Elements of mechanics - Parker

The present Work is written for the use of those students whose knowledge of Mathematics is limited to an acquaintance with Elementary Geometry, the solution of Simple Quadratic Equations in Algebra, and a few fundamental propositions in Plane Trigonometry.

The many real and perplexing difficulties which the study of Mechanics presents to the beginner are in a great measure due to the fact that he has not, at the very outset, acquired clearness of ideas about the first principles which form the groundwork of the subject. There is, therefore, some fear lest his work should degenerate into a mere manipulation of symbols and the application of rules, uninteresting and unsuggestive, because little understood. For this reason great care has been taken in explaining and illustrating the nature of the Resolution of Forces and the Principle of Moments in Statics, and the applications of Newton's Second Law of Motion in Dynamics. Also the numerous examples which are worked out are solved, wherever possible, without making use of mathematical formulae, but by the direct application of first principles.

It is right to mention that many suggestive examples have been borrowed from papers set to Students at the various Universities, and more especially from those proposed at Examinations of Engineering Students in the University of Dublin. The older method of treating Statics before Dynamics has been followed, partly because the subject, if taught in this manner, is, in the author's opinion, more easily grasped by the beginner, and also on account of the much greater practical importance of Statics, especially to the mechanical engineer. Those, however, who prefer to base the principles of Statics on Newton's Laws of Motion should, in the first instance, read Chapters XI and XII, also Articles 223-225 and 230-232; and, if this course is followed, the book can be used with equal advantage whichever method is adopted.

In conclusion, the author will feel grateful for any corrections or suggestions from teachers or students with a view to improvements in a future edition.

1. Mechanics. The science which treats of the action of forces on bodies is called Mechanics.

This subject is generally divided into two branches.

(1) Statics, which deals with the laws to which forces are subject when the bodies on which they act are in a state of rest.

(2) Dynamics, which treats of forces which by their action on bodies produce motion.

Some writers include Statics in Dynamics; and therefore if taken in this sense, the term "Dynamics" might be used instead of what is generally called Mechanics.

2. Hatter is something (we know not what) of whose existence we form an intuitive judgment, from the fact that the mind is conscious of such qualities as solidity, extension, inertia, &c.

3. A Body is a limited portion of matter; and a body whose size or bulk is so small that its mass may practically be regarded as concentrated into a mathematical point is called a Particle.

4. A Rigid Body is such that its parts always preserve the same relative positions with respect to one another when external forces are applied to it; or, in other words, there is no change produced in its shape by the action of external forces.

There is no body existing in nature which is absolutely rigid, for even in the case of a block of iron or stone there is a slight strain or change in shape produced by the action of external forces; but these changes are, under ordinary circumstances, so small that, for present purposes, we may regard them as non-existent.

5. Mass. The quantity of matter in a body is called its mass. In the British Isles, the unit of mass is the pound, the standard pound mass being that of a lump of platinum preserved in the Exchequer Office, London, several exact copies of which are kept in various places, so that if the original were lost it might be restored.

In France, the unit of mass is the gramme or gram, a standard kilogramme or 1000 grammes being carefully preserved, as in the case of the English pound.

6. Force. When we raise a weight off the ground or commence to wheel a truck which has previously been at rest, we are conscious of exerting a muscular effort to produce the motion; or, if the body be already moving, we find that a similar effort has to be made either to bring it to rest or to produce any other alteration in its motion. Our first conception of force is thus obtained from muscular effort exerted by a living agent.

7. Definition. A Force is anything which tends to change the state of rest or uniform motion of a body.


8. In order to measure a force, we must have (1) some test to apply to determine when two forces are equal;(2) some unit of measurement with which to compare other forces.

9. Equal Forces. Two forces are said to be equal if, when they are applied to a particle in opposite directions, the particle remains at rest.

From this it follows that, when it is said that the magnitude of one force is twice that of another, it is meant that the first, when applied to a particle in any direction, will exactly balance or equilibrate two forces, each equal to the second, acting on the same particle in the opposite direction. Similarly, we can get a clear idea of what is meant by a force whose magnitude is three, four, or five times that of another force; and hence, when some definite unit of force is laid down, the magnitude of any other force can be expressed in terms of that unit.

10. Weight. The Earth attracts the mass of every body towards its centre with a force which is called the weight of the body. The weight of a body is slightly different at different parts of the Earth ; for, on account of the Earth being slightly flattened at the poles,* a body near the poles is closer to the centre of the Earth than when near the equator, and therefore the Earth attracts it with a greater force; and hence the weight of a body increases slightly as we approach the poles.

We shall see subsequently that the force with which the Earth attracts a body is directly proportional to the mass of the body, and hence that the weight of a body is a measure of its mass.

11. Measurement of Forces. The statical unit of force is a force equal to that exerted by the attraction of the Earth on a mass of one pound. This is called "a force of one pound," which really means " a force equal to the weight of one pound."

13. Action and Reaction. When one body presses against another the latter body reacts against the former with an equal and opposite force. Thus, when a book rests on a table, the table exerts an upward force on the book equal and opposite to that exerted by the book on the table. That Action and Reaction are equal and opposite is included in Newton's Third Law of Motion.

It is very important that the student should, at the outset, clearly realize this duality of forces, viz. that every action is accompanied by an equal and opposite reaction, and that therefore force cannot exist independent of matter.

14. Perfectly Smooth Surfaces. When two bodies with perfectly smooth surfaces are in contact, the reaction between them is normal or perpendicular to their common surface. That this must be the case can be seen if we place a polished slab of wood or metal on very smooth ice: it will be found that very little resistance is experienced in moving it along the ice. The fact that any resistance, no matter how small, is experienced is due to the fact that there is no body whose surface is perfectly smooth; from this we conclude that if the surfaces of the ice and slab were perfectly smooth there could be no force between them parallel to their common surface, and that, therefore, the only reaction between two such bodies is perpendicular or normal to that surface.

15. Tension. When a force is applied to a body by means of a string, it is called tension. Thus, if we tie one end of a light string (whose weight may be neglected) to a body, and, with the hand, pull the other end with a force of 10 lbs. wt., then the body is pulled by the string with a force of 10 lbs. wt., and there is also an equal and opposite force exerted by the string on the hand (Art. 13); the string is then said to have a tension of 10 lbs. wt. throughout its length. It is also evident that this tension is not altered if the string be passed round one or more smooth pegs : that is, the tension of a light string passing round one or more smooth pegs is the same throughout its length.

16. Definition. When a number of forces acting on a body keep the body at rest, the forces are said to be in Equilibrium.

It is necessary in Mechanics to make the following assumptions, which may be regarded as axioms:

17. AXIOM 1. Transmissibility of a Force. If a force acts on a rigid body, its point of application can be transferred to any other point in the body in the line of action of the force without altering its effect.

Thus a force represented by AB acting on a rigid body (fig. 1) will be unaltered in effect if its point of application be changed from A to any other point A' in the body situated on AB; that is, if A'B' be cut off equal in length to AB, then a force represented by AB' is in every way identical with one represented by AB.

18. AXIOM 2. If a system of forces acts on a body, any other system which in itself would keep the body in equilibrium can be introduced or removed without any alteration in the system.

As an example of this axiom, we often introduce two equal and opposite forces into a system, or remove two equal and opposite forces from a system, without any alteration of effect.