Frayne James F - Mechanics (The Basics)

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MECHANICS

The Basics

James F. Frayne

(>>> p241) 271 Change Userkare

MECHANICS

The Basics

James F. Frayne

MONTANA PUBLISHERS

Copyright 2020 James F. Frayne

This book or any portion thereof may not be reproduced or used in any manner whatsoever without the express written permission of the publisher, except for the use of brief quotations in a book review or scholarly journal.

All characters appearing in the present-day part of this work are fictitious. Any resemblance to real persons, living or dead, is purely coincidental.

First Printing: 2020

ISBN: 979-8-590-00823-0

Montana Publishers

Ordering information:

Special discounts are available on quantity purchases by corporations, associations, educators, and others. For details, contact the publisher at the e-mail address detailed below.

U.S. trade bookstores and wholesalers:

Please contact publishers at: e-mail EAYG2021@hotmail.com or direct to www.jamesfrayne.co.uk

By the same author:

Tall Grows the Grass (Autobiographical Novel)

First Son of Khui (Historical Novel)

SHE (Anthology of Women in Mathematic and Science)

Easy as you Go! A Mathematical Companion

(Volume 1: A L)

Easy as you Go! A Mathematical Companion

(Volume 2: M- Z)

A-Star Question Bank

(Mathematics with or without solutions)

Mathematics & Statistics for Biology, Psychology & Chemistry

Mathematics by Stages (Angles to Vectors)

Mathematics by Stages (Circles and Curves)

Mathematics by Stages (Advanced Topics)

Hell Bank Notes (A Pictorial Catalogue)

Romancing the Wood

(History behind the American Wooden Nickel)

The Indian Hundi

(Favourite negotiable instrument in days gone by)

Hidden Stories behind Paper Money around the World

Selected Biology Advance Level Topics (Volume 1: A J)

Selected Biology Advance Level Topics (Volume 2: K Z)

Greenhouse Effect

Jenny Two-tails and her Friends (Book for Young Children)

Applications of Genetics

Atoms and Fundamental Particles

For more details please visit authors website at: http://jamesfrayne.co.uk

Contents

.

Preface

Mathematics and Physics

The relationship between mathematics and physics has been a subject of study of philosophers, mathematicians and physicists since Antiquity, and more recently also by historians and educators.

Considered a relationship of great intimacy, mathematics has been described as " an essential tool for physics " and physics has been described as " a rich source of inspiration and insight in mathematics ".

ref: Wikipedia Text under CC-BY-SA license

Mechanics

Mechanics (Greek :) is the area of physics concerned with forces applied to objects resulting in displacements , or changes of an object's position relative to its environment. This branch of physics has its origins in Ancient Greece with the writings of Aristotle and Archimedes .

During the early modern period , scholars such as Galileo , Kepler , Newton and the 12 th century Jewish-Arab scholar Bad az-Zaman Abu l-Izz ibn Isml ibn ar-Razz al-Jazar (al-Jazari for short! ) laid the foundations for what is now known as classical mechanics , that branch of physics , along with applied mathematics, that deals with particles that are either at rest or are moving with velocities significantly less than the speed of light.

Brachistochrone Problem

Mechanics is therefore a convenient symbiotic relationship between mathematics and physics. The Brachistochrone Problem is a counter-intuitive problem which vexed the best of scholars for a long time since it was posed by Johann Bernoulli in 1696.

Early in the study of plane geometry it is taught that the shortest distance between two points is a straight line . However, the shortest time to travel between two points may not occur along a straight line.

Suppose a ball at point O is allowed to drop and then to slide down to point B not directly under O . Along what path should the ball travel to reach point B in the least amount of time? A straight line? An ellipse? A parabola?

In both mathematics and physics , a brachistochrone curve (from Ancient Greek 'shortest time'), or curve of fastest descent, is the one lying on the plane between a point O and a lower point B , where B is not directly below O , on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time.

The path of quickest descent is the one taken by the ball and is conveniently expressed in terms of time t by the two equations:

x = a(t sin t) and y = -a(1 cos t)

where a is a positive constant.

The curve represented by these equations is called a cycloid .

An amazing fact about this curve is that no matter where the ball is released on the cycloid, it will take the same amount of time to reach point B. Some curves such as this one are more conveniently described using two equations rather than one. To work with such equations, called parametric equations , it is also convenient to use vectors.

The brachistochrone curve is the same shape as a tautochrone curve ; both are cycloids . However, the portion of the cycloid used for each of the two varies. The curve is independent of both the mass of the test body and the local strength of gravity. The curve of fastest descent is not a straight or polygonal line but a cycloid . .

So, whats in the book?

Now, how does all this relate to what is in this book?

First of all, it must be stressed that with the possible exception of the Calculus, there should be no mathematics that has not essentially been covered in prior mathematical courses. It is important to know that what is required of the student is simply the APPLICATION of that which should have been addressed in the past

Various problems have been supplemented by picture diagrams as well as vector and free-body diagrams (force diagrams). Also included are a variety of tables and graphs where appropriate.

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