Classical mechanics

Classical Mechanics is the study of the effects of forces on bodies. It is also known as Newtonian Mechanics, named after Sir Isaac Newton.

Introduction

Newtonian mechanics can describe the relationships of force, matter and motion. It is limited in it's scope as it cannot be used to describe phenomena at the atomic level or objects approaching the speed of light. It can be used to describe motion simply and accurately in an inertial frame of reference. During the twentieth century Quantum Mechanics was developed to describe atomic size systems and other phenomena in electromagnetism, and Albert Einstein developed Relativity.

Motion

Any body that moves from one point to another has an average velocity(v_av) which is a measure of the rate of change of displacement($\Delta$x) with time. In equation form:

$v=\Delta x/\Delta t\,\!$ ,

The instantaneous velocity is then the limit of the average as the time interval($\Delta$ t) approaches zero:

$v=dx/dt\,\!$,

In a one dimensional system the term speed could be used instead of velocity however in more dimensions the difference between a vector quantity (like velocity which has a magnitude and a direction) and a scalar quantity (such as speed which only has a magnitude) is very important.

If the velocity of a body changes with time the body has acceleration(a). Acceleration is related to velocity in the same way as velocity is to displacement:

$a=\Delta v/\Delta t\,\!$ , and

$a=dv/dt\,\!$,

One of Newton's inventions, calculus, which was simultaneously and independantly invented by Gottfried Wilhelm Leibniz, is useful in mechanics. Acceleration is the derivative of velocity (with respect to time), which is the derivative of displacement (with respect to time).

Newton's laws of motion

Newton's laws of motion help to analyze the principles of dynamics, the relationship of motion to the forces that cause it. These three laws were first published in 1687 in Philosophiae Naturalis Principia Mathematica. The original latin form was as follows:

• Lex I: Corpus omne perseverare in status suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.

• Lex II: Mutationem motis proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

• Lex III: Actioni contrariam semper et aequalem esse reactionem: sive corporum duorum actiones sin se mutuo semper esse aequales et in partes contrarias dirigi.

Or, in an english, more didatic form:

• First Law: A body acted on by no net force moves with constant velocity and zero acceleration.

This means that a body in motion has a property called inertia, a tendency to keep moving unitl another force causes it to stop.

• Second Law: If a net force acts on a body, the body accelerates. The force equals the mass of the body multiplied by the acceleration.

This relation of force and motion is a fundamental law of nature. If there is a constant net force acting on an object it's velocity will continue to increase at a constant rate of acceleration.

• Third Law: If body A exerts a force on body B, then body B exerts a force on body A. This force will have an equal magnitude and opposite direction.

This is less formally stated as; every action has an equal and opposite reaction. It's important to remember these two forces act on different bodies. For example a ball thrown in the air is being pulled towards the centre of the earth by a force due to gravity and is exerting a force of equal magnitude pulling the earth towards the ball. The acceleration on the earth is negligable because it has a much larger mass as stated in the second law.

These laws are only valid in an inertial frame of reference, or, as Newton, in an absolute space.

While Newton's laws can be stated very easily it can be complicated applying them to real-world situations where there are many different forces acting on any object. When two objects interact in contact with each other there are contact forces in action. Usually a normal (perpendicular) force and a friction force. The friction force always acts in a direction against the direction of travel.