Levers are all around us ... and within us, since the basic physical principles of the lever are what allow our tendons and muscles to move our limbs - with bones acting as the beams and joints acting as the fulcrums.
Archimedes (287 - 212 B.C.E.) once famously said "Give me a place to stand, and I shall move the Earth with it" when he uncovered the physical principles behind the lever. While it would take a heck of a long lever to actually move the world, the statement is correct as a testament to the way it can confer a mechanical advantage.
[Note: The above quote is attributed to Archimedes by the later writer, Pappus of Alexandria. It's likely that he never actually ever said it.]
How do they work? What are the principles that govern their movements?
How Levers Work
A lever is a simple machine that consists of two material components and two work components:
- A beam or solid rod
- A fulcrum or pivot point
- An input force (or effort)
- An output force (or load or resistance)
The beam is placed so that some part of it rests against the fulcrum. In a traditional lever, the fulcrum remains in a stationary position, while a force is applied somewhere along the length of the beam. The beam then pivots around the fulcrum, exerting the output force on some sort of object that needs to be moved.
The ancient Greek mathematician and early scientist Archimedes is typically attributed with having been the first to uncover the physical principles governing the behavior of the lever, which he expressed in mathematical terms.
The key concepts at work in the lever is that since it is a solid beam, then the total torque into one end of the lever will manifest as an equivalent torque on the other end. Before getting into the how to interpret this as a general rule, let's look at a specific example.
Balancing on a Lever
The picture above shows two masses balanced on a beam across a fulcrum.
In this situation, we see that there are four key quantities that can be measured (these are also shown in the picture):
- M_{1} - The mass on one end of the fulcrum (the input force)
- a - The distance from the fulcrum to M_{1}
- M_{2} - The mass on the other end of the fulcrum (the output force)
- b - The distance from the fulcrum to M_{2}
This basic situation illuminates the relationships of these various quantities. (It should be noted that this is an idealized lever, so we're considering a situation where there is absolutely no friction between the beam and the fulcrum, and that there are no other forces that would throw the balance out of equilibrium, like a breeze.)
This set up is most familiar from the basic scales, used throughout history for weighing objects. If the distances from the fulcrum are the same (expressed mathematically as a = b) then the lever is going to balance out if the weights are the same (M_{1} = M_{2}). If you use known weights on one end of the scale, you can easily tell the weight on the other end of the scale when the lever balances out.
The situation gets much more interesting, of course, when a does not equal b, and so from here on out we'll assume that they don't. In that situation, what Archimedes discovered was that there is a precise mathematical relationship - in fact, an equivalence - between the product of the mass and the distance on both sides of the lever:
M _{1} a = M _{2} b
Using this formula, we see that if we double the distance on one side of the lever, it takes half as much mass to balance it out, such as:
a = 2 b
M _{1} a = M _{2} b
M _{1}(2 b) = M _{2} b
2 M _{1} = M _{2}
M _{1} = 0.5 M _{2}
This example has been based upon the idea of masses sitting on the lever, but the mass could be replaced by anything that exerts a physical force upon the lever, including a human arm pushing on it. This begins to give us the basic understanding of the potential power of a lever. If 0.5 M_{2} = 1,000 lb., then it becomes clear that you could balance that out with a 500 lb. weight on the other side, just by doubling the distance of the lever on that side. If a = 4b, then you can balance 1,000 lb. with only 250 lbs. of force.
This is where the term "leverage" gets its common definition, often applied well outside the realm of physics: using a relatively smaller amount of power (often in the form of money or influence) to gain a disproportionately greater advantage on the outcome.
Types of Levers
When using a lever to perform work, we focus not on masses, but on the idea of exerting an input force on the lever (called the effort) and getting an output force (called the load or the resistance). So, for example, when you use a crowbar to pry up a nail, you are exerting an effort force to generate an output resistance force, which is what pulls the nail out.
The four components of a lever can be combined together in three basic ways, resulting in three classes of levers:
- Class 1 Levers: Like the scales discussed above, this is a configuration where the fulcrum is in between the input and output forces.
- Class 2 Levers: The resistance comes between the input force and the fulcrum, such as in a wheelbarrow or bottle opener.
- Class 3 levers: The fulcrum is on one end and the resistance is on the other end, with the effort in between the two, such as with a pair of tweezers.
Each of these different configurations has different implications for the mechanical advantage provided by the lever. Understanding this involves breaking down the "law of the lever" that was first formally understood by Archimedes.
Law of the Lever
The basic mathematical principles of the lever is that the distance from the fulcrum can be used to determine how the input and output forces relate to each other. If we take the earlier equation for balancing masses on the lever and generalize it to an input force (F_{i}) and output force (F_{o}), we get an equation which basically says that the torque will be conserved when a lever is used:
F_{i}a = F_{o}b
This formula allows us to generate a formula for the "mechanical advantage" of a lever, which is the ratio of the input force to the output force:
Mechanical Advantage = a/ b = F_{o}/ F_{i}
In the earlier example, where a = 2b, the mechanical advantage was 2, which meant that a 500 lb. effort could be used to balance a 1,000 lb. resistance.
The mechanical advantage depends upon the ratio of a to b. For class 1 levers, this could be configured in any way, but class 2 and class 3 levers put constraints on the values of a and b.
- For a class 2 lever, the resistance is between the effort and the fulcrum, meaning that a < b. Therefore, the mechanical advantage of a class 2 lever is always greater than 1.
- For a class 3 lever, the effort is between the resistance and the fulcrum, meaning that a > b. Therefore, the mechanical advantage of a class 3 lever is always less than 1.
A Real Lever
The equations represent an idealized model of how a lever works. There are two basic assumptions that go into the idealized situation which can throw things off in the real world:
- The beam is perfectly straight and inflexible
- The fulcrum has no friction with the beam
Even in the best real world situations, these are only approximately true. A fulcrum can be designed with very low friction, but it will almost never reach a friction of zero in a mechanical lever. As long as a beam has contact with the fulcrum, there will be some sort of friction involved.
Perhaps even more problematic is the assumption that the beam is perfectly straight and inflexible. Recall the earlier case where we were using a 250 lb. weight to balance a 1,000 lb. weight. The fulcrum in this situation would have to support all of the weight without sagging or breaking. It depends upon the material used whether this assumption is reasonable.
Understanding levers is useful in a variety of areas, ranging from technical aspects of mechanical engineering to developing your own best bodybuilding regimen.