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Given an object being pulled down to earth by a force of 1000 lb:

If a machine were to lift it off the ground for me, and I grabbed on to the rope before it was released, there is no way in the world I could hold it in place.

But, if I had a sufficiently long (and weightless) lever, I could achieve this with just a finger, with a force of less than a pound.

So where do the other 999 lbs of force come from? What triggers it to be applied now when I apply my finger as opposed to before I did?

(I understand the rules of torque and its equations; I am looking for and understanding of how & why. It does not help to be told that "by the rule, this must occur, and therefore the rule makes sense" - that is obviously circular. And saying, "well, observation bears it out" defends it but fails to explains it.)

Edit 1: To clarify, I'm just asking assuming there is a better answer than "it is a basic law of nature: torque causes force to appear out of nowhere, and cannot be explained in terms of other more basic laws or attributed to some supplier of force". When I ask to understand, I mean as opposed to an axiom, i.e. a starting point of reasoning, accepted but not understood or derivable from other accepted/proven rules. Of course, it's definitely possible that it is an "axiom".

Edit 2: After seeing several answers, I think the following is what is really irking me: Shouldn't there be, intuitively, conservation of force? How can there ever be something from nothing? Unless we define a new type of spontaneous force 'lever force', outside of the known four, then it must arise from the known four. But the known four transfer exactly what was applied to them (along the same line), no more no less. (Applying 1lb of force to the top atom in a stack of atoms will never increase its force on the next atom by 2lb after equilibrium is reached - it will be increased by exactly the 1lb being applied to it.)

Edit 3: In response to the suggestion that the fulcrum provides the force, please consider the following example: A Ferris wheel with a 1000lb load in one seat. It falls straight to the bottom. And ultimately, like is being suggested, it won't fall to the ground, because the center spoke supports it. However, it will hang straight down - i.e. the center spoke is not keeping the object an extra $r$ (where $r$ is the radius of the wheel) feet in the air! Clearly, any time you raise an object above where it was supported by another object, it requires a force equal to that of gravity. A table-top will never assist you in raising the object above the table top, though it will keep it off the ground (should you release it). Think of the table-top as the seat-containing-the-object hanging straight down - any higher is like lifting the object off the table, and should require a force equal to gravity. Except it doesn't. Hence my question.

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+1 for a non-trivial, thought provoking question. I'm just not sure what kind of answer you're actually seeking. –  Alfred Centauri Oct 24 '13 at 21:22
    
Not an expert, and probably not a complete answer, but if the finger, weight and lever were one object, the fulcrum would be at the balance point. If the finger and the weight were a planet and star orbiting in a simple system, the fulcrum would be at the barycentre. Even if you remove gravity being the helping force by making the lever weightless (why?), the lever's force of "holding itself together" (dunno the technical term) is added. –  AlbeyAmakiir Oct 24 '13 at 23:07
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Re: to everyone claiming there is no conservation of force. In equilibrium, there is conservation of force, as Newton's Second Law demands that $$\sum_i\mathbf{F}_i=\mathbf 0.$$ Thus if I exert a force on part of a lever, and there is no movement, this part of the lever must in turn be acting on other parts of the thing. –  Emilio Pisanty Oct 25 '13 at 17:59
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While I have not read very much of it, your question reminds me of Spivak's writings on physics. You may find Lecture 1 of this link relevant, as well as the Prologue of Physics for Mathematicians. –  Emilio Pisanty Oct 25 '13 at 18:07
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(cont.) Now physicists all agree that Newton's Three Laws are the basis from which all of mechanics follows, but it you ask for an explanation of the lever in terms of these three laws, you will almost certainly not get a satisfactory answer. You might be told something about conservation of angular momentum... " –  Jeff Oct 25 '13 at 18:53

11 Answers 11

The extra force in a lever comes from the distribution of energy over a smaller distance of action. We can look at it very generally from the perspective of conservation of energy, or the lossless transmission of work through a machine.

When we move something against a resistance, with a fixed force, we are putting in energy and doing work. This work is the product of force and distance: $w = F \times d$.

Devices which use pulleys, levers or hydraulics to generate a larger force all work by "trading" to a smaller $d$ and larger $F$. Since the machine is efficient (little energy is lost in the machine), the amount of work $w$ done on the machine by you is almost fully transmitted to the load. Since the distance $d$ by which the load is moved is smaller, the force is greater.

For instance a lifting machine whereby we push against a force, and move a meter, such that the load lifts by 1 cm, gives us a 100 fold advantage. The tradeoff is that although we moved something by a meter, the load only moved one centimeter. We paid for more force by sacrificing range of action.

We do not have to appeal to energy to see that large forces are generated; another way is to draw free-body diagrams of the parts of the machine, and the force applied to it, and to the load. Balancing the forces requires that a large force must be present on the load.

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@Jeff : This is the correct answer. There is conservation of energy, not conservation of force. Whatever lever you use, if you want to elevate a mass of $1000 lb$ from $1$ meter, you have to provide the same energy. –  Trimok Oct 25 '13 at 6:36
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This works in the other direction as well. Fundamental to how a bicycle works. A bicycle might be the most intuitive example because everybody rides a bike. When you're in a low gear, it's easy to spin. But you have to spin like crazy to move very far. The opposite is also true. On a high gear it's very hard to move, but when you move only a little bit the bike moves quickly –  Cruncher Oct 25 '13 at 13:12

Re: your second edit

No, there is no conservation of force. Forces can and do arise spontaneously. Imagine two balls colliding head on in free space (so no gravity, friction etc.). Suppose they collide elastically and bounce off in some other direction without losing any energy (for simplicity). Also assume for simplicity that we are sitting at the center of mass of the two bodies, so there is zero net momentum. They come in, collide, and bounce off symmetrically. This is just a convenience, you are free to look at this in any other frame of reference. You can approach this kind of ideal experiment with something like an air hockey table, or on a space walk. The idealisations aren't really important here, it's just to keep things simple.

During the collision there are huge forces acting on the bodies to change their velocities. Where did these forces come from? The answer is they didn't "come from" anywhere. Force isn't a substance which is stored inside things waiting to be used up (this is an Aristotlean misconception that many people have without realizing it, and it may be what's at the root of your problem). It didn't come from the energy of the bodies since the collision was elastic, i.e. the energy didn't change. And it didn't come from the momentum of the bodies as the total momentum is also conserved since there are no external forces acting on the system. In fact, after the collision the state of the balls is pretty much exactly the same as it was before, only their direction of motion is different. But the laws of physics don't have any special direction built in to them - they are "rotationally invariant," so there is actually no property of the objects which could be "used up" in this example to create a force.

In fact there is a way of understanding this process without invoking forces at all by thinking in terms of the electrostatic potential energy of the atoms in both of the balls. This is a more "modern," fundamental way to do physics and forces are derived as a consequence.

Force is not conserved, but momentum and energy are. Force is the rate of change of momentum. So you can think of momentum as a fluid. A force is a flow of momentum into or out of a body. (And incidently Newton's third(?) law, the action-reaction law, just says that as much momentum leaves one thing as goes into another. Dead simple.) Asking for a conservation of force is like asking for a conservation of flow. It's just the wrong level of description. Fluid is conserved, not the flow of the fluid. Flows are temporary things which arise as needed to make the fluid go from one place to another, without changing to total amount. Similarly forces arise as a way of redistributing momentum so that the total amount of momentum never changes.

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Are you saying: just like two balls bounce of each other because of this spontaneous force between the atoms on the surface of each object when they 'meet' (I forget the name, electromagnetic?), so to there is a new force that spontaneously appears on the end of levers? Wouldn't that be defining a new existing 'spontaneous' force outside of the four basic ones? Aren't all other forces equal transfers (gravity presses a ball down with 1lb of force, it in turn presses down on the item below it with exactly 1lb of force) or equal and opposite? –  Jeff Oct 25 '13 at 6:48
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@Jeff The forces involved in the lever are electromagnetic as well, but that level of detail is not necessary to understand that force is the flow of momentum. All forces arise in this way, though the momentum involved in any given situation may ultimately be stored in different particles/fields. –  Michael Brown Oct 25 '13 at 6:54
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@Jeff I suppose you might also be confused by the colloquial way people talk about there being four forces. They mean types of forces. That obviously doesn't mean you can only ever draw four arrows (or four pairs) on a free body diagram, or that if one force (pair) appears, another has to go away. :) All of the forces that arise in day to day experience are ultimately either electromagnetic or gravitational. The other two types only really matter inside the nuclei of atoms. –  Michael Brown Oct 25 '13 at 7:02
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As explained by the others the extra force comes from the fulcrum. –  Michael Brown Oct 25 '13 at 12:57
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I think by "conservation of force" he means "for very action there is an equal and opposite reaction". –  DJClayworth Oct 25 '13 at 17:34

That extra force comes from the fulcrum of the lever (the thing which supports the lever on the ground).

This fulcrum balances the downward forces on the lever in total (1000 lb of the object + 1 lb from your finger).

So Archimedes' claim about being able to lift the world with a long enough lever would only be possible if he had a strong enough fulcrum and a support (like a rigid wall) to keep the fulcrum in place.

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I like your point, except that it doesn't provide 1000lb when the lever is only a foot. So it can't be the fulcrum, since the same fulcrum doesn't consistently have the same impact; it must be something else to do with the lever itself. –  Jeff Oct 24 '13 at 19:13
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Well its like, the lever supports the weight of the load, the fulcrum supports the total force the lever applies downward(force you exert from your hand + weight of the load, i.e. total force of lever + load system), and the ground likewise supports the force fulcrum exerts downward. –  udiboy1209 Oct 25 '13 at 4:37
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Now it does matter how long the lever is because the force of the fulcrum cannot make the lever rotate(it is applying torque in the wrong direction), so you need the 1lb of your finger to make the lever rotate, and so you need a long enough lever to make 1lb sufficient. –  udiboy1209 Oct 25 '13 at 4:41
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@Jeff: The fulcrum isn't pushing on your finger, it's pushing on the lever. So it applies the exact force that the lever is applying to it. The lever, in turn, is applying normal forces to both you and the heavy object. The thing that changes as you move further away from the fulcrum is that the normal forces are less and less direct (less and less normal, really), so the lever is able to transmit to your finger a smaller and smaller proportion of the normal force from the fulcrum. So the forces (reactions) decrease accordingly. –  ruakh Oct 25 '13 at 14:30
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@Jeff perhaps it will help you to think of it this way: The force on the fulcrum doesn't come out of nowhere, it comes off the ground. As the fulcrum presses up with more force, the ground has to push up with less force, by an equal amount. There's your "conservation of force." –  Kevin Oct 25 '13 at 15:52

Jeff, you write:

And that is what an axiom is, accepted but not understood or derivable from other accepted/proven rules.

In this case, that is simply not true. You can derive the fact that increasing the length of the lever increases your applied force just using the high-school physics expression $w=Fd$. Here's how:

  1. First, note that your question is logically equivalent to the question "why is it easier to unscrew a bolt using a longer wrench?". If you do not understand why they are equivalent, I will clarify with a comment.

  2. Suppose it takes $E$ Joules to unscrew a particular bolt, which comes loose with one full turn. Since $w=Fd$, if you use a wrench of length $R$, you have $E=2\pi RF_h$. Hence the force you apply with your hand is $F_h=E/(2\pi R)$.

  3. However, the bolt still resists movement with the force $F_b=E/(2\pi r)$ irregardless of what shape of wrench you use, where $r$ is the outer radius of the bolt. The ratio $F_b/F_h=R/r$. This is exactly the lever force equation.

Thus, if you accept that $w=Fd$ (as confirmed by experiments), you have no choice but to subsequently conclude that levers allow you to generate extra force. And indeed, in reality one finds they do. It's all self-consistent.

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I guess I just don't understand how this force can appear out of nowhere, I expect intuitively for there to be "conservation of force". Forgive my lack of knowledge, but where else do we find something from nothing? However, I am forced to admit that you are absolutely right that it can be derived from w=Fd and conservation of energy. I still am having a really hard time wrapping my head around "force from nowhere", I'll need some time to mull over your answer. –  Jeff Oct 25 '13 at 4:31
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The real problem is that your intuitive belief that there should be "conservation of force" is simply incorrect, and not borne out by reality. Don't worry too much, though; there are a great many things which seem intuitively reasonable at first glance, but which upon deeper examination turn out to be completely false. Human intuition can occasionally be a double-edged sword. Playing with levers may also help :) –  DumpsterDoofus Oct 25 '13 at 21:45

First, it's important to differentiate between lifting the weight and keeping it from rotating. Your finger doesn't hold it up, the fulcrum does that. All your finger does, is to apply a force to keep it from rotating downward.

So, ok, why do you need less force the further out on the lever you go? The answer lies in geometry, the definition of "work", and the law of conservation of energy. Moving the lever further out by some distance translates into moving it a much smaller amount close to the fulcrum. So, if you do some work (W = F*d) far out from the fulcrum, you'd expect the same amount of work to be done by the lever (a rigid body) near the fulcrum. But, to do the same amount of work when d is smaller requires a larger F. Hence, the force applied far from the fulcrum becomes magnified up close, simply due to geometry and conservation of energy.

I hope that helps.

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But even when I do no work and just keep it in place, so work isn't a factor, a lever matters. More force just "appears out of nowhere" because I have a longer lever. Obviously the fulcrum isn't enough, since if my finger was removed, the object drops. But for some reason, how far away I put my finger determines how much force seemingly "appears out of nowhere". Why? How? –  Jeff Oct 24 '13 at 19:41
    
This is tricky. Here's how I see it... F = ma. If there's no movement, there's no acceleration. In that case, is there a force? Are we talking about an object made up of atoms that are constantly moving, or a single monolithic rigid object? If the former, then we can think that there's always small movements going on, so my explanation is perfectly fine. If the latter, then no movement means the net force = zero. A force only shows up when something else that's moving causes the lever to move, and again my explanation works. –  Owens Oct 24 '13 at 20:05
    
I stand corrected. But still, if that were all, then even if I were using a 1ft lever, I should still need the same exact amount of work to "keep it in place" (i.e. that millimeter of movement that the car experiences and that we call "in place"): at the cars end d is constant, F is constant (opposing gravity), and so once again, this force is seemingly increasingly appearing out of nowhere, the further my 1lb moves from the fulcrum, since I am increasingly able to oppose gravity's constant force on the car. –  Jeff Oct 24 '13 at 20:21
    
I'm confused by what you're describing, sorry. When you say you have a car, a 1 ft lever, a 1 lb force... I'm not sure how all that fits together. But anyway, I'll say a couple of things. 1) the force that balances the whole thing is applied back from the fulcrum, and 2) you have to be careful to not assume that a mathematical model is the same as physical reality. This is where a lot of problems come from. In reality we dont have perfectly rigid bodies. There's no such thing as two particles pushing each other and also staying immobile. There's always some change in momentum. –  Owens Oct 24 '13 at 22:05

To gain insight into this problem without appealing to "force amplification" etc, replace your finger with a 1 lb weight and consider the center of mass (COM) of the system assuming the 1000 lb weight and 1 lb weight are connected with a rigid and massless beam.

Assuming the system is free and parallel to the ground with no fulcrum, gravity acts on the COM as if the system were a 1001 lb point weight, accelerating the system without imparting a rotation to the beam.

Now, if the fulcrum is placed under the beam at the location of the COM, the fulcrum provides a counter force through the COM such that there is no acceleration of the COM and no rotation of the beam.

Of course, if the fulcrum is placed elsewhere along the beam, the force from the fulcrum is not through the COM and thus, the COM accelerates and the beam rotates.

So, from this perspective, the question of "where does the extra force come from" doesn't apply. Either the force of the fulcrum goes through the COM or it doesn't.

Now assuming the fulcrum is placed under the COM, replace the 1 lb weight with the 1 lb force from your finger...

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COM is an observation of where placing a fulcrum will not lead to rotation. It is a fact and a rule, but not an understanding as to why moving it one inch allows a 1lb weight to suddenly "generate" enough force to slowly lift 1000lb, or suddenly not be able to "generate" the 1000lb of force to keep it in place. That force seemingly still "appears out of nowhere", for "no apparent reason". COM is an observation and a rule of where this force (i.e. torque) will or won't occur, not a reason for how or why it occurs. –  Jeff Oct 24 '13 at 20:10
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@Jeff, COM is more general concept than that, a fact that you seem to discount too readily. Moreover, and importantly, it isn't clear to me what you're looking for or what it would mean to have an "understanding" that doesn't involve observations, facts, and rules. What would such an "understanding" be based on? Revelation? –  Alfred Centauri Oct 24 '13 at 20:26
    
I'm just asking assuming there is a better answer than "it is a basic law of nature: torque causes force to appear out of nowhere, and cannot be explained in terms of other more basic laws or attributed to some supplier of force". You're right, it's definitely possible, and I guess probably true. And that is what an axiom is, accepted but not understood or derivable from other accepted/proven rules. –  Jeff Oct 24 '13 at 20:42
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@Jeff, force is an abstraction from observation. Consider that, in other formulations of mechanics, e.g., Lagrangian mechanics, the concept of force is nowhere to be found. The differential equations of motion are derived from the Lagrangian and the principle of stationary action rather than $F = ma$. –  Alfred Centauri Oct 24 '13 at 21:03
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I agree with this answer, but it might also be worth mentioning that irrespective of the location of the weights or the fulcrum, the lever-weight combination applies a downward force of 1001lb. The forces involved never change. The only thing that changes is the centre of mass which causes the lever to lean left or right depending on its location. That extra inch doesn't suddenly generate extra force, it just moves the apparent location where it's applied. –  Philip C Oct 25 '13 at 8:38

There is no extra force. Draw a free body diagram of the system before and after you push on the lever. Before, the weight is reacted where it rests on the ground. After, the weight is reacted at the fulcrum. The only additional forces are the 1 lb you apply with your finger and the 1 lb reaction to that at at the fulcrum.

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Of course there is no extra force/energy generated here:

a lever simply does to mechanical work what a magnifying lens does to light and EM energy. It concentrates energy in a smaller space, so that the local effect is magnified but the total energy remains the same.

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Here's an example with force diagrams: suppose we have a mobile hanging from the ceiling, consisting of a rigid 60cm rod with a 5kg weight at one end and a 1kg weight at the other end. If the wire connecting the rod to the ceiling is attached 10cm from the 5kg-weight end, the forces will all be in equilibrium and the rod will be horizontal. We know this is true, because we have built and observed such mobiles.

force diagram of a balanced mobile hanging from the ceiling, as described above

(I'm using "kgf" units for forces to make the numbers tidier, and as usual for this sort of problem I'm assuming the masses of the rod and wires are negligible.)

Now, here's what I think is the crux of your question. As indicated on the above diagram, at one end of the rod, the rod is applying a 5 kgf rigid-body force to oppose the 5 kgf applied by gravity to the weight; at the other end of the rod, it's applying a 1 kgf rigid-body force, ditto. But if we draw a force diagram for the pivot, we see something different:

force diagram for the pivot in isolation, showing that near the pivot the downward forces on both sides of the rod must be equal

Gravity applies 6 kgf of downward force at the center of mass, which is the pivot. The pivot must be dividing that force equally between both sides of the rod, because otherwise the rod would be accelerating. So, a short distance $\delta$ away from the pivot in either direction, the rod must be resisting 3 kgf of gravity. How does 3 kgf a short distance from the pivot become 5 kgf a long distance in one direction, and 1 kgf a longer distance in the other direction?

force diagram for the left cantilever, showing that the spring force opposes 2 kgf of the gravity force on the weight, allowing the forces at the pivot to be in equilibrium

In real life the rod is not perfectly rigid. The diagram above shows the left cantilever arm: its left end is displaced downward by some distance $x$, producing a spring force $kx$ which can cancel out 2 kgf of the gravitational force, leaving 3 kgf at the pivot point. But this is not quite right: this analysis would predict that the rod would bend upward at the lighter weight, but this is not true (imagine a very flexible rod, like a fishing pole: it would bend downward at both weights, wouldn't it?) If you think about that for a while, you'll understand that what I said above — about the pivot dividing the forces equally between the two cantilever arms — must be wrong! Here's what's really going on:

redrawn force diagram for the entire mobile, showing that the spring force acts to oppose the entire mass of each weight at each end, and transfer that force to the pivot

The spring force due to the bending rod balances out the entire gravitational force of each weight, at both ends, and transfers it to the pivot; the pivot experiences 5 kgf of downward force due to the left-hand cantilever, and 1 kgf due to the right-hand cantilever. The wire connecting the pivot to the ceiling provides 6 kgf of upward force, leaving the entire system in equilibrium.

But wait: now we no longer have any explanation for why the rod isn't accelerating (angularly, about the pivot)! So let's think about what it would look like if the pivot weren't in the correct position for there to be no net torque.

enter image description here

I haven't redone the curve — it ought to be bent considerably more on the left, and less on the right — but I hope you can still see that there is now more than 5 kgf of downward force applied to the left side of the pivot, and less than 1 kgf on the right, so the left-hand side will start to accelerate downward. In effect, the spring forces have transfered some of the gravitational force due to the right-hand weight to the left side of the pivot!

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+1: nice answer –  Michael Brown Oct 26 '13 at 15:10

Perhaps it will help you if you reduce this to an edge case and build from there.

Say we have an (infinitesimally thin) 1kg load, resting right on top of an (infinitesimally thin) fulcrum. Do we even need a lever? No, it's balancing on its own. The fulcrum is providing 10N of force, straight up.

Now move the load an atom's breadth to the right. It's going to fall on its own because the fulcrum isn't directly under it any more, so we'll add a lever to extend the fulcrum's reach. But now there's the load on one side and nothing on the other, so we need to add something on the side opposite the load to keep it balanced. If I make the lever a meter long, do I need to put 10N of force on the other end to keep the load up? Obviously not, I barely need anything (0.1µg?). The fulcrum is still providing the full 10N to keep the load aloft, it just needs a little bit of an extension so it can provide that force directly under it (+𝜺 for the speck of dust).

Now slide the load over a mm. We need to extend the fulcrum's lifting force a bit more, so we slide the lever over too. The fulcrum is still providing all 10N of force we need to lift the load, the lever is still just extending the fulcrum's reach, we just need a slightly larger counterweight (~1g) to balance the torque.

OK, slide the load to 20cm. We need to extend the fulcrum's reach even further, so we slide the lever over as well. There's still an unbalanced torque on the lever, so to keep the load from falling we need to add some more counterweight (2.5N/250g). Despite the increase in counterweight, the fulcrum is still providing the upward force for the load, just from a longer distance.

We can repeat this process to any lever ratio, and it will always boil down to the same thing: the fulcrum is providing the upward force to keep the load in the air, and the counter-force on the lever arm is only there to keep the torques balanced so the fulcrum can do its job.

Don't think of a lever as amplifying the force you apply, think of it as a way to let the fulcrum do the lifting.

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Imagine you have a balance scale. This is a type of lever. You've got a 10lb object in one pan, and you are pushing down on the other one with 10 lb of force to keep it level. The weight and your hand together are pulling down on the pin at the center of the scale with 20 lb of force, same as if you had two 10 lb weights on each side.

Now imagine it is an unequal arm scale. The weight is 100 lb, and is only 1 inch away from the pin, and your hand is 10 inches away, still applying 10 lb of force. Since you are pushing down with 10 lb of force, and the object is pushing down with 100, the pin has to withstand 110 lb of force total.

Now make it an even more unequal arm scale. The weight is 5 feet away from the center pin, and you're almost a mile away. The weight is now 1000 lb, you are pushing down with a force of 1 lb, and the center pin is supporting 1001 lb.

If you've got a different kind of lever where you're pushing up, the fulcrum may be supporting 999 lb instead of 1001 lb, but it's still holding up the bulk of the weight, and will break away if it can't handle that much force.

(I don't know why other people are saying there's no conservation of force - it's not called that, but that's what Newton's Third Law is - nothing's accelerating, so all vectors sum to zero.)

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hyperphysics.phy-astr.gsu.edu/hbase/conser.html for conservation laws. Newton's third law always adds up to 0, there cannot be a force value other than 0 for an isolated system. In addition it does not work for situations where the fields carry momentum –  anna v Oct 26 '13 at 11:02

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