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Elementary statics - Smith
ELEMENTARY STATICS
BY J. HAMBLIN SMITH,
TORONTO: ADAM MILLER & CO., 1890.
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PREFACE
This treatise was originally designed to explain the part of Statics required in the Previous Examination and the Second Examination for Ordinary Degrees in the University of Cambridge. It is now published in such a form, that, while serving its primary purpose, it may meet the requirements of Students in Schools, especially those who are preparing for the Local Examinations. It may also be regarded as an introduction to the works on Mechanics, which will appear in due course in Rivington's Mathematical Series.
The Examples have been selected from Papers set in University Examinations. The propositions requiring a knowledge of Trigonometry are marked with Roman numerals.
For some explanations of the Elementary Definitions I am indebted to the late Dr. Whewell's work on "The Philosophy of the Inductive Sciences;" and a special acknowledgment is due from me to Dr. Parkinson, for permission to make free use of his treatise on Mechanics.
In this, as in my other publications, I have been assisted in no slight degree by Mr E. J. Gross, of Gonville and Caius College, who has taken the most lively interest in revising and correcting all that I have written.
The Examples have been selected from Papers set in University Examinations. The propositions requiring a knowledge of Trigonometry are marked with Roman numerals.
For some explanations of the Elementary Definitions I am indebted to the late Dr. Whewell's work on "The Philosophy of the Inductive Sciences;" and a special acknowledgment is due from me to Dr. Parkinson, for permission to make free use of his treatise on Mechanics.
In this, as in my other publications, I have been assisted in no slight degree by Mr E. J. Gross, of Gonville and Caius College, who has taken the most lively interest in revising and correcting all that I have written.
CONTENTS.
CHAPTER I.
Definitions
CHAPTER II.
On Component and Resultant Forces
CHAPTER III.
The Parallelogram or Forces
CHAPTER IV.
On the Triangle and Polygon of Forces
CHAPTER V.
On the Resolution of Forces
CHAPTER VI.
On Parallel Forces
CHAPTER VII.
On the Equilibrium of a Body Movable round a Fixed Point
CHAPTER III.
On the Centre of Gravity
CHAPTER IX.
Of Moments
CHAPTER X.
Of Mechanical Instruments
Answers to Examples
CHAPTER I - Definitions
1. Matter is that, which can be perceived by the senses of sight and touch.
A Body is any portion of matter.
A Rigid Body is one, in which the different portions are held together in invariable positions with respect to each other.
A Particle or Material Point is a portion of matter, indefinitely small in all its dimensions: so that its length, breadth, and thickness are less than any assignable linear-magnitude.
2. Rest. When a body or particle constantly occupies the same position, it is said to be at rest.
Motion. When the position of a body or particle is being changed continuously, it is said to be in motion.
3. Force. Any cause, which changes or tends to change the state of rest or motion of a body or particle, is called force.
4. Statics is the science, which treats of the conditions, under which forces, acting on matter, produce rest
5. Line of Action. The line of action of a force is the line, in which a particle would begin to move in consequence of the action of the force.
6. The Forces, with which we are chiefly concerned in this treatise, may be roughly divided into three classes:
(1) PRESSURES. (2) TENSIONS. (3) ATTRACTIONS.
Of the first and second kinds of force we have illustrations in many actions of our daily life. Whether we push, pull, or lift a body, we bring into action a force acting by pressure or by tension. Imagine a gimlet to be firmly fixed in a block of wood. If we push the gimlet, we apply to the block a force acting by pressure. If we pull the gimlet, we apply to the block a force acting by tension.
Hence we obtain the following definitions:
Pressure. If one body be forced against another, each body is subjected to a force acting at the point of contact: such force is called pressure.
Tension. When a body is pulled by means of a string or rod, the force exerted along the string or rod is called tension.
If we consider a string as a line of consecutive particles, when a force is applied at each end of the string, each particle of the string is pulled in opposite directions by the forces, which the consecutive particles on either side of it are compelled to exercise upon it. These forces are called tensions^ and are the same at every particle of the string.
Suppose an engine, attached to a truck by a coupling-chain, to be just on the point of moving the truck. Each link of the chain is then acted upon by two equal and opposite forces, which act by means of the other links on either side of any particular link. The force, with which the part of the chain on one side of any particular link resists the force exerted along the chain on the other side of the link, is called the tension of the chain.
7. Attraction is a force less easily conceived than pressure or tension, because it arises from the action of one body on another at a distance from it.
Such is the influence of the magnet on the needle: such is the influence by which the Earth attracts to itself all bodies about it: and such is the influence by which the Sun and Planets attract each other.
8. All bodies fall, if unsupported, or tend to fall, if supported, towards the surface of the earth.
The direction, in which a particle would fall freely at any place, is called the vertical line at that place.
A plane perpendicular to this vertical line is said to be horizontal.
If a ball of lead be suspended at one end of a string, and we hold the other end of the string, we must exert a certain force to sustain the ball, equal to the force, with which the Earth attracts the ball. This latter force is called the Weight of the ball. Hence we obtain the following definition:
Weight or Gravity is the name given to the force with which the earth attracts a body.
The tendency of bodies to the Earth results from their attraction or Gravitation to the Earth. This tendency is only a particular instance of the attraction, which is exerted by every body upon those about it ; and this attraction of one body to another arises from the attraction of every particle of matter to every other, which is called Universal Gravitation.
9. Volume is the amount of space occupied by a body. The Volume, Bulk, or Solid Content of a body is measured by the number of times a certain cubical unit must be repeated, to fill up the space occupied by the body.
Thus, when we say that the volume of a body is 8 cubic inches, we mean that a cubical unit, which we call a cubic inch, must be repeated 8 times, to fill up the space occupied by one body.
10. The quantity of matter in a body is called the Mass of the body. The mass of a body is proportional to the Volume and the Density conjointly.
This definition gives us the meaning of Density, for it shows us that if twice the original quantity of matter be forced into a vessel of given capacity, the density will be doubled.
The mass of a body will be the same at all parts of the earth's surface, but the weight of a body differs in different latitudes. But at any given place the weight of a body is a practical measure of its mass.
11. Equilibrium. If several forces acting on a particle, or on a body, are so related, that no motion of the particle or the body takes place, the forces are said to be in equilibrium.
Two forces, which, acting in opposite directions, keep each other in equilibrium, are necessarily and manifestly equal. If we see two boys pulling at two ends of a rope, so that neither of them in the smallest degree prevails over the other, we have a case in which two forces are in equilibrium. If three hooks be fixed in a log of wood, two at one end and one at the other, and if the efforts of two boys pulling at ropes, attached to the two hooks at one end, be just counteracted by the effort of a man pulling at a rope, attached to the hook at the other end, we have a case in which three forces are in equilibrium: and this illustration may be extended to four five or more forces.
Again, if a number of rings be inserted round a block of wood, if a rope be attached to each ring, and a boy set to pull at each rope, it is easy to conceive such a disposition of the forces exerted by the boys, that no motion of the block may take place. Here then we have a case, in which a number of forces, not acting in parallel directions, are in equilibrium.
12. When two men pull at a rope in the same direction, we know that the force, which they exert, is equal to the sum of the forces, which they would separately exert. When two stones are put in a basket suspended by a string, their weights are added and the sum is supported by the string. Thus we see that forces, acting together in the same direction, may be added together to obtain their combined effect.
Since two opposite forces, which balance each other, are equal, each force is measured by that which it balances; and since forces are capable of addition, a force of any magnitude is measured by adding together a proper number of such equal forces.
Thus a heavy body, which, appended to an elastic spring, will draw it through one inch, may be taken as the unit of weight. Then, if we remove this body, and find a second heavy body, which will also draw the spring through one inch, this second body is also a unit of weight. In like manner we might go on to a third and a fourth equal body ; and adding together the two, or the three, or the four heavy bodies, we have a force twice, or three times, or four times the unit of weight. And with such a collection of heavy bodies, or weights, we can readily measure all other forces ; for, since forces that keep a body at rest must be equal in their opposite effect, we conclude that any statical force is measured by the weight which it will support,
13. To measure forces we fix upon some definite force for our standard, or unit, and then any other force is measured by the number of times it contains this unit, and this number is called the measure of the force.
14. Two forces are commensurable when a force can be taken as the standard of measurement, such that it is contained in each an exact number of times.
15. Method of estimating forces.
The three elements specifying a force, all of which must be known in order to estimate the effect of the force, are
(1) The point of application of the force.
(2) The direction in which the force acts.
(3) The magnitude of the force.
16. Method of representing forces.
Forces may be represented by straight lines: for (1) A straight line can be drawn from any point, and thus it will represent a force with respect to the point of application.
(2) A straight line can be drawn in any direction, and thus it will represent the direction of a force.
(3) A straight line can be drawn of such a length, as to contain as many units of length as the given force contains units of force, and thus it will represent the magnitude of a force.
A Body is any portion of matter.
A Rigid Body is one, in which the different portions are held together in invariable positions with respect to each other.
A Particle or Material Point is a portion of matter, indefinitely small in all its dimensions: so that its length, breadth, and thickness are less than any assignable linear-magnitude.
2. Rest. When a body or particle constantly occupies the same position, it is said to be at rest.
Motion. When the position of a body or particle is being changed continuously, it is said to be in motion.
3. Force. Any cause, which changes or tends to change the state of rest or motion of a body or particle, is called force.
4. Statics is the science, which treats of the conditions, under which forces, acting on matter, produce rest
5. Line of Action. The line of action of a force is the line, in which a particle would begin to move in consequence of the action of the force.
6. The Forces, with which we are chiefly concerned in this treatise, may be roughly divided into three classes:
(1) PRESSURES. (2) TENSIONS. (3) ATTRACTIONS.
Of the first and second kinds of force we have illustrations in many actions of our daily life. Whether we push, pull, or lift a body, we bring into action a force acting by pressure or by tension. Imagine a gimlet to be firmly fixed in a block of wood. If we push the gimlet, we apply to the block a force acting by pressure. If we pull the gimlet, we apply to the block a force acting by tension.
Hence we obtain the following definitions:
Pressure. If one body be forced against another, each body is subjected to a force acting at the point of contact: such force is called pressure.
Tension. When a body is pulled by means of a string or rod, the force exerted along the string or rod is called tension.
If we consider a string as a line of consecutive particles, when a force is applied at each end of the string, each particle of the string is pulled in opposite directions by the forces, which the consecutive particles on either side of it are compelled to exercise upon it. These forces are called tensions^ and are the same at every particle of the string.
Suppose an engine, attached to a truck by a coupling-chain, to be just on the point of moving the truck. Each link of the chain is then acted upon by two equal and opposite forces, which act by means of the other links on either side of any particular link. The force, with which the part of the chain on one side of any particular link resists the force exerted along the chain on the other side of the link, is called the tension of the chain.
7. Attraction is a force less easily conceived than pressure or tension, because it arises from the action of one body on another at a distance from it.
Such is the influence of the magnet on the needle: such is the influence by which the Earth attracts to itself all bodies about it: and such is the influence by which the Sun and Planets attract each other.
8. All bodies fall, if unsupported, or tend to fall, if supported, towards the surface of the earth.
The direction, in which a particle would fall freely at any place, is called the vertical line at that place.
A plane perpendicular to this vertical line is said to be horizontal.
If a ball of lead be suspended at one end of a string, and we hold the other end of the string, we must exert a certain force to sustain the ball, equal to the force, with which the Earth attracts the ball. This latter force is called the Weight of the ball. Hence we obtain the following definition:
Weight or Gravity is the name given to the force with which the earth attracts a body.
The tendency of bodies to the Earth results from their attraction or Gravitation to the Earth. This tendency is only a particular instance of the attraction, which is exerted by every body upon those about it ; and this attraction of one body to another arises from the attraction of every particle of matter to every other, which is called Universal Gravitation.
9. Volume is the amount of space occupied by a body. The Volume, Bulk, or Solid Content of a body is measured by the number of times a certain cubical unit must be repeated, to fill up the space occupied by the body.
Thus, when we say that the volume of a body is 8 cubic inches, we mean that a cubical unit, which we call a cubic inch, must be repeated 8 times, to fill up the space occupied by one body.
10. The quantity of matter in a body is called the Mass of the body. The mass of a body is proportional to the Volume and the Density conjointly.
This definition gives us the meaning of Density, for it shows us that if twice the original quantity of matter be forced into a vessel of given capacity, the density will be doubled.
The mass of a body will be the same at all parts of the earth's surface, but the weight of a body differs in different latitudes. But at any given place the weight of a body is a practical measure of its mass.
11. Equilibrium. If several forces acting on a particle, or on a body, are so related, that no motion of the particle or the body takes place, the forces are said to be in equilibrium.
Two forces, which, acting in opposite directions, keep each other in equilibrium, are necessarily and manifestly equal. If we see two boys pulling at two ends of a rope, so that neither of them in the smallest degree prevails over the other, we have a case in which two forces are in equilibrium. If three hooks be fixed in a log of wood, two at one end and one at the other, and if the efforts of two boys pulling at ropes, attached to the two hooks at one end, be just counteracted by the effort of a man pulling at a rope, attached to the hook at the other end, we have a case in which three forces are in equilibrium: and this illustration may be extended to four five or more forces.
Again, if a number of rings be inserted round a block of wood, if a rope be attached to each ring, and a boy set to pull at each rope, it is easy to conceive such a disposition of the forces exerted by the boys, that no motion of the block may take place. Here then we have a case, in which a number of forces, not acting in parallel directions, are in equilibrium.
12. When two men pull at a rope in the same direction, we know that the force, which they exert, is equal to the sum of the forces, which they would separately exert. When two stones are put in a basket suspended by a string, their weights are added and the sum is supported by the string. Thus we see that forces, acting together in the same direction, may be added together to obtain their combined effect.
Since two opposite forces, which balance each other, are equal, each force is measured by that which it balances; and since forces are capable of addition, a force of any magnitude is measured by adding together a proper number of such equal forces.
Thus a heavy body, which, appended to an elastic spring, will draw it through one inch, may be taken as the unit of weight. Then, if we remove this body, and find a second heavy body, which will also draw the spring through one inch, this second body is also a unit of weight. In like manner we might go on to a third and a fourth equal body ; and adding together the two, or the three, or the four heavy bodies, we have a force twice, or three times, or four times the unit of weight. And with such a collection of heavy bodies, or weights, we can readily measure all other forces ; for, since forces that keep a body at rest must be equal in their opposite effect, we conclude that any statical force is measured by the weight which it will support,
13. To measure forces we fix upon some definite force for our standard, or unit, and then any other force is measured by the number of times it contains this unit, and this number is called the measure of the force.
14. Two forces are commensurable when a force can be taken as the standard of measurement, such that it is contained in each an exact number of times.
15. Method of estimating forces.
The three elements specifying a force, all of which must be known in order to estimate the effect of the force, are
(1) The point of application of the force.
(2) The direction in which the force acts.
(3) The magnitude of the force.
16. Method of representing forces.
Forces may be represented by straight lines: for (1) A straight line can be drawn from any point, and thus it will represent a force with respect to the point of application.
(2) A straight line can be drawn in any direction, and thus it will represent the direction of a force.
(3) A straight line can be drawn of such a length, as to contain as many units of length as the given force contains units of force, and thus it will represent the magnitude of a force.
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