The principles of elementary mechanics

This book is especially designed to treat of the principles of Rational Mechanics, and not to present a system of analysis. The analysis employed in die demonstration of principles is of an elementary character, the Calculus being entirely avoided. A few problems are solved which very properly belong to the Calculus, but the solutions have been effected by means of the well known properties of certain curves and the principles of elementary geometry. As examples of this mode of reasoning, reference is made to the following problems: The determination of the centre of gravity of a circular arc; The time of vibration of a simple pendulum ; and The quantity of flow of a liquid through a weir. A few problems are solved which involve a knowledge of Conic Sections, but these may be omitted, if desired, without detriment to the other portions of the work.

The Articles are as independent of each other as they can be practically, and at the same time present the subject in a connected manner. This feature will enable the teacher the more easily to select particular portions of the work when the whole cannot be taken.

The manner of applying the principles of the subject is shown by means of numerous problems, examples, and exercises. The problems are of a general character and are accompanied by a full solution. The examples are numerical, and are intended to be special applications of the formulas and principles contained in the chapter of which they form a part. The exercises are a novel feature of the work. They are intended particularly to draw out and fix in the mind the general principles of the subject. The answers to the questions under this head are not always explicitly given in the text, but the principle involved in the answer is sufficiently explained there. Additional questions will doubtless suggest themselves both to the teacher and student, and in some cases conditions may be added to those given in the exercise. Thus, in the 5th Exercise, on page 23, the question may be raised whether the weight of the rope is to be considered; and if so, whether the velocity is to be uniform or variable; also, in the latter case, whether the acceleration be increasing or decreasing.

The abstract relations which exist between a force and the motion which it produces in a body, are considered early in the work. I do not consider this order as in the least essential, but I have usually presented the subject in this way to my classes, regardless of the order given in the text-book. No part of abstract mechanics possesses a greater interest than this, and the questions pertaining to momentum and energy which grow out of these relations have provoked a great deal of discussion - among students in mechanics. In regard to Momentum and Vis Viva, much of the difficulty which arises in the mind of the student in regard to them would be removed, if they are considered, as they really are, mutually independent of each other, having no common unit between them, but each having its own peculiar unit.

The demonstration of the formula for centrifugal force may appear to be unnecessarily lengthy, but I trust that the student will gain a clearer conception of the mode of action of the forces, and be better satisfied with the logic of the demonstration, by following the proof here given, than by certain of the shorter methods. The demonstration by some of the latter methods is defective, although the same result is reached by them.

The principles of energy, which plays such an important part in modem physics, have been explained, and the principles of both Kinetic and Potential Energy used in the solution of problems.

Rest. - Bodies are said to be at rest when they remain in the same place in reference to surrounding objects.

Motion. - A body is said to be in motion relative to another body when it changes' its position with respect to that body. We may mention here two different kinds of motion, each complying with the above definition :(1) motion of translation when the change of position is that of the body as a whole with reference to other bodies external to it; in this case all points of the body describe paths equal in magnitude and parallel in direction. (2) If a point moves so that its direction is continually changing, the path is a curve. The point will have curvilinear motion,

3. Kinematics is the science of motion. - It does not consider the cause of the motion, but determines its measure, and the relations between different motions.

Rest and motion are relative terms. - A body may be at rest in reference to some objects, and in motion in reference to others. Thus, a person sitting on the deck of a ship may be at rest in reference to the objects on the ship, while he is moving with the ship over the water.

The path of a body is the line traced by its central point. - If all the points of a body move in parallel lines, any one of the lines may he taken as the path. Unless otherwise stated, we will assume that the body is reduced to a mere particle. The path is also called the space over which a body moves.


Velocity is rate of motion. - When a body passes over equal successive portions of space in equal times, its rate is uniform. In all other cases it is variable. Thus, if a body moves uniformly from A to B in four seconds, the spaces passed over each second will be one-fourth of AB, or A-1 = 1-2 = 2-3, etc., and any one of these spaces is the velocity.

When the motion is uniform, the velocity is the space passed over in a unit of time, and the velocity is said to be constant.

If s = the space passed over uniformly,
   t = the time of the movement, and
   V = the velocity,
then we have, according to the definition,

v = s/t

from which we find, and

s = v*t and t = s/v

The unit of velocity is understood to be one foot per second, unless otherwise stated. If other units are given, their equivalent value may be found in feet per second.

 Variable Velocity is that in which the rate of motion is constantly changing. - The true measure in this case cannot be the space passed over during any single second, but it is the space which would he passed over during a second if the body moved uniformly at the rate which it had at the instant considered.

We are familiar with this fact. We say that a train of cars moved at the rate of, say, forty miles per hour, when it may have moved at that rate for an instant only, and in coming to rest it may have moved at all conceivable rates less than forty miles per hour.

Angular Velocity is the rate of angular movement. It is measured by the circular arc having unity for its radius, which would he generated by the extremity of the radius if it turned about the centre at the same rate as that of the lody.

 Harmonic Motion. - If a body moves at a uniform rate around the circumference of a circle, the foot of the perpendicular from the body upon the diameter will appear to move to and fro along the diameter with a variable velocity. Thus, in Fig. 8, if the point move uniformly around the circumference AGB, the foot of the perpendicular will move from A towards B, thence from B towards A, and so on, The motion of the point O is said to be harmonic, for the law of its movement is similar to that of a musical
string, or a tuning fork.

Periodic Motion is that in which the motion repeats itself. - Thus, in Fig. 8, the point that moves to and fro along the diameter AB has a periodic motion. A pendulum, as it vibrates to and fro,-is another example.

Rotary Motion is motion about an axis. It is measured by its angular velocity. See Article 12.

The point about which it moves may also have a progressive velocity. Thus, the wheels of a carriage have a rotary motion about the axles, while the axles have a progressive movement. The moon revolves about the earth as a centre, and the earth not only revolves on its axis, but also revolves around the sun.