- 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
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What happens when you combine a Wilberforce pendulum with parametric resonance side to side? You should be able to find parameters for which all three energy modes resonate.
I would have explained the parametric resonance part as following: When pulling the rope at the midpoint you increase the potential energy by moving the weight straight up, the increase of speed is due to the conservation of angular momentum. When elongating the rope at the highest point, the change in height and thus in potential energy is smaller due to the angle compared to the gravitational field. This diffrence in energy between pulling and releasing adds energy to the system. This energy is constantly exchanged between potential (at the highest points) and kinetic (at the lowest point), so as more energy is added the pendulum goes higher and swings faster. What do you guys think about this?
It'd be interesting to come up with an experimental design that can separate the angular versus radial contributions to the pumping efficiency move on a swing.
Maybe 2 perpendicular linear actuators on a rigid pendulum would do it.
I make springs for a living so might be able to help you out with some big boys! That said what about gas springs for something body weight sized?
Steve, Thank you! You and Destin Sandlin do a great job sharing the observational discoveries that you stumble upon from your everyday life. Just like your video on "Bizarre liquid jets explained - the Kaye effect", https://youtu.be/WC6tjcDOShU. Or even the Chain Fountain and Mould Effect. And when Destin asks his question about "Strange water phenomenon", youtu.be/p7EiT9tKiig. And then does a follow up over five years later to give in depth explanations with his video about "The WALKING WATER Mystery", youtu.be/KJDEsAy9RyM. There are many more examples. Your work brings a little extra joy into our lives! Thank you!
p.s. Near the end of your Kaye Effect video you state, "So next time you notice something strange, dig a little deeper. You might find something amazing. And share it with me, because I'd like to make a video about it." Well, a couple of years ago I noticed an interesting crystallization in an oil blend. It would make these cool porcupine balls of crystals. Not something I would expect in oils. Cool to see. I think they call it lipid crystallization. If you're interested in doing something with it, I'd be happy to share details.
https://youtu.be/MUJmKl7QfDU?t=339 at 5:39 I actually believe that you still add energy when you lenghen the string because you let gravity pull the weight instead of countering it. Like if you used the Height of the weight as Potential Energy. Am i wrong or Are you ?
Love what you do btw !! thanks a lot :D
Thanks for this awesome video! I made a game based on this (physics with spring pendulum) but I didn't know what parametric resonance was, why it worked, or how to really explain it besides intuition. This knowledge will help I'm sure! :) It's by no means polished but If you're interested, here's the game in question: https://evn.itch.io/chains-chars-clj
Anyone else notice the stand at 3:10 rocking when he pulls the string.
is it an illusion that, when you just use the rope and pule, it looks like what I imagine is the exact opposite to "The Mould effect"?