Lever Paradox Explained

In this video, we dive into Steve Mould’s fascinating lever paradox. We’ll break down why his intuition, though compelling, leads to a surprising result — and why it’s wrong. You’ll learn what actually happens, why Steve’s reasoning feels so natural, and how it all connects to the key difference between gravitational and inertial mass. Along the way, we’ll explore how levers transform inertia, making them more than simple machines of mechanical advantage. Plus, we’ll resolve the paradox using both linear and rotational physics, showing how mechanical advantage scales not only force but also inertia.

Special Thanks to Steve Mould for allowing me to use clips from his original video (link below!)

Chapters:
00:00 Intro & Steve’s Lever Paradox
01:27 Why Steve’s Intuition Feels Right
01:42 The Role of Gravitational vs. Inertial Mass
02:49: The Core of the Paradox
03:50 Mechanical Advantage and Inertia Transformation
06:14 From Masses to Forces: Changing the Perspective
07:10 Solving the Paradox with Rotational Dynamics
11:08 Final Thoughts: Why Our Intuition Fails, and Takeaways

🧪 More on inertial mass vs. gravitational mass:
#Physics #Levers #SteveMould #MinutePhysics #ScienceExplained

[bryantkaye]

Excellent video! Such a subtle point that there’s a difference between inertial and gravitational mass. I have PhD in Applied Physics and I never thought about that levers could change one differently than the other

Anyone that says this is wrong or elementary likely doesn’t understand it

[DePisceEpiscopi]

Very nice explanation, had a great time watching!

[DrJulianNewmansChannel]

Nice video; the irony of 1:27ff is that that exact reasoning can actually be translated to the case of the lever to derive the correct r^2 factor for rotational inertia: If you were applying no force to counteract gravity, then the mass would effectively be in freefall, corresponding to a downward acceleration of g and hence an angular acceleration of g/r -- and so, if the force required to balance the effect of gravity is proportional to r, then the extra force required to achieve an angular acceleration of g/r in the opposite direction is also proportional to r.

PS: As much as I admire and respect Veritasium in general, I don't think the video you've referenced in the description is so helpful. Please forgive the 'self-advertising', but you can watch my video entitled "A potential misconception about F=ma" to see why.

[PhysicsFellow]

Does this not also break down further to the Law of Energy conservation?

In Steve's example, he talks of the same applied Force, but it's applied over the same distance, F x D = E.

If we negate the lever and apply the same energy to launch a 2 mass with a spring of set compression for example, then launch a 1 mass with the same soring compression, the one mass will launch at root 2, adding a level doesn't change this as energy has to be conserved.

[zekejanczewski]

I think you're sort of overcomplicating it. It's just that the angular moment is different. That's it. That's the paradox.

[ThisTimeWithMaths]

This is an interesting take! I ended up with similar equations, but did so in a completely different way and came to completely different (but mathematically equivalent) conclusions. You were more rigorous than me, though.

So far as I understand it, your argument here is that when we use F=ma to understand the motion on the "in" side, the m is actually an "inertial" mass given by m_in=m_out/s^2. But my question is, why create this fictitious perceived mass at all when we could just say that there is no mass on the "in" side, and hence F=ma doesn't apply? Your approach seems to be exactly the kind of thinking that lead to the paradox in the first place; an assumption that the relationship between any pair of force and acceleration is described by F=ma.

This is a genuine question. Are there situations where it is aids our intuition to conceptualise it in terms of a fictitious mass on the input side?

Or maybe the contrapositive makes more sense to investigate: is there a situation where a fictitious mass is also inappropriate? i.e. can a hand applying a force ever have a non-linear relationship with its acceleration? I'll comment later if I think of something.

[sabriath]

seems like you overdid it and the back of my mind is telling me you are wrong somewhere but I cannot pinpoint it. The "paradox" is simply the work performed that was not calculated correctly. When you move a mass that is twice the distance away, it moves much further in actual space when distance must be accounted for, that's work. It's the same thing when you are giving yourself mechanical advantage through a pulley system in order to pull a car out of the mud, you have to pull on more rope over a longer period of time but the strength needed is extra rope and time is easily felt and seen, which means you are going to be able to feel the angular difference. If no movement was involved at all and you were just holding it, then the weights would feel exactly the same.

[RickMacmurchie]

Not a paradox, it is mechanical advantage, something taught to elementary school students.