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:

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.

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 $$F_g=G\frac{m_1m_2}{r^2}$$, where $$G$$ is a constant of proportionality.