The mighty mathematics of the lever

A famous Ancient Greek once said, “Give me a place to stand, and I shall move the Earth.” But this wasn’t some wizard claiming to perform impossible feats. It was the mathematician Archimedes describing the fundamental principle behind the lever. The idea of a person moving such a huge mass on their own might sound like magic, but chances are you’ve seen it in your everyday life. One of the best examples is something you might recognize from a childhood playground: a teeter-totter, or seesaw. Let’s say you and a friend decide to hop on. If you both weigh about the same, you can totter back and forth pretty easily.

But what happens if your friend weighs more? Suddenly, you’re stuck up in the air. Fortunately, you probably know what to do. Just move back on the seesaw, and down you go. This may seem simple and intuitive, but what you’re actually doing is using a lever to lift a weight that would otherwise be too heavy. This lever is one type of what we call simple machines, basic devices that reduce the amount of energy required for a task by cleverly applying the basic laws of physics. Let’s take a look at how it works.

Every lever consists of three main components: the effort arm, the resistance arm, and the fulcrum. In this case, your weight is the effort force, while your friend’s weight provides the resistance force. What Archimedes learned was that there is an important relationship between the magnitudes of these forces and their distances from the fulcrum. The lever is balanced when the product of the effort force and the length of the effort arm equals the product of the resistance force and the length of the resistance arm.

This relies on one of the basic laws of physics, which states that work measured in joules is equal to force applied over a distance. A lever can’t reduce the amount of work needed to lift something, but it does give you a trade-off. Increase the distance and you can apply less force. Rather than trying to lift an object directly, the lever makes the job easier by dispersing its weight across the entire length of the effort and resistance arms. So if your friend weighs twice as much as you, you’d need to sit twice as far from the center as him in order to lift him. By the same token, his little sister, whose weight is only a quarter of yours, could lift you by sitting four times as far as you.

Seesaws may be fun, but the implications and possible uses of levers get much more impressive than that. With a big enough lever, you can lift some pretty heavy things. A person weighing 150 pounds, or 68 kilograms, could use a lever just 3.7 meters long to balance a smart car, or a ten meter lever to lift a 2.5 ton stone block, like the ones used to build the Pyramids. If you wanted to lift the Eiffel Tower, your lever would have to be a bit longer, about 40.6 kilometers. And what about Archimedes’ famous boast?

Sure, it’s hypothetically possible. The Earth weighs 6 x 10^24 kilograms, and the Moon that’s about 384,400 kilometers away would make a great fulcrum. So all you’d need to lift the Earth is a lever with a length of about a quadrillion light years, 1.5 billion times the distance to the Andromeda Galaxy. And of course a place to stand so you can use it. So for such a simple machine, the lever is capable of some pretty amazing things.

And the basic elements of levers and other simple machines are found all around us in the various instruments and tools that we, and even some other animals, use to increase our chances of survival, or just make our lives easier. After all, it’s the mathematical principles behind these devices that make the world go round.

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