A treatise on analytical statics - Todhunter

In this book will be found all the Propositions which usually appear in treatises on Theoretical Statics. To the different chapters Examples are appended, which have been principally selected from the University and College Examination Papers; these will furnish ample exercise in the application of the principles of the subject.

Some of the Examples in the earlier chapters assume results which are obtained at a later part of the book; the student who has no previous acquaintance with the subject may therefore, on his first perusal of the book, omit the more difficult Examples of the first six chapters.

In the first three chapters and in the ninth chapter I have made considerable use of Mr. Pratt's Treatise on Mechanical Philosophy, which was placed at my disposal by the Publishers.

In the second edition the work has been thoroughly revised and has received large additions; these additions have been made with the view of rendering the subject more readily intelligible by explaining and illustrating those parts which were found by the experience of teachers to be difficult for beginners.

CHAPTER I - INTRODUCTION

A body is a portion of matter limited in every direction, and is consequently of a determinate form and volume. A material particle is a body indefinitely small in every direction; we shall speak of it for shortness as a particle.

2. A body is in motion when the body or its parts occupy successively different positions in space. But we cannot judge of the state of rest or motion of a body without comparing it with other bodies, and for this reason all motions which come under our observation are necessarily relative motions.

3. Force is that which produces or tends to produce motion in a body.

4. When several forces act simultaneously on a body, it may happen that they neutralise each other; when a body remains at rest though acted on by forces, it is said to be in equilibrium; or, in other words, the forces are said to maintain equilibrium.

5. Mechanics is the science which treats of the laws of rest and motion of bodies. Statics treats of the laws of the equilibrium of bodies, and Dynamics of the laws of motion of bodies.

6. There are three things to consider in a force acting on a particle: the position of the particle: the direction of the force, that is, the direction in which it tends to make the particle start; and the intensity of the force. As the dimensions of a particle are indefinitely small its position may be determined in the same manner as that of a point in geometry, and the direction of the force may be determined in the same manner as that of a straight line in geometry. We proceed then to consider the magnitude or intensity of a force.

7. Forces can be measured by taking some force as the unit, and expressing by numbers the ratios which other forces bear to this unit. Two forces are equal when being applied in opposite directions to a particle they maintain equilibrium. If we take two equal forces and apply them to a particle in the same direction we obtain a force double of either; if we unite three equal forces we obtain a triple force ; and so on.

When we say then that a force applied to a particle is a certain multiple of another force, we mean that the first force may be supposed to be composed of a certain number of forces equal to the second and all acting in the same direction. In this way forces become measurable quantities, which can be expressed by numbers, like all other quantities, by referring them to a unit of their own kind. Forces may also be represented by straight lines proportional in length to these numbers, drawn from the point at which the forces act and in the directions in which they act.

8. Experience teaches us that if a body be let free from the hand, it will fall downwards in a certain direction; however frequently the experiment be made, the result is the same, the body strikes the same spot on the ground in each trial, provided the place from which it is dropped remain the same. The cause of this undeviating effect is assumed to be an affinity which all bodies have for the earth, and is termed the force of attraction. If the body be prevented from falling by the interposition of a table or of the hand, the body exerts a pressure on the table or hand. Weight is the name given to the pressure which the attraction of the earth causes a body to exert on another with which it is in contact.



9. A solid body is conceived to be an aggregation of material particles which are held together by their mutual affinities. This appears to be a safe hypothesis, since experiments shew that anybody is divisible into successively smaller and smaller portions without limit, if sufficient force be exerted to overcome the mutual action of the parts of the body,

10. A rigid body is one in which the particles retain invariable positions with respect to each other. No body in nature is perfectly rigid; every body yields more or less to the forces which act on it. If, then, in any case this compressibility is of a sensible magnitude, we shall suppose that the body has assumed its figure of equilibrium, and then consider the points of application of the forces as a system of invariable form. By body, hereafter, we mean rigid body.

11. When a force acts on a body the effect of the force will be unchanged at whatever point of its direction we suppose it applied, provided this point be either one of the points of the body or be invariably connected with the body. This principle is known by the name of the transmissibility of a force to any point in its line of action ; it is assumed as an axiom or as an experimental fact. We may shew the amount of assumption involved in the axiom, by the following process.