The Lever Paradox

When I was making this double decker newton's cradle I released something very strange about levers.

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[mahdidiab]

After watching the video, now i understand levers less

so i guess you convinced me that i didn’t understand levers from the start

[niedrigbreit-podcast]

Good old Steve explains something, I go "aah yes, makes sense" and then he's like "yeah, but that's absolutely not how it works"

[peterk]

There is such an urge to treat dynamic systems as purely a series of still frames. This beautifully shows the danger in that.

[RDGES]

This helps me understand why playing a grand piano feels very different in touch to an upright. One of the main differences in the action is that grand pianos have much longer keys (the black and white bits on the outside look the same but the pivot point and total length of the leavers are much longer in the grand piano). So although the down weight (minimum weight to depress the key) and the up weight (maximum weight for ley to go up) can be similar, the levers mean a grand piano feels very different at playing speeds.
There are many other differences in the mechanism that help a grand act faster and repeat faster too. But key length has an impact and I think it is similar to the experiment in this video with waggling different levers.

[Renslay]

This video reminds me of the famous quote: “Aristotle said a bunch of stuff that was wrong. Galileo and Newton fixed things up. Then Einstein broke everything again. Now, we’ve basically got it all worked out, except for small stuff, big stuff, hot stuff, cold stuff, fast stuff, heavy stuff, dark stuff, turbulence, and the concept of time”.

[MichielPlooij]

You may want to look up the term "reflected inertia". It explains why the inertia does not scale linearly with mechanical advantage, but quadratically. And actually, it can be generalized to reflected impedance, as all mechanical impedance scales quadratically with mechanical advantage. So this includes stiffness and damping. This is very important in gearboxes for instance. Also, the same principle has been used in theater equipment that makes you fly over the podium, to counteract gravity without adding much inertia.

[PetruRatiu]

So, as a civil engineering student (25 years ago), I took a lot of semesters of statics and dynamics and so on in uni. Apparently I remembered enough of them to scoff at your "black box" step and mutter "surely the inertia moments of the two systems is different so they behave differently in a dynamic system", but other than that I almost followed your logic as well. I think the intuition fail when discussing levers (and pulleys and inclined planes...) is that the way we're taught about them (at around 10-12 years old in most countries, I believe) is always in a static setup, and the intuition that's drilled into our brains is about their behaviour at equilibrium so we're always reaching for that intuition before thinking about conservation of energy. Not sure what a good framing would be to help shift the viewer intuition that dynamic problems behave differently than static ones. In uni I've been taught the virtual work principle to analyze dynamical equilibrium, not sure what a good explanation would look like.

Unrelated, at some point you touch about how we perceive forces in a somewhat indirect way: I'd point out that 1kg feels WAY heavier if you hold it at an arms length so the brain is doing some sort of proxy measurement there to estimate the force (my engineer's brain would reach for the virtual work concept again, but maybe it's wrong).

[蔡博宇-uk]

I’m a high school student from Asia and a passionate physics enthusiast. Here are some of my humble thoughts: I view a lever as a converter of equal torque. On one end, it inputs a unit torque, and on the other end, it outputs the same unit torque. As for rotational speed, it’s analogous to pushing a 1 kg object and a 2 kg object over a distance of 1 meter using a force of 1 newton. Pushing the 2 kg object takes more time because performing the same amount of work does not imply the same power during the process.( In rotational motion, the moment of inertia corresponds to inertial mass in translational motion.)

[DarkblueIbanez]

When you said that the experiences were the same that immediately felt wrong to me. I've worked construction a decent amount and I knew instinctually that was wrong somehow. But I couldn't figure out why. This was such a cool video. I love how you take intuition and convert it into understanding.

[TheActionLab]

Great video Steve! I love the thought process through the whole video. A simple Newton's cradle already requires solving multiple differential equations simultaneously while making major assumptions like non-dispersive hertzian springs. So when you introduced a lever to the whole thing I knew it was going to get complicated :)

[popepasawat]

This topic comes up a lot in tennis rackets. There's a measurement called "swing weight" (unit is kg/m^2). It represents how hard it is to accelerate (swing) a racket rather than how hard to hold it still horizontally. Two rackets can have the same weight and balance point but different swing weights. The one with smaller mass near the tip will be harder to accelerate than the one with larger mass near the handle despite "mass x distance" being the same.

[loganbrown]

I think there is a relatively simple and intuitive explanation for this which is that the mechanical advantage of the lever "applies twice".

If you halve the mass and double the distance, the mass will accelerate the same amount as it did originally for a given applied force. This makes sense intuitively. The end you are holding on to, however, will accelerate at half the rate of the mass itself due to the mechanical advantage of the lever. That means that it will feel harder to move the mass that is small and further out.

Due to this effect, if you double the length of the lever the apparent mass will go up by 4x, but the apparent strength of gravity will go down by 2x leading to a total increase in force of 2x. If you also halve the mass, it will then feel 2x as heavy as it did before instead if 4x, meaning that the force is the same, but the mass is doubled.

[intoxicary]

I understand levers. Once I asked my father what this thing was and where wanted me to put it. He said it was a leeverite. I asked him what's a leeverite. He said, Leeeverite there.

[aaronl]

I love the fact that you have the humility to admit you don't fully understand something and reach out to someone else to help out, and you put it on YouTube.

[swiebelunknown]

I am studying mechatronics in the Netherlands, and the timing of your video is amazing. I just finished my exam with a good grade and the subject was exactly this it's called dynamics, as in calculate the forces on and in an object while in acceleration. Good video!!

[SeanDaSuzy]

A perfect example of how anything non-linear (in this case the sqrt(2) factor when doubling the leverage and halfing the mass) can become quickly confusing in terms of physics intuition. Well illustrated and explained, well done all of you guys!

[eagle-from-aut]

Even with the incorrect assumptions made about the black box, you could still tell. You could push both lever ends down to the same distance, and let go at the same time. One will take longer to fall than the other. I think this is identical to your explanation but i'm not entirely sure.

[nickrp]

As far as system improvements: The single biggest thing will be increasing the rigidity of your pivot. Holding the anchor in a rubber jaw vice clamped to the table provides lots of opportunity for flex and thus damping. That thing needs to be attached rigidly to something very heavy. I am picturing fixing your pivot bar into a concrete block or some such to give it a solid base. It is like with blacksmithing anvils. You want it heavy and rigidly connected to the floor so the hammer rebounds and all the energy goes into the work piece.

The fishing line / rocker pivot design is fantastic; large radius line contact with little movement and no gap. I might glue some magnets to the sides to replace the fishing line, but you are not going to do better than that line contact with any small ball bearing or shaft pivot.

It might also be worth trying to make a curved lever, something that looks a bit like a recurse bow pointing toward the balls so that the pivot is in line with the contact faces, that would also make it possible to balance so it is not trying to tip forward.

[sophiachalloner]

9:41 this reminds me of something from class. We were talking about pendulums, and the weight at the end had no effect on the time to swing back and forth. Only the length of the string.
I think the same principle is happening here.

This whole thing reminded me why I like Lagrange's formalism so much: It's hard to reason about forces as soon as the system becomes even a little bit complex, but it's much easier to reason about kinetic and potential energy.