Fields of Study

Classical Mechanics

Summary

Simple machines are mechanical devices that make tasks—typically, moving an object— easier by redirecting or amplifying an applied force or by increasing the object’s velocity or displacement. The classical simple machines include the lever, wheel and axle, pulley, inclined plane, wedge, and screw. Gear trains are also sometimes classified as simple machines, and they can be thought of as having the properties of a lever (movement about a fixed fulcrum) and a wheel and axle (transfer of rotational movement).

Principal Terms

What Is a Simple Machine?

Simple machines are devices that make tasks—typically, moving some target object—easier by redirecting or amplifying some input force into a new output force. A lever is a very simple and effective machine, consisting of nothing more than a rigid plane and a fulcrum on which the lever is balanced. The ancient Greek mathematician Archimedes (ca. 287–212 BCE) came up with the concept of the simple machine. He is famous for having said that, with a place to stand and a lever, he could move the entire world.

Generally, simple machines are distinguished from other, more complex machines by virtue of being the simplest possible ways of generating mechanical advantage—that is, multiplying a force. The six classical simple machines are the lever, wheel and axle, pulley, inclined plane, wedge, and screw. The gear is also a simple machine, often described as a type of special lever combined with a wheel and axel.

In many instances, more complex machines—sometimes called “compound machines”—can be thought of as being an assembly of several simple machines. For instance, a hand-crank can opener works using wedges (the cutting edge), wheels and axles (the crank), and levers (the grips).

Inclined plane—a flat, rigid surface raised at an angle, in other words, a ramp. The ramp that extends behind some moving trucks is a classic example of an inclined plane. A ramp directs and amplifies a forward push into the lifting or lowering of heavy objects, but it reduces distance. It helps move objects vertically by making it easier to push or pull an object across its surface (not acting directly against gravity), but an object must be moved farther as a result.

Lever—a tool that redirects and amplifies (or reduces) an input force. It consists of a stiff plank, bar, or rod—anything straight and rigid—balanced over a fulcrum. The fulcrum is usually wedge shaped or round, but anything that the lever can be pushed or pulled against can work. An impromptu lever can easily be made by laying a sturdy stick across a round rock. Even an action as simple as using a screwdriver to pry open a paint can lid is actually a form of lever use. The rim of the can serves as a fulcrum for the steel rod of the screwdriver. Three types of levers exist and depend upon the location of the fulcrum with respect to the load and the effort. In a first-class lever, the fulcrum is between the load and the effort, as in a see-saw. The second-class lever places the load between the fulcrum and the effort. Lastly, in a third-class lever, the effort is between the load and the fulcrum. Pressing down on a lever with a fulcrum near the target force will result in an amplified upward force.

Pulley—uses one or more wheels (often grooved to keep the rope from slipping) and rope to redirect or amplify an input force. The simplest version consists of little more than a rope draped over a surface that it can slide easily over, such as a smooth metal hook, and serves only to redirect force. For instance, a rope thrown over a rafter in a barn will allow a person to pull down on one end of the rope, aided by gravity, to lift a load upward at the other. This is an impromptu example of a simple pulley system, with the rafter acting as an anchored pulley. More complex arrangements with moving pulleys produce a mechanical advantage, amplifying an input force. In exchange, the person applying the force has to move a greater amount of rope. A two-pulley system, with one anchored and one able to move freely, is the simplest possible way of arranging ropes over wheels to produce a mechanical advantage. The more movable pulleys there are in a system, the greater the advantage.

Screw—an inclined plane with a spiraling groove wrapped around a cylinder, typically with a pointed edge. Screws amplify force and direct a circular force into a linear force along the line of the central column. For example, turning a screw into wood with a screwdriver turns the circular motion of a wrist into linear motion into the wood. Screws are generally used to secure objects tightly together, as in common household construction projects. Other uses for screws include clamping or crushing objects (as in a vice), excavating holes (a drill), and even moving air (a fan). Screws work by using the inclined plane to redirect and increase a turning force (torque) applied to the cylinder into a vertical force parallel to the cylinder.

Wedge—a triangular object with one thick, flat end and two sloped sides that come to a point at the other end made from two inclined planes fused together along their bottoms to create a sharp point. Wedges amplify and redirect an input force applied to their thick end into an amplified force pushing out to either side of the sharp end, perpendicular to the input force. Thus, wedges are often used to cut or pry things apart. Axe heads, for instance, amplify the downward force of a swing and redirect it to either side as the sharp end of the head is driven into wood, forcing the wood apart and chopping it in two. Most cutting surfaces work the same way, but on different scales. Other examples of wedges also railroad spikes, and chisels. Wedges may also be used to secure objects in place, as with railroad spikes or doorstops.

Wheel and axle— a broad disk mounted on a stiff rod so that when one rotates, the other does too. Typically, the rod is attached to the center of the disk. When the wheel spins, the force is amplified and transmitted to the axle. Spinning the wheel transfers the motion to the axle. If the wheel is bigger than the axle, then the force is reduced but the axle moves faster. If the wheel is smaller than the axle, then the force is amplified but the axle spins more slowly. Spinning the axle rather than the wheel reduces the force but moves the wheel a greater distance. Note that gears can be considered an extension of the same basic ideas as the wheel and axle.

Gears—a wheel with teeth around the outer edge. These teeth mesh with the teeth of another gear, or with a chain that meshes with the teeth of another gear. Two or more gears connected in this way form a gear train. This is a simple machine that takes the rotational force applied to the input gear and transfers, redirects, and amplifies or reduces it. This depends on the relative sizes of the gears and the direction of the force transfer. Gears are commonplace in all manner of machinery, particularly those involving a rotational motor. The main types of gears are bevel, spur, rack and pinion, and worm.

Power and Work Remain Constant

A force is said to do work if it moves an object. An object will move if the net force on it—the sum of all forces acting upon it—results in a positive force in any direction. If the net force acting on an object is positive in any direction, the object will move in that direction. Therefore, a person standing absolutely still on the surface of the earth is experiencing a net force of zero. The force of gravity is performing no work on him or her. If a person is falling straight down, the net force is positive in the direction of gravity and the gravitational force of the earth is doing work.

Simple machines are used many ways in our lives. The tail of a hammer is a lever; a ramp is an inclined plane; pulleys are often used to lift heavy objects; a bicycle has many simple machines including a wheel and axle; a bench vise works because of a screw; to split wood, you might use a mallet and wedge; and a power drill chuck and chuck key are a gear system.

A heavy box sitting on a ramp experiences a net force of zero. However, if somebody applies a horizontal force to the box, sliding it upward and away, the net force will be positive in the diagonal direction of movement, because an object moving up or down a ramp always experiences a net force parallel to the incline.

A heavy bale of hay resting on the floor of a barn experiences a net force of zero. If somebody attaches a pulley system and starts lifting the bale upward, however, the net force is positive in the direction of movement.

An axe (wedge) lying flat on a log experiences a net force of zero. If someone swings the axe straight down, the net force is positive in the direction of gravity, and the gravitational force of the earth is doing work on the axe.

In the International System of Units (SI), work and energy are both measured in joules (J). One joule is equal to the work performed (or energy transferred) when a force of one newton (N) moves something a distance of one meter. Since work is what happens when energy is transferred, the law of conservation of energy applies.

The work input into a machine must equal the work output because the energy on either end of the machine must also remain constant. Power, measured in watts (W), is simply the rate of work over time, and it must also remain equal on either side of a machine. One watt is equivalent to one joule per second (J/s).

In physics, work (W) is equal to the product of the strength of the force (F) applied, the displacement of the object from its original position (s), and the cosine of the angle between the directions of force and displacement (θ)

w = F · s · cosθ.

This formula is useful for understanding the force/distance tradeoff inherent to the way machines work. For the same fundamental reasons that energy can only be transformed, not created or destroyed, the total work performed at either end of a simple machine must remain constant. (In other words, “work” and “energy transfer” are essentially the same thing.) To keep the work value constant, a simple machine that amplifies force via mechanical advantage must also reduce the displacement (total distance) caused by that force. Force and displacement are inversely related—increasing one by a certain factor will decrease the other by the same factor.

The formula for work, above, helps one understand the force-distance trade-off inherent in simple machines. For the same reasons that energy can only be transformed, not created or destroyed, the total work done at either end of a simple machine must remain constant. To keep the work value constant, a simple machine that amplifies force via mechanical advantage must reduce the displacement caused by that force. (The angle between the force and displacement determines whether that force caused the displacement.)

The relationship between the two forces acting on a simple machine, effort and load, is called the mechanical advantage (MA). Theoretical mechanical advantage for each simple ma-chine is calculated differently. Levers are calculated by the ratio of effort arm length (d₁) to load arm length (d₂). A wheel and axle is calculated by the ratio of wheel radius (r_w) to axle radius (r_a). Mechanical advantage of inclined planes is calculated by the ratio of length (L) to height (h). Pulleys are calculated by multiplying the number of moveable pulleys by two. A wedge is calculated similar to the inclined plane as a ratio of length (L) to width (w). The mechanical advantage of a screw is calculated by a ratio of the screw circumference (πd) to thread lead (L). And a gear system is calculated by the ratio of the driven gear’s number of teeth to the driver gear’s number of teeth.

Pushing a box up a ramp, for example, will not move the box as far as the same push would along flat ground, but the push will carry greater force. A pulley system might enable a farmer to lift a heavy hay bale, but the bale will be displaced a shorter distance per unit of force applied. If a smaller driver gear is meshed with a larger gear, the smaller gear turns at a faster rate and in the opposite direction of the larger gear but with less force.

Imperfect Machines

A perfect machine would be a machine unimpeded by friction or design flaws, capable of transmitting a force perfectly. In the real world, there is no perfect machine. Even the simplest machines fail to transmit forces perfectly; some energy is always lost to friction. Thus, a distinction is made between ideal mechanical advantage (which assumes a perfect machine) and actual mechanical advantage (to take into account friction and other forces).

The difference between a theoretically perfect machine and its real-world counterpart is measured in terms of its efficiency. Efficiency is the ratio of the actual, measured performance of a machine to its theoretically perfect performance. In other words, efficiency is the ratio of the actual mechanical advantage to the ideal mechanical advantage. A perfect machine would have an efficiency value of 1.

Wedges lose efficiency due to friction between the sloping sides and the surfaces they move against. Gears lose efficiency from the friction between their meshed teeth. In the case of ramps, the smoother the ramp, the easier an object would move on it, as less energy would be wasted overcoming the friction between them. Unfortunately, less friction would also mean that the load could slide down the ramp more easily, negating the work.

Calculating Mechanical Advantage in Simple Machines

When a machine amplifies the force put into it, it is said to provide a mechanical advantage. Perhaps the simplest way of generating mechanical advantage is with a lever—a simple machine consisting of a stiff plane balanced on a fulcrum. A playground see-saw is a classic lever. Archimedes (ca. 287–212 BCE), an ancient Greek mathematician and engineer, proved the law of the lever. This law states that the mechanical advantage of a lever is dependent on the relative position of the fulcrum. If it is closer to the output end (where the load is), then the lever will produce a mechanical advantage. If closer to the input, it will instead reduce the input force. The mechanical advantage offered by a lever is directly related to how close the fulcrum is to the output end.

In the real world, there is no such thing as a perfect machine. Even the simplest machines do not transmit forces perfectly; some is lost to friction, or resistance, between moving surfaces. The actual mechanical advantage of a machine is measured against the ideal mechanical advantage, or the theoretically perfect performance, of the same machine. The degree to which a machine achieves its ideal mechanical advantage is its efficiency.

Simple Machines Are Everywhere

Simple machines permeate every aspect of everyday life. One of the first uses of the screw was to transfer water. According to legend, the Archimedes’ screw was developed by famed Greek inventor Archimedes (ca. 287–212 BCE) on a visit to Egypt as a method of lifting water into irrigation ditches. Also called a “screw pump,” the device is a large screw with a broad thread fitted tightly inside a pipe. As the screw is turned, the threads pull water upward. Similar devices are still used to move water, grain, and other substances.

The “pulling” action of a screw can also be seen in the propellers of ships and propeller planes. These spin blades act like the threads on a screw. Indeed, Leonardo da Vinci (1452–1519) designed a helicopter-like device that used a large, broad-threaded screw to be spun by hand using a lever and to “pull” the device upward into the air. (Unfortunately, it did not work.)

Almost every cutting tool, from a knife to a sword to a pair of scissors, uses a wedge to push apart the surface it is cutting. Wedges can also be used to pin things in place by being driven into a material; instead of splitting the material, the amplified force allows the wedge to be driven securely into place. Old iron railroad spikes are simple bars with a wedged end that were used to hold rail tracks in place.

Gears are extremely common, albeit often hidden from view. Any motor-powered device almost certainly uses a gear train to transfer and amplify the force generated by the motor. Electric screwdrivers do so, for example, as do combustion engines.

Every staircase is like an unevenly built ramp, designed to let people ascend without tiring by moving diagonally instead of straight upward against gravity. Ramps have been used since antiquity to help move large loads and are common in construction and shipping.

Pulleys are common, especially in industries such as shipping, where large loads often need to be lifted. Some examples of pulleys are less obvious. Consider the hook-and-loop strap on a winter glove’s cuff. The strap goes through a slot and bends back over itself; pulling on the end of the strap makes the plastic slot act as a tiny movable pulley

Due to the simple construction of the wheel and axle and its easy amplification of force, it has found countless applications throughout modern society. A mechanical winch used to lift a bucket from a well is a wheel-and-axle system, while a doorknob acts as a wheel and axle to unlatch a door. Numerous motor-powered devices feature wheels and axles connected to gear-and-pulley systems to transmit and redirect the force of the motor to its desired ends.

More importantly, the principles of power, work, force and mechanical advantage, along with the basic structures of classical simple machines, form the basis for a deeper understanding of much of the more complicated machinery one may need to interact with, such as automobile engines.