Applied mechanics embracing strength and elasticity of materials

The subject of Applied Mechanics is one which covers a very wide field, and it would not be possible adequately to cover the ground in a single volume. In the present work the author has attempted to compress into one volume of moderate dimensions sufficient material for a two years' course in the subject. To carry out this object the author has endeavoured to be as clear and concise as possible, and he has written the text on the assumption that the student will spend a considerable time in working out the numerous exercises which are given.

The illustrations, which are very numerous, have all been specially prepared for this work, they have been' made as small as possible consistent with clearness, and they have been set up with the text in such a manner as to be in close connection with it and to economise space as much as possible.

A special feature has been made of the exercises, which will be found in groups in the various chapters. Of the 780 exercises given, 600 are original, and the author has given as much attention to these as to the text. The remaining 180 exercises have been selected with great care from the examination papers of various examining bodies. Many of the exercises will be found to amplify the text, and thus add to the scope of the book.

The author would here desire to impress upon the student the great importance of working a large number of exercises. A student may imagine, after hearing a lecture, or after reading the text on a part of the subject, that he knows it thoroughly, and that he may therefore leave it, but he will generally find, if he proceeds to apply his knowledge to a practical example, that some important point has escaped his attention or has not been thoroughly understood. This applies to the clever student as well as to the student of ordinary ability. Besides, the working of exercises is essential for thoroughly impressing the subject on his mind. Another matter of very great importance to the student is the cultivation of neatness and accuracy and the systematic arrangement of his work.

The majority of the exercises given involve numerical answers, and these will be found at the end of the book. Some teachers who may use this book in their classes may object to their students having the answers to the exercises beforehand, but such teachers may, if they choose, make simple alterations in the data of the exercises before giving them to their students, and thus in an easy way have their own set of good exercises. The answers given at the end of the book will, however, be useful to students who may be studying privately, and also to conscientious and industrious students who may desire to get thoroughly familiar with the subject by working examples.

A good and enthusiastic teacher interested in his subject does not as a rule follow strictly any particular text-book, not even if he has written it himself, and many of the best teachers seldom refer to any text-book in their lectures. It is, however, very important that a student should form as good a library of his own as he can afford, and the author of this book hopes that it will not be unworthy of a place in such a library, especially in the initial stages of its formaation.

CHAPTER III - WORK AND ENERGY

Work. When a force acting on a body causes that body to move, the force is said to do work. Also, if a body is moved against a resistance, work is done in overcoming the resistance. The amount of work done depends on the magnitude of the force and also on the distance through which it acts.

In measuring work the unit which is generally used by engineers is the work done when a force of one pound acts through a distance of one foot, this unit being called a foot-pound. If the unit taken be the work done when a force of one ton acts through a distance of one foot, it is called a foot-ion. The foot-ton is used in measuring large quantities of work. For measuring small quantities of work the inch-pound, or the work done when a force of one pound acts through a distance of one inch, is frequently used.

The work done by a force is found by multiplying the magnitude of the force by the distance through which it acts.

Work by an Oblique Force. If a force acting on a body acts in a direction inclined to that of the body's motion the force may be resolved into two components, as explained in Chapter IV., one acting in the direction of the body's motion, and the other perpendicular to that direction. The latter component does no work, and the work done by the former is its magnitude multiplied by the distance through which the body moves.

Work in Raising a System of Weights. When a number of weights are raised through different heights, or when all the parts of one weight are not raised through the same height, the amount of work done is obtained by multiplying the total weight lifted by the distance through which the centre of gravity of the system is raised.

Diagram of Work. If a straight line OC (Fig. 22) represents to scale the distance S through which a body moves under the action of a force, and if OB drawn at right angles to OC represents to scale the magnitude P of the force, then the area of the rectangle BC will represent to scale the work done by P in acting through the distance S.

Turning Moment Work in Turning. When a force P acting on a body causes that body to rotate about a fixed axis, the line of action of the force being in a plane perpendicular to that axis, the product of P, the magnitude of the force, and the perpendicular distance R of its line of action from the axis is called the turning moment or torque of the driving force P. If P is in pounds and R is in feet, the turning moment PR is in pound-feet or foot-pounds ; but if P is in pounds and R is in inches, PR is in pound-inches or inch-pounds. If the line of action of P is not in a plane perpendicular to the axis of rotation, but makes an angle with that plane, then the turning moment is PR cos 0.

 Rate of Work Horse-power. The working power of any agent depends on the amount of work which it can do in a given time. Watt found that a good working horse could do 33,000 foot-pounds of work in one minute, and he introduced this as the unit for measuring the working power of steam-engines. A steam-engine or any working agent is said to be of one horse-power when it can do 33,000 foot-pounds of work in one minute, or 550 foot-pounds in one second.

Evidently the simple rule for finding the horse-power of any working agent or the horse-power transmitted by any piece of machinery is to divide the number of foot-pounds of work done or transmitted per minute by 33,000, or horse-power equals work per second divided by 550. Horse-power is a measure of the rate of doing or transmitting work.

Electrical Units and their Mechanical Equivalents. The electromotive force, or electric pressure of an electric current, is measured in volts, and the strength of the current, or the rate of flow of the electricity across a section of the conductor, is measured in amperes. The power of a current of 1 ampere at an electrical pressure of 1 volt is called a watt. Volts x amperes = watts. 1 horse-power = 746 watts. 1 kilowatt = 1000 watts. 1 electrical unit or 1 Board of Trade unit = 1000 watt-hours.

Machines. For the purposes of this Article a machine may be defined as a contrivance for overcoming a force applied at one point by means of another force applied at another point. In books on mechanics it used to be the practice to call the former force the weight and the latter force the power , but since the force to be overcome is not necessarily that of gravity, it is better to call it the resistance, and since the term power is used in connection with rate of work, it is better to use the term effort instead of power when referring to the driving force in a machine. In this Article the effort will be denoted by P, and the resistance by W.

The points at which the effort and resistance act may be called the driving point and working point respectively.

In a machine in which the displacement of the driving point bears a constant ratio to the displacement of the working point, this ratio is called the velocity ratio of the machine. In a machine in which this ratio is variable, the velocity ratio of the machine for any given positions of its parts is the ratio of the displacement of the driving point to the displacement of the working point when these displacements are in definitely small.


Energy. In mechanics the term energy means capacity for doing work.

Potential Energy is energy due to the relative position of one body to another, or of one part of a body to another part when the two bodies or the parts of the same 1body are under the action of a force or forces tending to alter their relative positions. For example, a body which is allowed to fall towards the earth may be made to do work; hence before it begins to fall it possesses potential energy, or energy due to its position in relation to the earth. A compressed spiral spring has potential energy, because if it is allowed to resume its unstrained form it can be made to do work. Likewise compressed air possesses potential energy. The energy stored in a piece of coal is potential energy, and under favourable conditions the atoms of the constituents of the coal and the atoms of the oxygen of the air will rush together and produce heat which may be converted into work.