Frictionless plane
Encyclopedia
The frictionless plane is a concept from the writings of Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

. In his 1608 The Two New Sciences, Galileo presented a formula that predicted the motion of an object moving down an inclined plane
Inclined plane
The inclined plane is one of the original six simple machines; as the name suggests, it is a flat surface whose endpoints are at different heights. By moving an object up an inclined plane rather than completely vertical, the amount of force required is reduced, at the expense of increasing the...

. His formula was based upon his past experimentation with free-falling bodies. However, his model was not based upon experimentation with objects moving down an inclined plane, but from his conceptual modeling of the forces acting upon the object. Galileo understood the mechanics of the inclined plane as the combination of horizontal and vertical vectors; the result of gravity acting upon the object, diverted by the slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

 of the plane.

However, Galileo's equations do not contemplate friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

, and therefore do not perfectly predict the results of an actual experiment
Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

. This is because some energy is always lost when one mass applies a non-zero normal force
Normal force
In mechanics, the normal force F_n\ is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.The normal force is one of the components of the ground...

 to another. Therefore, the observed speed
Speed
In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

, acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...

 and distance traveled should be less than Galileo predicts. This energy is lost in forms like sound and heat. However, from Galileo’s predictions of an object moving down an inclined plane in a frictionless environment, he created the theoretical foundation for extremely fruitful real-world experimental prediction.

There are no frictionless planes anywhere to be found; there, by definition, cannot be. However, if there were, one can be all but certain that objects on them would behave exactly as Galileo predicts. However illusory, their value in the design of engines, motors, roadways, even the design of tow truck beds to name a few examples, is noteworthy.

Effect of friction on Galilean predictions

Actual results from objects moving down inclined planes can be predicted very accurately by calculating the effect of friction upon the result predicted by Galileo. This is done
with the simple formula F(fr)=μ(k/s)*F(n), where the force of friction is equal to the static
Static
Static has several meanings:* Static electricity, a net charge of an object** The triboelectric effect, e.g. from shoes rubbing carpet* White noise, a random signal with a flat power spectral density** Noise , a type of radio interference...

 or kinetic
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

 friction coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

 times the vertical normal force
Normal force
In mechanics, the normal force F_n\ is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.The normal force is one of the components of the ground...

 of the object against the plane. Unless in a vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

, a (usually) small amount of potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

 is also lost to air drag.

Other scientific applications

Similar conceptual tools are in constant use in the physical sciences. From string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

 to cosmology
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

, theories are proposed and predictions are made based upon assumption about the properties of objects that have never been observed. Many of these objects are far harder to imagine than inclined plane with a frictionless surface. John Dalton
John Dalton
John Dalton FRS was an English chemist, meteorologist and physicist. He is best known for his pioneering work in the development of modern atomic theory, and his research into colour blindness .-Early life:John Dalton was born into a Quaker family at Eaglesfield, near Cockermouth, Cumberland,...

 proposed the positive proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

 and negative electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

 structure of atoms almost a century before anyone had seen one, yet groundbreaking advances in chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

, physics, and eventually quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 and quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

 were only possible because of that postulate. Similarly, no one has directly observed dark energy
Dark energy
In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...

 or captured any quantity of dark matter
Dark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...

. No one has observed a tiny string that was the indivisible foundation of all matter in the universe. Yet, theories that imagine such objects are not tautological; they entail predictions that can be, and have been, disproved. Thus, they serve as valuable tools in the human endeavor to understand the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...

, despite the fact that they have not or cannot be found within it.

Non-scientific applications

Neither is the frictionless plane a concept unique to science. Many other fields have used conceptual understandings of problems with limited or no direct real-world applicability to great effect. Of particular note is a reference to Galileo’s plane in Thomas Hobbes
Thomas Hobbes
Thomas Hobbes of Malmesbury , in some older texts Thomas Hobbs of Malmsbury, was an English philosopher, best known today for his work on political philosophy...

Leviathan
Leviathan
Leviathan , is a sea monster referred to in the Bible. In Demonology, Leviathan is one of the seven princes of Hell and its gatekeeper . The word has become synonymous with any large sea monster or creature...

.
In Leviathan
Leviathan
Leviathan , is a sea monster referred to in the Bible. In Demonology, Leviathan is one of the seven princes of Hell and its gatekeeper . The word has become synonymous with any large sea monster or creature...

, Hobbes begins by considering the condition of man in the state of nature
State of nature
State of nature is a term in political philosophy used in social contract theories to describe the hypothetical condition that preceded governments...

; without government, without morals, without order. He famously describes the state of nature as the state of war, a state of all against all, where life was solitary, poor, nasty, brutish, and short. However, Hobbes makes clear that the justification for the government described in Leviathan, whose primary purpose is to keep man out of the state of nature, and from the corresponding fear of a violent death, does not depend upon man ever having lived in this condition. Instead, just like the frictionless plane, his description of the state of nature is valuable for how it informs humanity's current condition, not only for the accuracy of its description:

"But there is another saying not of late understood, by which they might learn truly to read one another, if they would take the pains; and that is, Nosce Teipsum, Read Thy Self: which was not meant, as it is now used, to countenance, either the barbarous state of men in power, towards their inferiors; or to encourage men of low degree, to a sawcie behaviour towards their betters; But to teach us, that for the similitude of the thoughts, and Passions of one man, to the thoughts, and Passions of another,whosoever looketh into himselfe, and considereth what he doth, when he does Think, Opine, Reason, Hope, Feare, &c, and upon what grounds; he shall thereby read and know, what are the thoughts, and Passions of all other men...for these the constitution individuall, and particular education do so vary, and they are so easie to be kept from our knowledge, that the characters of mans heart, blotted and confounded as they are, with dissembling, lying, counterfeiting, and erroneous doctrines, are legible onely to him that searcheth hearts.
And though by mens actions wee do discover their designee sometimes; yet to do it without comparing them with our own, and distinguishing all circumstances, by which the case may come to be altered, is to decipher without a key, and be for the most part deceived."

More recently, John Rawls
John Rawls
John Bordley Rawls was an American philosopher and a leading figure in moral and political philosophy. He held the James Bryant Conant University Professorship at Harvard University....

, while very much disagreeing with Hobbes' conception of the state of nature, used this same methodology in describing an admittedly artificial original position
Original position
The original position is a hypothetical situation developed by American philosopher John Rawls as a thought experiment to replace the imagery of a savage state of nature of prior political philosophers like Thomas Hobbes. In it, the parties select principles that will determine the basic structure...

 from which his Theory of Justice derives.

Frictionless plane as a pragmatic methodology

Galileo died 40 years before Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's Philosophiae Naturalis Principia Mathematica
Philosophiae Naturalis Principia Mathematica
Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687. Newton also published two further editions, in 1713 and 1726...

which described normal force
Normal force
In mechanics, the normal force F_n\ is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.The normal force is one of the components of the ground...

 (Newton's third law of motion), inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

 (Newton's first law of motion), and most importantly, Newton's law of universal gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

; he died nearly three centuries before Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

 published his theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. Nevertheless, by thinking about the forces (as he understood them) acting upon an object on an inclined plane, Galileo came to understand the mechanics of the situation in a very fundamental way. From that understanding, he was able to extrapolate the general formula.

At bottom, the frictionless plane is a method of understanding otherwise opaque phenomena to make them receptive to experimentation and understanding. Galileo did not solve the inclined plane by performing experiments, considering the results, then attempting to reverse engineer a calculation that could accommodate those results. From what we now know, such an equation would have been incorrect, for any result is the combination of gravitational and frictional forces. Instead, he began thinking about how gravity works upon free-falling objects, and what about that force is the same, and what about it is different when an object moves down an inclined plane. It is this methodology that is so remarkable, and gives the frictionless plane its immense practical value.
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