When someone asks you to consider the concept of a ​machine​ in the 21st century, it's a virtual given that whatever image leaps into your mind involves electronics (e.g., anything with digital components) or at least something powered by electricity.

Failing that, if you are a fan of, say, 19th-century American westward expansion toward the Pacific Ocean, you may think of the locomotive steam engine that powered trains in those days – and represented a genuine marvel of engineering at the time.

In reality, ​simple machines​ have existed for hundreds and in some cases thousands of years, and none of them require high-tech assembly or power outside of what the person or people using them can supply. The aim of these various types of simple machines is the same: to generate additional ​force​ at the expense of ​distance​ in some form (and maybe a little time, too, but that's quibbling).

If that sounds like magic to you, it's probably because you're confusing force with ​energy,​ a related quantity. But while its true that energy cannot be "created" in a system except from other forms of energy, the same is not true of force, and the simple reason for this and more await you.

Before tacking how objects are used to move other objects about in the world, it's good to have a handle on basic terminology.

In the 17th century, Isaac Newton began his revolutionary work in physics and mathematics, one culmination of which was Newton introducing his three fundamental laws of motion. The second of these states that a net ​force​ acts to accelerate, or change the velocity of, masses: ​F_net​ = m​a​.

• It can be shown that in a closed system at ​equilibrium​ (i.e., where the velocity of anything that happens to be moving is not changing), the sum of all forces and torques (forces applied about an axis of rotation) are zero.

When a force moves an object through a displacement d, ​work​ is said to have been done on that object:

The value of work is positive when the force and the displacement are in the same direction, and negative when it is in the other direction. Work has the same unit as energy does, the meter (also called the joule).

Energy is a property of matter that manifests in many ways, in both moving and "resting" forms, and importantly, it is conserved in closed systems in the same way force and momentum (mass times velocity) are in physics.

Clearly, humans need to move things, often long distances. It is useful to be able to keep distance high yet force – which requires human power, which was all the more glaring in pre-industrial times – somehow low. The work equation appears to allow for this; for a given amount of work, it should not matter what the individual values of F and d are.

As it happens, this is the principle behind simple machines, although often not with the idea of maximizing the distance variable. All six classical types (the ​lever,​ the ​pulley​, the ​wheel-and-axle​, the ​inclined plane​, the ​wedge​ and the ​screw​) are used to reduce applied force at the cost of distance to do the same amount of work.

The term "mechanical advantage" is perhaps more alluring than it should be, as it almost seems to imply that physics systems can be gamed to extract more work without a corresponding input of energy. (Because work has units of energy and energy is conserved in closed systems, when work is done, its magnitude has to equal the energy put into whatever motion occurs.) Sadly, this is not the case, but ​mechanical advantage (MA)​ still offers some fine consolation prizes.

For now, consider two opposing forces F₁ and F₂ acting about a pivot point, called a ​fulcrum​. This quantity, ​torque​, is computed simply as the magnitude and direction of the force multiplied by the distance L from the fulcrum, known as the ​lever arm​: ​T = F​​L​. If the forces F₁ and F₂ are to be in balance, ​T₁​ must be equal in magnitude to ​T₂​, or

This can also be written ​F₂/F₁ = L₁/L₂​. If F₁ is the ​input force​ (you, someone else or another machine or source of energy) and F₂ is the ​output force​ (also called the load or the resistance), then the higher the ratio of F2 to F1, the higher the mechanical advantage of the system, because more output force is generated using comparatively little input force.

The ratio ​F₂/F₁,​ or perhaps preferably ​F_o/F_i,​ is the equation for MA. In introductory problems, it is usually called ideal mechanical advantage (IMA) because the effects of friction and air drag are ignored.

From the above information, you now know what a basic lever consists of: a ​fulcrum,​ an ​input force​ and a ​load​. Despite this bare-bones arrangement, levers in human industry come in remarkably diverse presentations. You probably know that if you use a pry bar to move something that offers few other options, you've used a lever. But you've also used a lever when you've played the piano or used a standard set of nail clippers.

Levers can be "stacked" in terms of their physical arrangement such a way so that their individual mechanical advantages sum up to something even greater for the system of as a whole. This system is called a compound lever (and has a partner in the pulley world, as you'll see).

It is this multiplicative aspect of simple machines, both within individual levers and pulleys and between different ones in a compound arrangement, that makes simple machines worth whatever headaches they may occasionally cause.

A ​first-order lever​ has the fulcrum between the force and the load. An example is a "​see-saw​" on a school playground.

A ​second-order lever​ has the fulcrum at one end and the force at the other, with the load in between. The ​wheelbarrow​ is the classic example.

A ​third-order lever,​ like a second-order lever, has the fulcrum at one end. But in this case, the load is at the other end and the force is applied somewhere in between. Many sporting implements, such as baseball bats, represent this class of lever.

The mechanical advantage of levers can be manipulated in the real world with strategic placements of the three requisite elements of any such system.

Your body is loaded with interacting levers. One example is the bicep. This muscle attaches to the forearm at a point between the elbow (the "fulcrum") and whatever load is being borne by the hand. This makes the bicep a third-order lever.

Less self-evidently perhaps, the calf muscle and Achilles tendon in your foot act together as a different sort of lever. As you walk and roll forward, the ball of your foot acts as a fulcrum. The muscle and tendons exert upward and forward force, counteracting your body weight. This is an example of a second-order lever, like a wheelbarrow.

A car with a mass of 1,000 kg, or 2,204 lb (weight: 9,800 N) is perched on the end of a very rigid but very light steel rod, with a fulcrum placed 5 m from the center of mass of the car. A person with a mass of 5- kg (110 lb) says she can counterbalance the weight of the car by herself by standing on the other end of the rod, which can be extended horizontally for as long as is needed. How far from the fulcrum must she be to achieve this?

Balance of forces require that F₁L₁ = F₂L₂, where F1 = (50 kg)(9.8 m/s²) = 490 N, F₂ = 9.800 N, and L2 = 5. Thus L1 = (9800)(5)/(490) = ​100 m​ (a little longer than a football field).

A pulley is a kind of simple machine that, like the others, has been in use in various forms for thousands of years. You've probably seen them; they can be fixed or movable, and include a rope or cable wound around a rotating circular disk, which has a groove or other means of keeping the cable from slipping sideways.

The main advantage of a pulley is not that it boosts MA, which remains at the value of 1 for simple pulleys; it is that it can change the direction of an applied force. This might not matter much if gravity weren't in the mix, but because it is, virtually every human engineering problem involves fighting or leveraging it in some way.

A pulley can be used to lift heavy objects with relative ease by making it possible to apply force in the same direction gravity acts – by pulling down. In such situations, you can also use your own body mass to help raise the load.

As noted, since all a simple pulley does is change the direction of the force, its utility in the real world, while considerable, is not maximized. Instead, systems of multiple pulleys with different radii can be used to multiply applied forces. This is done through the simple act of making more rope necessary, since F_i falls as d rises for a fixed value of W.

When one pulley in a chain of them has a larger radius than the one that follows it, this creates a mechanical advantage in this pair that is proportional to the difference in the value of the radii. A long array of such pulleys, called a ​compound pulley​, can move very heavy loads – just bring plenty of rope!

A crate of recently arrived physics textbooks weighing 3,000 N is lifted by a dock worker, who pulls with a force of 200 N on a pulley rope. What is the mechanical advantage of the system?

This problem really is as simple as it looks; ​F_o/F_i​ = 3,000/200 = ​15.0.​ The point is to illustrate what remarkable and powerful inventions simple machines, despite their antiquity and lack of electronic glitz, truly are.

You can treat yourself to online calculators that let you experiment with a wealth of different inputs in terms of lever types, relative lever-arm lengths, pulley configurations and more so you can gain hands-on feel for how the numbers in these kinds of problems play. An example of such a handy tool can be found in the Resources.