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Elementary mechanics of solids
ELEMENTARY MECHANICS OF SOLIDS
BY W. T. A. EMTAGE
MACMILLAN AND CO., NEW YORK, 1900
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Elementary mechanics of solids
PREFACE
This book has been written after many years' experience in teaching Theoretical Mechanics to students of a great variety of ages and attainments.
I attach great importance to the value of carefully selected and carefully explained examples, and throughout the book numerous examples will be found worked out and accompanied by notes on the processes employed in their solution. In addition to these, there are nearly five hundred questions for exercise. Many of these questions are taken from examination papers, the source being always clearly stated.
In the teaching of any branch of science the value of experimental illustrations has now come to be fully recognized; and I have described upwards of forty experiments which may all be performed by teacher or student with the help of very inexpensive apparatus. At the same time more elaborate apparatus may be used, when it is available, to illustrate much of the subject matter.
The book will be found to contain all the subjects in the syllabus of the Elementary Stage of Theoretical Mechanics of Solids of the Board of Education, South Kensington; while, to increase its usefulness and to adapt it to the requirements of students for other examinations, the theoretical proofs of many propositions have been added.
It may be read without any mathematical attainments beyond an ability to solve easy algebraical equations, except that in a few instances easy quadratics and the properties of similar triangles have been employed.
I attach great importance to the value of carefully selected and carefully explained examples, and throughout the book numerous examples will be found worked out and accompanied by notes on the processes employed in their solution. In addition to these, there are nearly five hundred questions for exercise. Many of these questions are taken from examination papers, the source being always clearly stated.
In the teaching of any branch of science the value of experimental illustrations has now come to be fully recognized; and I have described upwards of forty experiments which may all be performed by teacher or student with the help of very inexpensive apparatus. At the same time more elaborate apparatus may be used, when it is available, to illustrate much of the subject matter.
The book will be found to contain all the subjects in the syllabus of the Elementary Stage of Theoretical Mechanics of Solids of the Board of Education, South Kensington; while, to increase its usefulness and to adapt it to the requirements of students for other examinations, the theoretical proofs of many propositions have been added.
It may be read without any mathematical attainments beyond an ability to solve easy algebraical equations, except that in a few instances easy quadratics and the properties of similar triangles have been employed.
CONTENTS.
CHAPTER I.
Force. Parallelogram and Triangle of Forces
CHAPTER II.
Resolution of Forces. Polygon of Forces
CHAPTER III.
Rotative Tendency of Force. Moments
CHAPTER IV.
Parallel Forces. Centre of Parallel Forces. Couples,
CHAPTER V.
Centre of Gravity. Mass. Density. Specific Gravity
CHAPTER VI.
Centre of Gravity (continued). States of Equilibrium
CHAPTER VII.
States of Matter. Elasticities
CHAPTER VIII.
Work. Power. Energy
CHAPTER IX.
Machines. Mechanical Advantage. Efficiency. Levers. Inclined Plane
CHAPTER X.
Pulleys. Wheel and Axle. Screw. Toothed Wheel, 151
CHAPTER XI.
Balance. Steel-yards
CHAPTER XII.
Velocity. Acceleration. Kinematical Equations
CHAPTER XIII.
Use of the Kinematical Equations. Acceleration due to Gravity
CHAPTER XIV.
Dynamical Measure of Force. Newton's First and Second Laws of Motion
CHAPTER XV.
Dynamical Measure of Weight. Attwood's Machine
CHAPTER XVI.
Impulse. Newton's Third Law of Motion
CHAPTER XVII.
Kinetic Energy
CHAPTER XVIII.
Potential Energy. Conservation of Energy. Perpetual Motion. Energy after Collision
CHAPTER XIX.
Relative Velocity and Acceleration, Composition of Velocities and Accelerations. Uniform Circular Motion
CHAPTER XX.
Simple Harmonic Motion. Pendulums
CHAPTER I.
FORCE. PARALLELOGRAM AND TRIANGLE OF FORCES.
Force - Suppose a piece of wood is placed on a smooth horizontal table, or, better still, to float on water, so that it will yield to "the application of the slightest push or pull in any direction. Now let two strings be attached to it, and let these both be pulled out horizontally. As a rule the wood will yield to the combined effect of the pulls in the strings, or the two forces applied to it, and will begin to move. It is easy to imagine in a general way what will be the effect produced.
(1) If the two pulls are inclined to one another, the wood will begin to move off along a line lying in the angle between them.
(2) If they are opposite to one another, the wood will move in the direction of the greater pull.
(3) The wood may begin to turn instead of moving away bodily, or even perhaps as well as moving away bodily.
Suppose the two pulls to be equal to one another, that is, so that the tensions in the two strings are equal, and suppose that they act in opposite directions along parallel straight lines, not along the same straight line. The combined effect of these two pulls will be to turn the wood round without moving it away bodily.
Figures 1, 2, 3 represent these three cases. The arrow-heads P and Q denote the pulls of the strings ; and M denotes the motion of the wood.
Now in certain circumstances the wood will not move at all. Let us consider what these are. The two pulls must be along the same straight line ; that is, the first string, the line joining the two points of attachment, and the second string must be all in one straight line; the two pulls must be equal to one another; they must act in opposite senses. It is only under these conditions that the wood will remain at rest. And, further, whenever these conditions are fulfilled, we may be sure that the combined effect of the two forces acting on the wood will be nothing.
In what we have just considered the pulls in the strings are examples of mechanical forces. Such forces are produced in numerous ways; and in general we may say: A force is that which moves, or tends to move, a body, or alters, or tends to alter, its state of motion.
Equilibrium - If a body is acted on by a set of forces in such a manner that it does not move it is said to be in equilibrium. Sometimes the forces are spoken of as being in equilibrium, or are said to form a system in equilibrium with each other, this meaning that their combined effect on anybody on which they may be acting is nothing.
Conditions for Equilibrium - We have just met with an example of forces in equilibrium, the case in which the forces are two in number. Let us now state, in general terms, the conditions which must necessarily hold when two forces are in equilibrium, and which are sufficient to ensure that the forces shall be in equilibrium. The conditions may thus be stated to be necessary and sufficient.
In saying that they are necessary, we say that if we know that the forces are in equilibrium, the conditions must hold, or must necessarily hold ; and in saying that they are sufficient, we say that if the conditions are known to hold, the forces must be in equilibrium, or the conditions suffice to ensure equilibrium. It should be remembered, then, that when conditions are said to be necessary and sufficient, two distinct statements are made: in each of them we know something, and something else follows as a result; and what we know in one case is what follows in the other, and vice versa. The two statements, or propositions, are thus converses of each other. We may now say that
The necessary and sufficient conditions between two forces in equilibrium are that they should be equal, and should act in opposite directions along the same straight line.
Thus, in the case of the wood pulled by two strings, when we say what conditions are necessary, we mean that if the wood acted on by the two pulls does not move, the pulls must be equal and act oppositely along the same straight line; and when we say what conditions are sufficient, we mean that if the pulls are equal and act oppositely along the same straight line, the wood will not move.
It is important to understand this about necessary and sufficient conditions, because it frequently happens that, in a case of this sort, the two sets of conditions are the same, and we thus have a compact way of stating what they are.
Tension of Strings - We have seen that a force may be caused to act on a body by attaching a string to it and pulling the string, as, for instance, with the hand. The pull exerted by the hand is transmitted along the string, and is applied to the body. The string is said to be in a state of tension. The pull all along its whole length, or the force which any piece of it exerts on the next piece, is the same, being equal to the force applied by the hand. This pull, which is exerted throughout the length of the string, is called the tension of the string; and we may say that the force acting on the body-is the tension of the string. A pull exerted by the hand in this way would not be a definite or constant force. A steady force of a definite magnitude may be obtained in various ways.
If a body is tied to the end of a string and hangs steadily from it, it produces by its weight a constant pull along the string, of a definite magnitude. If the string passes round a smooth pulley, that is, one which turns quite readily on its axle, the pull throughout the string will still be the same as in the vertical portion of it which is immediately above the body.
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FORCE. PARALLELOGRAM AND TRIANGLE OF FORCES.
Force - Suppose a piece of wood is placed on a smooth horizontal table, or, better still, to float on water, so that it will yield to "the application of the slightest push or pull in any direction. Now let two strings be attached to it, and let these both be pulled out horizontally. As a rule the wood will yield to the combined effect of the pulls in the strings, or the two forces applied to it, and will begin to move. It is easy to imagine in a general way what will be the effect produced.
(1) If the two pulls are inclined to one another, the wood will begin to move off along a line lying in the angle between them.
(2) If they are opposite to one another, the wood will move in the direction of the greater pull.
(3) The wood may begin to turn instead of moving away bodily, or even perhaps as well as moving away bodily.
Suppose the two pulls to be equal to one another, that is, so that the tensions in the two strings are equal, and suppose that they act in opposite directions along parallel straight lines, not along the same straight line. The combined effect of these two pulls will be to turn the wood round without moving it away bodily.
Figures 1, 2, 3 represent these three cases. The arrow-heads P and Q denote the pulls of the strings ; and M denotes the motion of the wood.
Now in certain circumstances the wood will not move at all. Let us consider what these are. The two pulls must be along the same straight line ; that is, the first string, the line joining the two points of attachment, and the second string must be all in one straight line; the two pulls must be equal to one another; they must act in opposite senses. It is only under these conditions that the wood will remain at rest. And, further, whenever these conditions are fulfilled, we may be sure that the combined effect of the two forces acting on the wood will be nothing.
In what we have just considered the pulls in the strings are examples of mechanical forces. Such forces are produced in numerous ways; and in general we may say: A force is that which moves, or tends to move, a body, or alters, or tends to alter, its state of motion.
Equilibrium - If a body is acted on by a set of forces in such a manner that it does not move it is said to be in equilibrium. Sometimes the forces are spoken of as being in equilibrium, or are said to form a system in equilibrium with each other, this meaning that their combined effect on anybody on which they may be acting is nothing.
Conditions for Equilibrium - We have just met with an example of forces in equilibrium, the case in which the forces are two in number. Let us now state, in general terms, the conditions which must necessarily hold when two forces are in equilibrium, and which are sufficient to ensure that the forces shall be in equilibrium. The conditions may thus be stated to be necessary and sufficient.
In saying that they are necessary, we say that if we know that the forces are in equilibrium, the conditions must hold, or must necessarily hold ; and in saying that they are sufficient, we say that if the conditions are known to hold, the forces must be in equilibrium, or the conditions suffice to ensure equilibrium. It should be remembered, then, that when conditions are said to be necessary and sufficient, two distinct statements are made: in each of them we know something, and something else follows as a result; and what we know in one case is what follows in the other, and vice versa. The two statements, or propositions, are thus converses of each other. We may now say that
The necessary and sufficient conditions between two forces in equilibrium are that they should be equal, and should act in opposite directions along the same straight line.
Thus, in the case of the wood pulled by two strings, when we say what conditions are necessary, we mean that if the wood acted on by the two pulls does not move, the pulls must be equal and act oppositely along the same straight line; and when we say what conditions are sufficient, we mean that if the pulls are equal and act oppositely along the same straight line, the wood will not move.
It is important to understand this about necessary and sufficient conditions, because it frequently happens that, in a case of this sort, the two sets of conditions are the same, and we thus have a compact way of stating what they are.
Tension of Strings - We have seen that a force may be caused to act on a body by attaching a string to it and pulling the string, as, for instance, with the hand. The pull exerted by the hand is transmitted along the string, and is applied to the body. The string is said to be in a state of tension. The pull all along its whole length, or the force which any piece of it exerts on the next piece, is the same, being equal to the force applied by the hand. This pull, which is exerted throughout the length of the string, is called the tension of the string; and we may say that the force acting on the body-is the tension of the string. A pull exerted by the hand in this way would not be a definite or constant force. A steady force of a definite magnitude may be obtained in various ways.
If a body is tied to the end of a string and hangs steadily from it, it produces by its weight a constant pull along the string, of a definite magnitude. If the string passes round a smooth pulley, that is, one which turns quite readily on its axle, the pull throughout the string will still be the same as in the vertical portion of it which is immediately above the body.
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