, I mean, what's parametric resonance?
Well, I mean, what's resonance?
(gentle pipe organ music)
Resonance. We talked about resonance before on this channel,
and my absolute favorite way to explain resonance
is with a flame tube.
And not just because it's flashy and I'm flashy,
but because I actually think
it has a lot of explanatory power.
I'll leave a link in the card
and the description to that video.
But actually for the purposes of this video,
the more instructive explanation is the traditional one.
The explanation that uses a child being pushed on a swing.
Resonance happens when you're adding energy to a system
and you're doing it at just the right frequency
so that the energy you put into the system
adds to the energy that's already there.
In the case of a child on a swing,
it's because you're periodically pushing the child
and that's adding to the energy of the swing.
In other words, every time you push the child,
the amplitude of the swing goes up,
but that's only because
you're getting the timing just right.
The frequency of your pushes
matches the frequency of the pendulum,
that is the child on the swing.
If the frequency of your pushes didn't match,
then eventually you'd be out of phase with the swing.
Like you'd start off pushing the child
at the height of the swing, but eventually at some point,
you would end up pushing directly against the child
at the low point of the swing,
and at that point you'd be taking energy
away from the child.
So resonance is when you get the timing just right
so that the energy you put into the system
adds to the energy that's already there.
It's like with this vibration generator
that I've attached a little bit of ruler to.
As I slowly increased the frequency,
eventually I reached the natural resonating frequency
of the ruler and the amplitude goes up.
In this case and the case of the child on the swing,
the way you add energy to the system is really obvious.
You're just shoving the thing repeatedly,
but there is another way to do it.
Take this pendulum setup, for example.
Can I without directly shoving the mass
get the amplitude of the swing to increase?
Well, one thing I can do is lengthen
and shorten the string by tugging here.
Watch what happens when I shorten
and lengthen the string at just the right frequency.
The amplitude of the swing starts to go up.
Actually it's quite dramatic.
You'll notice that the lengthening
and shortening of the string
happens at twice the frequency of the pendulum.
So like the pendulum's going one, two, three, four,
but the lengthening and shortening is one, two, three, four,
five, six, seven, eight.
This kind of resonance is called parametric resonance
because I'm periodically altering
one of the parameters of the pendulum;
in this case, the length of the pendulum.
Pendulum only really has one parameter.
I mean, the strength of gravity is another parameter,
but that's really hard to vary any reasonable frequency.
You might think that the mass is a parameter as well,
but actually the period of a pendulum swing
is independent of the mass.
So, how does lengthening
and shortening the string in this way
cause the pendulum to swing higher?
Well, of course, it's all in the timing.
And to see at what point the string needs to be lengthened
and at what point the string needs to be shortened,
have a look at the trajectory of the pendulum.
It forms this figure eight shape.
So that must mean that the spring is at its longest
at this point and at its shortest at these points.
When you begin to shorten the string
at this point in the swing,
well, that's also the point in the swing
when the pendulum is moving at its fastest,
which means it's also the point in the swing
when the tension in the string is at its highest
because of centrifugal force.
So when you shorten the string,
you're doing work against that tension.
You're doing work against centrifugal force.
And so, that work adds energy into the system.
You can see the effect of that work
because the pendulum speeds up.
It's like when you're swinging a ball round on a string
and you shorten the string, the ball speeds up.
The same thing happens here.
And as a result, the pendulum swings higher on this swing.
You then lengthen the string again at this point
which actually removes a bit of energy from the pendulum,
but not as much energy is lost
as the energy that you put in.
That's because when you lengthen the string,
the work done by the tension in the string isn't as high
because the tension isn't as high.
That's because you're only working against the tension
due to gravity not centrifugal force
because at this point the pendulum isn't moving,
so there is no centrifugal force.
So the net effect of a large energy gain here
and a small energy loss here is a net gain in energy.
And so, the amplitude of the pendulum swing goes up.
At this point, I want to clear something up about swings.
Like you might be thinking
that something like this is going on when you pump a swing.
Perhaps you're raising and lowering your center of mass
on a swing when you pump it, which is equivalent
to lengthening and shortening the pendulum.
And in fact, you can pump a swing in that way
and increase your amplitude like you can see me doing here.
That does seem to be
some sort of secondary mode of vibration there.
Ignore that.
The point is the typical way of pumping a swing,
which looks like this doesn't have anything to do
with raising and lowering your center of mass,
though I have seen that explanation online,
instead it's to do with the transfer of angular momentum,
which I won't get into in this video.
But as you rotate your body forwards and backwards,
that imparts angular momentum to the swing.
So that's parametric resonance,
but what about autoparametric resonance?
Well, with this set up,
the pendulum also has variable length because of the spring,
but unlike the pendulum
where I was manually changing the length of the string,
I have no control
over the shortening and lengthening of this pendulum.
In this case, that's controlled by the spring.
It's a combination of the spring constant and the mass
that gives you the frequency
of the lengthening and shortening of the spring.
And remember, if you want parametric resonance,
then the driving frequency needs to be double
the natural resonating frequency of the pendulum.
So, if we want the spring oscillations
to drive the pendulum oscillations,
we need to tune the mass
so that one frequency is double the other.
And that's what I must have hit upon by accident
when I was playing with the springs and the masses
for the other video that I'm working on.
Something really beautiful falls out of the mathematics.
Like the equations aren't super complicated.
They're not super simple either,
but they all cancel out to give this lovely relationship.
If you look at how much
does the length of the pendulum extend
when you release the mass,
if the extension is an extra third again,
then you end up with parametric resonance.
It's interesting to note
that energy is conserved in this system.
So, as the spring oscillations
power the pendulum oscillations,
the spring oscillations lose energy
and eventually you end up with just a pendulum.
And then the same thing happens in reverse
because as the pendulum swings,
it changes the length of the spring,
well, the pendulum starts to power
the spring oscillations,
again, until all the energy leaves the pendulum swing
and you only have spring oscillations.
And so, the oscillations go back and forth
from spring oscillations to pendulum oscillations.
This is actually really reminiscent
of two loosely coupled pendulums
that have the same period of oscillation.
Look, this one starts off oscillating,
but because of the way they're linked at the top,
it slowly drives the other one
until all the energy is gone from one
and is transferred to the other.
And then the same thing happens again.
Energy transfers all the way back
to the original pendulum.
And so, the oscillations go back and forth,
and back and forth, and back and forth.
This isn't an example of parametric resonance by the way.
This is just
your good old-fashioned bulk standard driven resonance.
Just because it's fun, here's another example
of a loosely coupled pair of oscillators
that actually looks like just one oscillator.
Well, it's probably more correct to say
that there's one oscillator with two modes.
Specifically an up and down mode and a twisty mode.
You could call them translational and rotational modes
if you're being fancy.
This is called a Wilberforce pendulum.
And look, you can see that it switches back and forth
between two modes.
Why does it do that?
Well, when you stretch a spring,
it naturally wants to untwist,
which causes the mass to twist.
So, as the spring expands and contracts,
the mass twists and untwists.
And because the frequency of the spring oscillations
matches the natural twisting frequency
of the mass and spring
because I fiddled with these screws until they did,
it eventually transfers all the energy
into the twisting mode.
Then the same thing happens in reverse.
The twisting and untwisting
repeatedly loosens and tightens the spring
causing it to lengthen and shorten.
Eventually all the twisting energy
is transferred back to up and down energy.
This is another example
of autoparametric resonance by the way
because when you twist and untwist a spring,
you change it's spring constant,
and that's one of the parameters of the spring oscillations.
What's interesting about this coupled pendulum setup
is that the transferring back and forth of energy
is just one mode of operating it.
There's actually a stable mode as well.
Actually two stable modes.
There's this one where the pendulums
are swinging exactly opposite each other
and this one where they're swinging
exactly in phase with each other.
Compare that to when the energy's being transferred
back and forth between the two pendulums,
you'll notice they're actually 1/4 out of phase
with each other.
So, if they're exactly in phase or exactly out of phase,
you get a stable mode,
but if they're 1/4 out of phase in either direction,
then it's unstable.
If these pendulums have stable modes,
we should expect the spring pendulum
to have stable modes as well.
And in fact, it does.
Here's one where it goes higher in the middle
and here's the other stable mode
where it goes lower than it normally would in the middle.
And in fact, the Wilberforce pendulum
should have stable modes as well.
Here's one
and here's the other.
I did think about setting up an actual swing,
but where there were springs instead of ropes or chains,
and I looked into it and the size of spring
is starting to get into dangerous territory.
Like a garage door spring kind of territory.
And those things will take your arm off.
So, I decided against that in the end.
And I don't recommend that you do it either
because, yeah, those huge springs, they can hurt you.
So instead, here's a Lego man on a tiny spring swing.
A really cool thing here
is that if you look at the mathematics,
well, eventually you drive an equation
called a Mathieu type equation.
And it's this equation that predicts the instability
of the spring pendulum.
But interestingly, it's the same equation
that predicts the stability of the inverted pendulum
that I showed in a previous video.
It's a really old video.
I also mentioned it in a more up-to-date video
about inverted liquid pendulums.
A link to both of those videos in the card
and in the description.
This video is sponsored by Wren,
a website where you can calculate your carbon footprint
and then offset your carbon footprint.
The process of calculating your carbon emissions
is really straightforward.
And then you can see the different ways
that you can make changes to reduce your carbon footprint,
but no one can reduce their carbon footprint to zero.
So, you can also offset your carbon.
You pay monthly based on your calculated carbon footprint.
You can do more or less, but it gives you the value.
And then that money is invested
in projects like planting trees.
I always think like carbon offsetting
is a difficult thing to get right.
And if you're gonna do it right,
you're gonna need a lot of transparency.
And I feel like Wren does that well.
They're very transparent
about the way the business operates.
Like they have a live feed of transactions,
they publish all their receipts
for all the projects that they fund.
We don't know what the solution will look like ultimately.
And it's my belief that with problems like this,
you have to try lots and lots of different things.
So I really like the fact
that they don't just fund the established projects
like planting trees and stuff like that.
They fund these new projects like Biochar Initiative
or establishing clean fuel supply in developing countries.
The money that I spend to offset my carbon
is also going towards finding new solutions to the problem.
They sent me this footage of one of their project locations.
And that's clearly a bird doing a poo.
I've partnered with Wren to plant an extra 10 trees
for the first 100 people that sign up using my special URL,
which is wren.co/start/stevemould.
The link is also in the description.
So check out Wren today.
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