Introductory Classical Mechanics/Introduction

Classical Mechanics is the study of large (relatively speaking) objects, as opposed to quantum mechanics, which studies particles and very small objects, or relativity, which pertains to very large objects (on the order of stars and galaxies) and objects moving very quickly, over about .5c. The field includes Newton's laws of motion and gravitation and Kepler's laws of planetary motion, in addition to other laws and the application of these laws to real world problems. Classical Newtonian mechanics is accepted as incorrect due to errors observed on large scales, such as the progression of the perihelion of Mercury, which motivated Einstein to create General relativity, or under certain other circumstances including blackbody radiation, which motivated Max Planck to author his papers generating the quantum hypothesis. The theory is accurate on large scales to an incredible degree of accuracy.

Units of Measurement

Physics today uses an internationally agreed system of units for measurement. These units are precisely defined and are known as SI Units, (SI is an abbreviation of the system's French name Système international d’unités). The system defines 7 base units, and several derived units. The derived units can all be defined in terms of the base units. The base units are as follows:

SI base units

Unit name	Unit symbol	Quantity	Definition (Incomplete)	Dimension symbol
metre	m	length	Original (1793): 1⁄10000000 of the meridian through Paris between the North Pole and the EquatorFG Current (1983): The distance travelled by light in vacuum in 1⁄299792458 second	L
kilogram	kg	mass	Original (1793): The grave was defined as being the weight [mass] of one cubic decimetre of pure water at its freezing point.FG Current (1889): The mass of the International Prototype Kilogram	M
second	s	time	Original (Medieval): 1⁄86400 of a day Current (1967): The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom	T
ampere	A	electric current	Original (1881): A tenth of the electromagnetic CGS unit of current. [The [w:CGS|] emu unit of current is that current, flowing in an arc 1 cm long of a circle 1 cm in radius creates a field of one oersted at the centre.IEC Current (1946): The constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2×10−7 newton per metre of length	I
kelvin	K	thermodynamic temperature	Original (1743): The centigrade scale is obtained by assigning 0° to the freezing point of water and 100° to the boiling point of water. Current (1967): The fraction 1⁄273.16 of the thermodynamic temperature of the triple point of water	Θ
mole	mol	amount of substance	Original (1900): The molecular weight of a substance in mass grams.ICAW Current (1967): The amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.	N
candela	cd	luminous intensity	Original (1946):The value of the new candle is such that the brightness of the full radiator at the temperature of solidification of platinum is 60 new candles per square centimetre Current (1979): The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×1012 hertz and that has a radiant intensity in that direction of 1⁄683 watt per steradian.	J
Note The original definitions of the various base units in the above table were made by the following authorities: FG = French Government IEC = International Electrical Congress ICAW = International Committee on Atomic Weights

In order to simplify the writing of very small or very large quantities of units, prefixes denoting an order of magnitude are used. These are also defined by by the Système Internationale and are thus called SI prefixes, or sometimes metric prefixes. They also help to simplify equations. The following table lists the officially defined prefixes.

Metric prefixes

Prefix Symbol 1000m 10n Decimal Short scale Long scale Since yotta Y 10008 1024 1000000000000000000000000 septillion quadrillion 1991 zetta Z 10007 1021 1000000000000000000000 sextillion trilliard 1991 exa E 10006 1018 1000000000000000000 quintillion trillion 1975 peta P 10005 1015 1000000000000000 quadrillion billiard 1975 tera T 10004 1012 1000000000000 trillion billion 1960 giga G 10003 109 1000000000 billion milliard 1960 mega M 10002 106 1000000 million 1960 kilo k 10001 103 1000 thousand 1795 hecto h 10002/3 102 100 hundred 1795 deca da 10001/3 101 10 ten 1795 10000 100 1 one – deci d 1000−1/3 10−1 0.1 tenth 1795 centi c 1000−2/3 10−2 0.01 hundredth 1795 milli m 1000−1 10−3 0.001 thousandth 1795 micro μ 1000−2 10−6 0.000001 millionth 1960 nano n 1000−3 10−9 0.000000001 billionth milliardth 1960 pico p 1000−4 10−12 0.000000000001 trillionth billionth 1960 femto f 1000−5 10−15 0.000000000000001 quadrillionth billiardth 1964 atto a 1000−6 10−18 0.000000000000000001 quintillionth trillionth 1964 zepto z 1000−7 10−21 0.000000000000000000001 sextillionth trilliardth 1991 yocto y 1000−8 10−24 0.000000000000000000000001 septillionth quadrillionth 1991

Newton's laws

Newton's legacy is founded upon the three concepts he contributed to the idealogical world. They are his three laws of motion, his law of gravitation, and calculus. They work hand in hand in defining Newtonian mechanics. Since his time, physics as a subject has always been taught in that order, simply because each concept builds upon the work of the one before.

Before jumping into the subject at hand, it is worth it to note that Newton did not create the three laws of motion from scratch. Physics, and mechanics along with it, came before Newton. He observed nature carefully, working upon the hypotheses and knowledge of his predecessors, such as Galileo and Kepler. His modification and research lead to the formation of his three laws of motion. Hence his famous saying:

If I have seen further it is by standing on the shoulders of Giants.

His formal approach starts with a few axioms. His model of the world was of masses and forces. They form a powerful theoretical framework where the real world can be simulated. However powerful it may seem, do treat them as merely a theoretical construct and not the real thing, for they must be discarded once they do not function.

Firstly, we must define force. A force is a push or a pull, very close to how they are normally defined. However, it would be rather difficult to visualise a force that acts at a distance or that acts on a mass uniformly.

The First Law

The first law states that A body at rest stays at rest and a body in uniform motion stays in uniform motion unless acted upon by an outside force.

This law is formulated as the fundamental principle in Newton's work, and it is also known as the law of inertia. It references Aristotle's work, which states the complete opposite. This law is important for its intellectual leap, not for the mathematical use, simply because this law is completely embedded in the second law. Where mathematics becomes a hindrance, as in Newton's time, it will be helpful.

The Second Law

Newton's second law of motion states that The rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the same direction. In mathematical terms, that is $\Sigma {\vec {F}}={\frac {d{\vec {p}}}{dt}}$ where ${\vec {F}}$ refer to the various forces acting on the body and ${\vec {p}}$ refers to the momentum of the body.

This forces us to define momentum. Momentum is a mathematical product of mass and velocity. Specifically, it is mass times velocity. Then, what are mass and velocity? Velocity is the speed AND direction of the body's motion. Mass, however, is something that is totally new. It is a quantity attached to all objects; a number that Newton's theory must enforce in order to work. Luckily, from the information given so far, it is possible to define it backwards. Since we know of the law of inertia, we can define mass, inertia mass, as how difficult it is to change the motion of a body. In fact, it is possible to define the unit mass as the amount of matter (same thing as mass) in a body which changes by a unit velocity when subjected to a unit force for a unit time.

$\Sigma {\vec {F}}={\frac {d{\vec {p}}}{dt}}={\frac {d}{dt}}(m{\vec {v}})$

$\therefore m{\frac {\Delta v}{\Delta t}}=\Sigma {\vec {F}}$ for constant mass and force over time.

$m={\frac {\Delta t}{\Delta v}}\Sigma {\vec {F}}$

The Third Law

Newton originally formulated the third law as To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. However, because of the fact that English evolves over time, we have to replace actions with forces in our modern context. It is therefore shortened to For every force there is an equal, but opposite, force

Please do note that the opposing force has to act on the other body, and is in most cases ignored.

The Law of Universal Gravitation

Two bodies exert a force on each other directly proportional to the product of their masses and inversely proportional to the square of the distance between them. That is $F_{g}=G{\frac {m_{1}m_{2}}{r^{2}}}$, where $G$ is a constant of proportionality.

See also

• Lagrangian formulation of classical mechanics

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