Mechanical Advantage of Classical Machines & Tools

Tools are a natural extension of our mastery of physics. By putting our knowledge to use, we are able to manipulate the world around us on a much larger scale. Tools and machines have allowed us to build great monuments, to settle otherwise inhospitable locations, and to launch ourselves into space.

But what defines a tool or machine? The simplest tools, such as the lever, offer us a mechanical advantage. Simple tools multiply the force we exert on a tool at the cost of increasing the distance we need to sustain the force. This trade-off occurs because simple tools (ideally) do not alter the amount of work done. Work is the product of distance and force, so by increasing the distance via a tool one can decrease the force required. The ratio between the force exerted by the tool and the force applied to the tool defines the mechanical advantage.

Renaissance scientists defined six classical simple machines or tools. One of the simplest is the lever.

The lever underpins many basic tools. Examples include using a claw hammer to extract nails, using a wheelbarrow to raise a heavier mass than you could normally lift, and swinging a hammer at a nail. These examples, respectively, also illustrate the three classes of levers, which depend on the position of the fulcrum and applied force.

As shown above, the mechanical advantage is the ratio of the length of the effort arm, the arm where the force applied, and the length of the load arm, where the mass we want to move is set. If the effort arm is four times the length of the load arm, we can lift a mass four times heavier than the force we apply. The cost of this is that the object will move 1/4 as much as the effort arm does.

Many other types of simple tools exist. The pulley system uses sets of pulleys to decrease the force needed to lift an object while increasing the length of rope to pull. As the number of pulleys increases, the mechanical advantage increases. This block and tackle system combined with levers underlies the mechanisms used by cranes for lifting heavy materials for building or excavating.

Another simple tool is the wedge. This tool allows one to split and lift objects. Examples include axe heads, chisels, and nails. They can also be used to hold objects still, such as a door wedge. In all cases a wedge derives its mechanical advantage from the ratio of its length to width. The thinner it is, the longer it takes to separate an object, but the less force is needed. Conversely when used to hold something fast, the longer the wedge is, the greater the frictional resistance to moving the object.

The wedge of course is related to the inclined plane, where rather than push the wedge into or under an object, the user pushes an object up the slope. This can greatly reduce the effort relative to lifting the object straight up. As with the wedge, the ratio of the length of the inclined plane to the height determines the mechanical advantage. Gentler slopes require less effort but take longer to reach the same height.

The screw is a familiar tool that coverts rotational motion, or motion in a circle, into linear motion. In the process, it reduces the force needed by spreading it out over the rotation of the screw. The ancient Greeks described it as an inclined plane wrapped around a cylinder. Its mechanical advantage is thus the ratio of the circumference of the screw to the height traverse by one rotation of the screw thread, also known as the lead. Wolfram|Alpha can tell us a great deal more about screws, from the details of a 1/4 inch bolt to the thread count of a coarse #6 bolt. A more comprehensive analysis of threads can be done using the threading formula.

The final classic tool is the wheel and axle. This tool combines features of the lever and the screw. Like the screw, it works with rotational force, but like the lever it takes a lesser force on the larger wheel and transfers it as a greater force about the much smaller axle. A classic example is the windlass, the device for raising a bucket of water from a well. The ratio of the radius of the wheel to the axle determines the mechanical advantage.

Run in reverse, the wheel and axle allows a powerful force, such as a car engine, to distribute the force across a much larger distance. This is one tool used to help cars and similar vehicles to reach high speeds. Another tool is the gear or rather the gear ratio.

Combinations of gears allow an engine or other power source to relay their output to a wheel without the need to adjust their normal rotational speed. This can be seen when shifting gears in a car or more transparently when adjust the gears on a modern bicycle. Gear ratios allow one to adjust the relative angular velocities of the wheel without making similar adjustments to the speed of the power source.

The machines we have just discussed were ideal, which means they lacked the imperfections, such as deflections or friction, of real devices. Working out the effects of friction on an inclined plane is a classic homework assignment for physics courses and a very real issue for anyone working with these tools. These effects cause a decrease in the actual mechanical advantage of these tools.

This leads to inefficiencies in converting the input force to the output force of the tool. The actual efficiency of these machines can be calculated by taking the ratio between actual mechanical advantage to the ideal mechanical advantage.

More complex machines are also limited by their efficiency. Some machines such as heat engines have strict efficiency limits due to the laws of thermodynamics. Such machines are limited in their maximum efficiency by the Carnot cycle. The Carnot cycle is an idealized process by which energy is extracted from a hot reservoir, such as a mass of boiling water, and converted into work while dumping waste heat into a cooler reservoir such as the ambient air.

The efficiency depends on the ratio of the cold reservoir to the hot reservoir. If the cold reservoir is 300 degrees Kelvin and the hot reservoir is 400 degrees Kelvin, the maximum efficiency is only 0.25 or 25 percent. As the ratio approaches zero, the efficiency approaches 1. But since one can not reach absolute zero, there will always be some inefficiency.

In the real world it is impossible to make an ideal Carnot cycle. Instead, other less ideal cycles are used. Examples include the Atkinson cycle (used in many hybrid electric cars), Brayton cycle (used in gas turbines and jet engines), Diesel cycle (used by diesel engines), Ericsson cycle, Otto cycle (which describes the typical spark ignition reciprocating piston engine), Rankine cycle (used in steam engines and thus solar thermal, coal, and nuclear power plants), and Stirling cycle.

One can run these cycles in reverse, using work to remove heat from the cold reservoir and heating the hot reservoir. This refrigeration cycle is behind refrigerators, freezers, and air conditioning units. For those interested in the latter, Wolfram|Alpha allows one to work out how much air conditioning a given home needs.

Humanity has developed many tools and machines for overcoming our limitations and multiplying our productivity. Wolfram|Alpha is one of those tools, allowing us to easily access and compute complex formulas wherever we may be.