The Lever Paradox

[00:01] calling the double decker Newton's cradle. The idea is pretty straightforward. With a normal Newton's cradle, the energy of the first ball is entirely transferred to the last ball. Whereas, in the double decker setup, I

[00:13] want the energy to transfer from the first ball through this lever and into the first ball of the lower Newton's cradle. But, as you can maybe tell, the issue is with tuning the lever. Specifically, the mass of the lever and

[00:28] the distance of the balls from the pivot of the lever. And it was while I was trying to optimize those parameters that I realized something, which is that I was originally going to be, "Hey, look at this weird Newton's cradle that I

[00:44] made." But, now it's going to be me trying to convince you that you don't understand levers, either. And then, maybe together we all get to the point where we understand levers again, with a bit of help from Matt Parker here and

[00:58] Henry from MinutePhysics. And if you're someone who continues to understand levers throughout this video, I'm sure you'll leave a comment to let us know, perhaps describing how obvious it all seems to you. That way, we know how

[01:11] optimizing the double decker Newton's cradle eventually, I promise. But, first, here's how I'm going to try and persuade you that there's at least about levers. To do that, I'm going to introduce you to a thought experiment

[01:26] involving a system of levers. And then, I'm going to go through a series of logical steps that seem very reasonable. And then, at the end, we'll have a situation where I've put one joule of energy into the system, but I've got two

[01:41] joules of energy out. Okay. So, suppose we have a 1 kg mass, and it's floating in space. And imagine we've got a rocket that we can use to push on this mass. that it's always going to push with exactly 1 N of force. So, we start the

[01:57] engine and we start pushing and we keep pushing until we've traveled a distance of 1 m. And so, of course, the mass speeds up and now it has some kinetic energy. So, how much work has the rocket done on the mass? Well, work done is

[02:11] done on the mass? Well, work done is force times distance. The force is 1 N, the distance is 1 m, and so the work done is 1 J. 1 * 1 is 1 and that 1 J of work turns into the kinetic energy of the mass. So, the 1-kg mass ends up with

[02:27] a kinetic energy of 1 J. And if you want to be belt and braces about it, you can also calculate it by starting with force equals mass times acceleration. It's a longer calculation, but you still end up with a kinetic energy of 1 J. And by the

[02:40] way, its velocity at the end is root 2 m/s. That will come in handy later. In the previous animation, I had the rocket kind of disappear for clarity, but of rocket's going to be moving at that speed as well and they'll just glide

[02:55] together. Now, let's imagine that the whole thing is mediated by a lever. It's the same thing. We push on the lever with a force of 1 N over 1 m. The mass feels the same force over the same distance and so it ends up with the same

[03:09] kinetic energy of 1 J. For the next logical step, we just need to remind ourselves of something we already know about levers. And for this, we're not floating in space anymore. We're back on Earth and we're just holding up a

[03:21] kilogram mass with our finger. A certain amount of force is required to do that, about 10 N here on Earth because the pull of gravity is about 10 m/s/s here. That's little g. I'm going to assume that little g is about 10 throughout

[03:35] this video. So, it's 10 N of force from the finger to hold up that 10 N weight force from the 1-kg mass. Now, here's the famous thing about levers. If you half the mass, but double the distance from the pivot, the finger doesn't feel

[03:49] any difference. The mass now has double the mechanical advantage, but it has half the mass, and they cancel out. If you half the mass, but double the distance, the turning force stays the same. That's the thing we know about

[04:01] levers. And so, your finger is required to apply the same force as before, about Your finger cannot tell the difference between the 1 kg mass that's the same distance away from the pivot as your finger, and the half kg mass that's

[04:14] as your finger. In fact, you could put the mass end of the lever in a black with your finger to be able to distinguish between a 1 kg mass and a There's no experiment that you could do to figure out which one it was. The

[04:30] finger has the same experience because the force is the same in both scenarios. going to do the same thing. We're going to move the mass twice as far away, and we're going to half it. That's half a kilogram. And again, the rocket will

[04:44] apply a force of 1 N over a distance of 1 m, which of course means that the mass will be pushed over a distance of 2 m because that side of the lever is twice as long. And so, the mass gains some kinetic energy. Now, the rocket can't

[04:56] distinguish between these two situations. The experience of the rocket should be the same in both cases, as we know from the levers on Earth. The reaction force of the rocket will be the same, and so the acceleration of the

[05:08] rocket will be the same. And so, the rate at which the lever rotates around rate at which the lever rotates around the pivot will be the same. Like this. Now, this end of this lever is traveling twice as fast as this end of this lever

[05:22] pivot. So, after the engines turn off, the half kg mass will be traveling at twice the speed of the 1 kg mass. So, how does that affect its kinetic energy? What happens to the kinetic energy of something when you half the mass and

[05:36] double the speed? Well, let's take a look at the equation for kinetic energy. You've got that V squared term in there. Let's write that out as V * V. So, we've got 1/2 MVV. And remember, we're a halfing the mass

[05:49] and doubling the velocity. So the kinetic energy is halved, then doubled, then doubled again. Now, if you half something, then double it, that cancels out, and we're just left with that final doubling. So instead of 1 J, it's double

[06:03] that. It's 2 J. If you want to use actual values, we can do that, because if you remember earlier, we got a value for the velocity of the 1 kg mass of root 2. But of course, we're doubling that. So it's 2 root 2. Squaring that

[06:17] gives you 8. The mass is half, so we divide by 2, and then half again because the equation for kinetic energy has a half in it. That gives you 2 J. But remember, the rockets are having an identical experience. 1 N over 1 m is 1

[06:34] J. So I've just shown that we can put 1 J of energy into a lever and get 2 J of energy back out. So that means one of two things. Either free energy technology is real and the government is suppressing it, or Steve doesn't

[06:48] understand levers. Or to be more precise, one of those logical steps, logical steps, let's put it in scare quotes, one of those logical steps uh was wrong. Now, to me, they all seem quite reasonable, rigorous even. But

[07:03] there was one that was a bit more hand-wavy than the others. Maybe you spotted it actually. It's at this point where I said, "The finger pushing down on the lever, it requires the same force to hold up a half a kilogram mass over

[07:17] to hold up a half a kilogram mass over here and the 1 kg mass over here." That bit's true. But then I said something like, "Yeah, in other words, the experience of the finger is the same in both cases." There's nothing that the

[07:29] finger could do to distinguish whether it's a half kilogram mass or a 1 kg mass. It sounds plausible, and it seemed reasonable to me when I first started thinking about it. But at the same time, I can't think of any physical concept or

[07:42] idea that would justify it beyond, you know, physics intuition. So, this seems went wrong and how we ended up getting although you need to apply the same force in this scenario that you do in

[07:56] this scenario, maybe it doesn't follow that the experience of, say, waggling the lever is the same for both scenarios. Well, brilliantly, that's something that we can test. So, I set up this blind test. It's not double blind

[08:09] because I can't be bothered to do good experimental design, to be honest. But, anyway, we dragged a few people in off the streets, and this is what we found the streets, and this is what we found out. Oh, I would say

[08:26] Interesting. It's pound down on that. Uh get it right. also waggles. also waggles. Test subject number two, Lucy Green.

[08:39] Yeah, one is obviously harder. Yeah. Yeah, yeah, yeah. Which Oh, definitely, this one. Okay. Yeah. Pound down. So, if I hold them both, I can convince myself that's the same force. Yeah. But, you're saying

[08:51] different. Yeah. Once it becomes a it changes. Aha. That's cool. I also tried the experiment, and so did my wife, and we

[09:04] had the same results. So, that seems pretty conclusive. Everyone that conducted the experiment felt that it was easier to waggle the mass that was at the same distance from the pivot as their hand, and harder to waggle the

[09:19] their hand, and harder to waggle the mass that was half the mass but twice as far away from the pivot. But, let's try and quantify easier to waggle. What does that mean? Well, I think it means two things. I think it means for a given

[09:31] force, the lever moves more rapidly when it's a 1 kg mass here than when it's a half kilogram mass here. In other words, the lever accelerates more rapidly for a given force. Or to put it another way, the smaller mass that's further away

[09:47] requires a larger force to get the same amount of acceleration. And so, to bring all this back to the lever in space, what we find is that when we apply a force of 1 N over 1 m, well, it takes longer in the case of the half kilogram

[10:01] that's twice as far from the pivot. In other words, it accelerates more slowly, and the velocity at the end is lower than my intuition would have expected. To wrap it up mathematically, we can ask the question, well, how much

[10:13] slower will it be going? Well, we know that the energy must be 1 J, and then from that. But, if we're trying to disprove the free energy hypothesis, we

[10:25] should probably prove that the half kilogram mass has a kinetic energy of 1 J from a force perspective first, which is actually really easy. The key is to notice that the force on the half kilogram mass is half a Newton, because

[10:38] From the work equation, we get half a Newton * 2 m is 1 J. You then get the velocity by rearranging the kinetic energy equation, and when you do that, you find that whatever the final velocity of the larger mass was, the

[10:54] smaller mass, instead of being double the velocity, it's root two times the velocity of the larger mass. So, that's where my intuition went wrong. These two scenarios don't have identical

[11:07] slowly. But, I still think it's a compelling line of reasoning. Like, when I presented it to smart people, it hasn't persuaded them that you can get 2 J from 1, but at the same time, they weren't able to immediately tell me

[11:20] where I'd gone wrong. You know, I bet I could persuade a few people to invest in a free energy device just by running them through my space lever slide deck. Be a good way to make a bit of money as well, not that I do it, obviously,

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[12:54] Odoo today. Just for completeness, by the way, where would the half kilogram mass need to be placed so that the rocket did have the same experience as the 1 kg mass that was the same distance from the pivot. If you do the maths, it

[13:07] turns out you're not sending it double the distance, you're sending it root two times the distance from the pivot. But it still feels counterintuitive to me. Like why is it that the force required to hold it down is the same, but the

[13:23] force to get the thing moving is different? I mean we know why it is but I suppose my question is why does my intuition go the other way? Why does my intuition say that if the force is the

[13:38] same to hold a mass stationary against gravity, then the force should be the same to get it to accelerate at some particular rate. Well, I've been thinking about that and and I think this is where it comes from. Suppose I'm

[13:51] holding a mass, maybe it's a tenth of a kilogram. That means I'm having to apply a force of one Newton to hold this thing against gravity. And I know from life experience, like I don't know the numbers, but just like qualitatively, I

[14:07] know that if I double that force, then the thing would accelerate upwards at 10 right? At little g. And so when I get a sense of like how hard is it to keep

[14:19] this thing stationary, I have an expectation that if I double that force, expectation that if I double that force, then it should accelerate at little g. then it should accelerate at little g. But when I press down on a lever with a

[14:33] certain force, right? To to keep it stationary, and then I double that force, my brain is expecting my hand to my brain is expecting my hand to accelerate downwards at little g. But it

[14:46] might not, right? It depends what's going on on the other side of the lever. And then I was thinking like, is there a neat way, like is there something I can point to and say, "Look, this is where it deviates from intuition." And I

[14:59] thought, "Well, it's because it's a spinning thing." You know, the physics of spinning things is counterintuitive cuz you've got the moment of inertia with this r squared term in it. Like, maybe that's why. Maybe it's just

[15:12] because the physics of spinning things is weird and it we don't have an intuition for it. But then I was talking to Henry from MinutePhysics about it, to Henry from MinutePhysics about it, and now I think I'm wrong.

[15:25] about this lever paradox puzzle thing, I also thought it must be because rotating down and did the calculations, I noticed that I didn't really have to rely on any Ultimately, the two different masses are distinguishable when you waggle the

[15:38] distances. The little mass on the twice as long lever moves twice as far for a me thinking about other places where an identical force applied over a fixed different distances, which made me think of pulleys. Like, you could hold up a 1

[15:53] kg object with a rope over a pulley like this, or a 0.5 kg object with a pulley experience would be the same. Feel free to pause and think about it. For you to pull your rope 1 m, the 0.5 kg object will move 2 m, while the 1 kg object

[16:07] only moves 1, just like the rocket lever thought experiment. And the rest of the levers. If you hid these two pulley setups inside boxes, you could tell on the rope, the equivalent of wiggling the lever. It's kind of like the pulley

[16:19] moment of inertia, or whatever the equivalent of that is for pulleys. And you can also contrive a similar, but not exactly the same situation with inclined more about the mechanical advantage of these scenarios. At least mechanical

[16:33] results in an object moving a different distance than your input force moved. ways of setting up a situation with inclined planes where you really can't different masses because the force you experience is the same, and they both

[16:45] about how mechanical advantage allows you to convert the magnitude of the you to convert the magnitude of the motion. advantage, which, interestingly, was the central premise of my last video about

[16:59] Brennan torpedoes. Total coincidence there. The famous thing about mechanical advantage is there's a trade-off between force and distance moved. I don't require much force to move this mass up and down because I have mechanical

[17:12] just to move a little bit, I have to move my hand a lot. And I guess we can just extend that. So, it's not just a trade-off between force and distance, it's a trade-off between force and

[17:25] distance over time. In other words, velocity. It's also a trade-off between force and distance over time squared. In apply that back to the waggly sticks, look, in this scenario, we actually have

[17:39] mechanical disadvantage. So, let's look at what happens if we give ourselves going to double the mass, but bring it closer to the pivot. And now, it's easier to waggle, but we have to waggle it further, and it's going to waggle

[17:53] faster, and it's going to accelerate more rapidly, too. On the other side, if you've got mechanical disadvantage, then the force is larger, as we could feel, but also the distance traveled is smaller, and the velocity is smaller,

[18:08] well. Hey, this is Henry butting in here again. I want to add one other reason I suspect your original puzzle feels thinking about the experience of applying a force for a given amount of

[18:20] seconds. But, if you tell me I'm going to apply a force over a given distance, sounds super similar to our brains, it's actually a very ambiguous statement. Lifting a 1 kg weight a meter could take 5 seconds or 5 hours, and those would be

[18:34] very different experiences. Yeah, that makes sense. viewers, you can let us know in the comments. But, what does all this mean for my double-decker Newton's cradle? Well, when you're trying to figure out

[18:46] where the ball needs to strike the lever, you're trying to figure out the position that would cause all of the kinetic energy of the ball to be transferred to the lever, just like you would if it was a ball colliding into

[18:59] another ball. Like, if it was just the ball and the lever, you'd want the ball spinning. And when you do the math, it turns out you have to match the moment of inertia of the lever and the ball about the pivot point. Now, when I was

[19:15] imagining all this before I'd started to build anything, in my mind, the lever had the same mass as one of the balls. And in that scenario, the collision point needs to be the square root of a third of the distance along the lever

[19:31] from the pivot. But, in reality, the lever is heavier than one of the balls. And my first thought was, "Well, that's easy because I know about levers and mechanical advantage. So, if the lever is twice the mass of the ball, then

[19:46] well, it's just as if I doubled the mass on the end of a lever. And so, in the same way, I need to move the collision point double the distance from that point double the distance from that magic square root of a third distance."

[20:01] then we triple the distance, and so on. I've been experimenting with a few different levers. The mass ratio of this particular one is footage is showing you what happens when you move out by that amount. And as you

[20:16] can see, after the collision, the ball continues to move forwards slightly. It hasn't stopped dead. So, we're too far away from the pivot. And if you remember earlier how I had to place the half mass at a distance of root two away from the

[20:31] pivot for the rocket to have the same experience. Well, we need to take the root of the mass ratio here, too. And when we do that, this is the position of the collision. And look, the ball stops dead, which is great. Really though,

[20:44] this lever just doesn't work very well when you bring all the balls in. This is a shorter lever that has a more traditional swivel pivot, whereas this lever works much better. Like, the back and forth bouncing doesn't last

[20:56] particularly long. But, what's important is about one ball comes away from the Newton's cradle after each go around, whereas before two or three balls were coming out, and it was all failing in that way. You might notice that the

[21:09] lever is being held in place in a weird way. I'm using this fishing wire here to hold it against the pivot point. And then this fishing wire is adjustable, and that's kind of what I used to hold it level. It works pretty well. The

[21:24] issue is it introduces another force. Like when the ball collides with the top of the lever, that's okay. But when the ball collides with the bottom of the lever, well, the elasticity in the fishing wire is additional force that

[21:37] the ball is going to encounter. My suspicion is that it's negligible, and this design has other benefits. And it works better than the other lever with the more traditional swivel pivot. And I tried different lengths of both as well.

[21:49] What's really weird though is that for this lever the optimum position of the theoretical optimum position. And it's nowhere near the fallacious theoretical optimum position, either. So at this point, I just maybe

[22:04] we need to be thinking about like materials. You know, what's the like materials. You know, what's the elasticity of the lever? Is it acting like a spring, and so we kind of need to worry about some of the energy coming

[22:18] back? I definitely think there could be some design improvements. Maybe if I had a CNC machine, I could create a more interesting pivot that's also like a hook for the lever to hang off or something like that. But anyway, it's

[22:33] not bad. You get one ball popping out each time, but it dampens very quickly. video about the Newton's cradle while I was making this one. In it, he shows a really clever way of analyzing the behavior of a Newton's cradle in terms

[22:48] of shock waves. What's really interesting is you can analyze it as shock waves in the way that James does, or you can analyze it as if you have these perfectly incompressible balls that collide perfectly elastically and

[23:01] either way, you get exactly the same behavior. I expect that the equivalence between these two different models hints at something deeper, but I can't think to think about my setup in terms of shock waves, well, what would that mean

[23:16] for the lever? Actually, it becomes about impedance matching. I made a whole video about that. I'll link to it in the description. I'll link to James's video definitely worth your time. But anyway, when you think about waves traveling

[23:30] through a system, if part of the system impedes the progress of the wave more or less than the rest of the system, then some of the wave energy gets reflected at that part. And by tuning the mass of the lever and the collision point, what

[23:44] we're really doing is matching the impedance of the lever to the impedance reflection. So whether you think about it in terms of perfectly rigid, you think about it in terms of traveling shock waves, the process of optimizing

[23:58] the lever collision ends up at the same result. But anyway, that's how a silly idea that I had about Newton's cradles made me realize that levers are stranger than they first appear. And so are pulleys.

[24:11] And inclined planes. If you have any ideas for how I could improve my setup, enjoyed this video. If you did, don't forget to hit subscribe. And the algorithm thinks you'll enjoy this video next.