Simple Machines: Gear

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: A gear is a wheel with teeth on its outer edge, meant to interlock with other gears. Multiple gears locked together form a gear train. This is a simple machine that can be used to transmit, redirect, and amplify an input force. Gear trains are an extremely common component in modern machines, including many engine systems.

PRINCIPAL TERMS

• actual mechanical advantage: the ratio comparing the input force of a machine to the output force, taking into account friction and other factors that limit the efficiency of real-world machines. A mechanical advantage of more than one indicates an amplification of force.
• efficiency: the measure of how effective a machine is at transforming or transferring energy, quantified as the ratio of the actual performance of the machine to an idealized, theoretical version of the same machine. A perfect machine would have an efficiency value of one.
• ideal mechanical advantage: the ratio comparing the input force of a machine to the output force, ignoring friction and other factors that limit the efficiency of real-world machines. A mechanical advantage of more than one indicates an amplification of force.
• joule: abbreviated J, the International System of Units unit of work and energy. One joule is equal to the work done by a force of one newton acting across a distance of one meter.
• net force: the sum of all of the forces acting on an object. Forces with equal magnitude but opposite directions will cancel each other. An object moves in the direction of the net force acting on it.
• newton: abbreviated N, the International System of Units unit of force. One newton is equal to the force required to accelerate a one-kilogram mass at one meter per second per second.
• power: the rate of work (energy transfer) over time. The International System of Units unit of power is the watt (W), which is equal to one joule of work or energy per second (J/s).
• work: a force successfully moving an object, or the successful transfer of energy. The International System of Units unit of work is the joule.

What Is a Gear?

A gear is 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.

What Is a Simple Machine?

Simple machines are devices which make tasks—typically, moving some target object—easier by redirecting or amplifying an input force into a new output force. Gears, such as those found in cars and bicycles, help to make work (especially involving movement) easier by increasing or reducing speed or force as well as changing rotational direction. These more complex machines use different-sized gears that, when meshed together, alter speed and force.

Generally, simple machines differ from other, more complicated 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. 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). Simple machines often act as the building blocks for more complex machines. For instance, a hand-crank can opener works using wedges (the cutting edge), wheels and axles (the crank), levers (the grips), and gears.

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. 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. 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. Power, measured in watts (W), is simply the rate of work over time. One watt is equivalent to one joule per second (J/s).

Work (w) is equal to the strength of the force (F) applied times the absolute distance the object is moved from its original position (displacement, s) times the cosine of the angle between the force and displacement (θ):

w = F · s · cosθ

This formula is useful for understanding the force/distance trade-off inherent to the way machines work. In order to keep the work value constant, a simple machine which amplifies force via mechanical advantage must also reduce the displacement (total distance moved) caused by that force. 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

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, for example. In the real world, a distinction is made between ideal mechanical advantage and actual mechanical advantage. The former assumes a perfect machine, unimpeded by friction or design flaws, which transmits and transforms a given force perfectly. The latter is based on actual measurements of the ratio of input to output forces. The mechanical advantage (MA) of a two-gear train is equal to the ratio of the size of the two gears, written as:

MA = d_out / d_in

The difference between a theoretically perfect machine and its real-world counterpart is referred to as its efficiency, which 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 to the ideal mechanical advantage. In the real world, gears lose efficiency from the friction between their meshed teeth.

Sample Problem

A student is building a simple gear train as an engineering experiment. She has access to a board on which she can anchor the gears, a small hand-crank she can attach to a gear of her choosing, and two gears, one large and one small. The large gear has a diameter of 30 centimeters, and the small gear a diameter of 15 centimeters. These gears are exceptionally well designed, so friction can be ignored. What is the mechanical advantage if the small gear is the input gear? What if the large gear is the input gear?

Answer:

Recall that the mechanical advantage (MA) of a two-gear train is equal to the ratio of the size of the two gears. In this case, diameter is given (which is equivalent to the gear’s number of teeth), so the diameter of the input gear is (di_n) and the output gear is (d_out). If the small gear is the input gear, then:

MA = d_out / d_in = d_large/ d_small

MA = 30 cm / 15 cm

MA = 2

Reversing the arrangement so that the large gear is the input gives:

MA = d_out / d_in = d_small/ d_large

MA = 15 cm / 30 cm

MA = 0.5

The train works via the force-distance relationship. When the small gear is the input, turning it results in reduced motion (rotational speed) but increased force (mechanical advantage) at the large gear. When the large gear is the input, the force at the small gear is reduced but the gear moves more (spins faster).

Gears Are Ubiquitous

Simple machines like gears permeate every aspect of daily life. 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.

More importantly, the principles of power, work, force, and mechanical advantage that govern the gear train, along with the rest of the simple machines, form the basis for a deeper understanding of much of the more complex machinery one may need to interact with.

Bibliography

Coolman, Robert. "What Is Classical Mechanics?" Live Science. Purch, 12 Sept. 2014. Web. 20 July 2015.

"The Elements of Machines." Inventor’s Toolbox. Museum of Science, 1997. Web. 20 July 2015.

Henderson, Tom. "Kinematics." Motion in Two Dimensions. N.p.: Physics Classroom, 31 Aug. 2012. Digital file.

Khan, Sal. "Mechanical Advantage." Khan Academy. Khan Academy, n.d. Web. 20 July 2015.

Simanek, Donald. "Kinematics." Brief Course in Classical Mechanics. Lock Haven U, Feb. 2005. Web. 20 July 2015.

"Simple Machine." Encyclopaedia Britannica. Encyclopaedia Britannica, 26 Aug. 2014. Web. 20 July 2015.