Dynamics for beginners - Lock
DYNAMICS FOR BEGINNERS
BY J. B. LOCK,
SENIOR FELLOW, ASSISTANT TUTOR AND LECTURER IN MATHEMATICS AND PHYSICS OF GONVILLE AND CAIUS COLLEGE; LECTURER IN PHYSICS IN THE UNIVERSITY OF CAMBRIDGE
LONDON; MACMILLAN AND CO.; 1887
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Dynamics for beginners
PREFACE.
This work has been written in the hope of supplying a want which I believe is very widely felt; viz. of a book which explains the elementary principles of Dynamics, illustrating them by numerous easy numerical examples in a manner suitable for use in Schools with boys of ordinary mathematical attainments.
I have adopted a suggestion, - due I believe to Mr Hay ward, F.R.S., of Harrow School - that the first part of the book should treat exclusively of Linear Dynamics; thus avoiding, at the beginning of the subject, all purely geometrical difficulties.
It will be found that, with the exception of one or two Articles, Sections I. and IV. (which may be read separately, and in which the fundamental Principles of Dynamics are explained) demand only a knowledge of Simple Equations in Algebra. In no case is any greater knowledge on the part of the Student assumed, than is denoted by Progressions in Algebra, the Trigonometry of one Angle, and (in Chapters IX. and X.) the Parabola.
I have ventured to suggest names for the units of velocity and acceleration, the use of which will be found to simplify considerably the language of the subject.
I should be greatly obliged to those who may make use of the book if they would point out any defects or obscurities in the text or would offer suggestions for its improvement.
I have adopted a suggestion, - due I believe to Mr Hay ward, F.R.S., of Harrow School - that the first part of the book should treat exclusively of Linear Dynamics; thus avoiding, at the beginning of the subject, all purely geometrical difficulties.
It will be found that, with the exception of one or two Articles, Sections I. and IV. (which may be read separately, and in which the fundamental Principles of Dynamics are explained) demand only a knowledge of Simple Equations in Algebra. In no case is any greater knowledge on the part of the Student assumed, than is denoted by Progressions in Algebra, the Trigonometry of one Angle, and (in Chapters IX. and X.) the Parabola.
I have ventured to suggest names for the units of velocity and acceleration, the use of which will be found to simplify considerably the language of the subject.
I should be greatly obliged to those who may make use of the book if they would point out any defects or obscurities in the text or would offer suggestions for its improvement.
TABLE OF CONTENTS.
SECTION I. RECTILINEAR DYNAMICS
CHAPTER I.
- Velocity
- Uniform Velocity
- Average Velocity
- Velocity not uniform
- Uniformly Increasing Velocity
- Acceleration
CHAPTER II.
- Mass and Force
- Mass-Velocity or Momentum
- Mass-Acceleration
- Weight
CHAPTER III.
- Inertia
- Stress
- Newton's Third Law of Motion
- Atwood's Machine
CHAPTER IV.
- Impulse
- Impact
- Elasticity
CHAPTER V.
- Change of Units
- C.G.S. System
SECTION II. DIRECTION
CHAPTER VI.
- Direction
- Parallelogram of Distances
- Resolution of Distances
- Projectiles
CHAPTER VII.
- Parallelogram of Velocities
- Triangle of Velocities
- Parallelogram of Acceleration
CHAPTER VIII.
- Parallelogram of Forces
- Smooth inclined Plane
- Chord of Quickest Descent
SECTION III. ILLUSTRATIONS
CHAPTER IX.
- Projectiles (continued)
- Path of a Projectile a Parabola
CHAPTER X.
- Oblique Impact
CHAPTER XI.
- Newton's Laws of Motion
- Uniform Motion in a Polygon
- Uniform Motion in a Circle
- Centripetal Force
- Area traced out at a constant Rate
CHAPTER XII.
- Relative Motion
CHAPTER XIII.
- The Hodograph
SECTION IV. ENERGY
CHAPTER XIV.
- Work
CHAPTER XV.
- Energy
- Potential Energy
- Kinetic Energy
CHAPTER XVI.
- Power
CHAPTER XVII.
- Energy is indestructible
EXAMINATION PAPERS
CHAPTER II - MASS AND FORCE
Consider some definite quantity of matter; say, a cubical lump of iron. This (i) consists of a certain kind of material (iron), (ii) has a definite shape, (iii) has a definite volume, (iv) contains a certain quantity of matter, (v) has a definite weight.
Each of these (i), (ii), (iii), (iv), (v) is a separate and distinct idea applicable to the lump of iron.
In dynamics we are chiefly concerned with the last two; viz., the quantity of matter, which we shall call the mass, and the weight.
We proceed to consider what is meant by mass.
We derive our idea of matter or mass from our muscular sense.
We also derive our idea of force from our muscular sense. When we exert our muscles we say we are applying force to something; and that to which we apply force we call mass.
In order to make an experiment with force we must have some means of applying a constant force to a given mass.
We know that when we compress a coiled steel spring in a certain way we have to exert a definite amount of force; also it is known that a spring can be made which will resist the same amount of force for a very considerable time.
The great difficulty in making an experiment which shall test the effect of a single force when applied to a certain mass, is the difficulty of getting rid of all other forces (such as friction, weight, etc.) so as to leave only the single force to be considered.
Let us imagine a perfectly smooth, perfectly horizontal sheet of ice. Upon this, place a lump of matter, such as a smooth block of stone in the form of a cube. Now take a steel spring arranged in some manner so that its state of compression can he easily observed (a spring letter-weighing machine for instance), and fastening this to the centre of one of the horizontal faces of the lump, apply a constant horizontal force by means of it to the lump, in the direction perpendicular to its face, taking care to keep the force constant and to apply it continuously and uniformly to the lump for an interval of, say, 10 seconds.
What happens? It will be found that by the force the lump has a certain amount of acceleration given to it; so that a velocity grows in the lump as long as the force is applied; and thus at the end of the 10 seconds the lump will have acquired a certain number of velos.
At the end of the 10 seconds let the force be withdrawn.
What happens? The lump has a certain number of velos; and it will be found that it will continue to move uniformly with that number of velos, so long as there is nothing in the nature of a force applied to it.
Suppose now that we try the experiment of Art. 28 with several lumps of the same size but of different material; say, one of lead, one of stone, one of cork; and suppose that by the aid of three exactly similar steel springs we can apply an equal constant horizontal force to each lump continuously for an interval of, say, 10 seconds.
What happens? It will be found that the lead, the stone, and the cork, have each a constant number of celos communicated by the force; but that the number of celos given to the cork is greater than the number given to the stone, and the number of celos given to the stone greater than the number given to the lead. So that by the end of the 10 seconds they will each have acquired a certain number of velos, the cork more than the stone, the stone more than the lead; also, that after the force has been withdrawn, each will continue to move with its own constant velocity, so long as there is nothing in the nature of a force applied to it.
DEF. We choose as our unit mass the mass of a certain lump of metal [called 1 lb. (avoirdupois)].
We call this unit mass a pound, or, 1 lb.
DEF. Force t is that which when applied to mass produces in it acceleration in the direction of the force; so that the force varies as the acceleration which it produces in a given mass; and also varies as the mass in which it can produce a given acceleration [see Art. 40].
DEF. We choose as our unit force that force which acting on a pound produces in it 1 celo.
We call this unit force a poundal.
The statement of Art. 31 is to be given the fullest possible interpretation; it asserts that
I. Force is that which produces acceleration in mass; therefore, whenever a mass has acceleration, it is under the action of some external force.
II. When no external force acts on a mass it has no acceleration, in other words, if at rest, the mass remains at rest, and if in motion, the mass continues to move with uniform velocity.
III. When the force applied to a certain mass is doubled or trebled, etc., then the acceleration produced is doubled or trebled, etc
IV. When the mass to which a certain force is applied is doubled, or trebled, etc., then the acceleration is halved, or divided by three, etc.
V. Every force applied to a mass produces in that mass its proper acceleration in its own direction, independently of any other motion which the mass may have.
Each of these (i), (ii), (iii), (iv), (v) is a separate and distinct idea applicable to the lump of iron.
In dynamics we are chiefly concerned with the last two; viz., the quantity of matter, which we shall call the mass, and the weight.
We proceed to consider what is meant by mass.
We derive our idea of matter or mass from our muscular sense.
We also derive our idea of force from our muscular sense. When we exert our muscles we say we are applying force to something; and that to which we apply force we call mass.
In order to make an experiment with force we must have some means of applying a constant force to a given mass.
We know that when we compress a coiled steel spring in a certain way we have to exert a definite amount of force; also it is known that a spring can be made which will resist the same amount of force for a very considerable time.
The great difficulty in making an experiment which shall test the effect of a single force when applied to a certain mass, is the difficulty of getting rid of all other forces (such as friction, weight, etc.) so as to leave only the single force to be considered.
Let us imagine a perfectly smooth, perfectly horizontal sheet of ice. Upon this, place a lump of matter, such as a smooth block of stone in the form of a cube. Now take a steel spring arranged in some manner so that its state of compression can he easily observed (a spring letter-weighing machine for instance), and fastening this to the centre of one of the horizontal faces of the lump, apply a constant horizontal force by means of it to the lump, in the direction perpendicular to its face, taking care to keep the force constant and to apply it continuously and uniformly to the lump for an interval of, say, 10 seconds.
What happens? It will be found that by the force the lump has a certain amount of acceleration given to it; so that a velocity grows in the lump as long as the force is applied; and thus at the end of the 10 seconds the lump will have acquired a certain number of velos.
At the end of the 10 seconds let the force be withdrawn.
What happens? The lump has a certain number of velos; and it will be found that it will continue to move uniformly with that number of velos, so long as there is nothing in the nature of a force applied to it.
Suppose now that we try the experiment of Art. 28 with several lumps of the same size but of different material; say, one of lead, one of stone, one of cork; and suppose that by the aid of three exactly similar steel springs we can apply an equal constant horizontal force to each lump continuously for an interval of, say, 10 seconds.
What happens? It will be found that the lead, the stone, and the cork, have each a constant number of celos communicated by the force; but that the number of celos given to the cork is greater than the number given to the stone, and the number of celos given to the stone greater than the number given to the lead. So that by the end of the 10 seconds they will each have acquired a certain number of velos, the cork more than the stone, the stone more than the lead; also, that after the force has been withdrawn, each will continue to move with its own constant velocity, so long as there is nothing in the nature of a force applied to it.
DEF. We choose as our unit mass the mass of a certain lump of metal [called 1 lb. (avoirdupois)].
We call this unit mass a pound, or, 1 lb.
DEF. Force t is that which when applied to mass produces in it acceleration in the direction of the force; so that the force varies as the acceleration which it produces in a given mass; and also varies as the mass in which it can produce a given acceleration [see Art. 40].
DEF. We choose as our unit force that force which acting on a pound produces in it 1 celo.
We call this unit force a poundal.
The statement of Art. 31 is to be given the fullest possible interpretation; it asserts that
I. Force is that which produces acceleration in mass; therefore, whenever a mass has acceleration, it is under the action of some external force.
II. When no external force acts on a mass it has no acceleration, in other words, if at rest, the mass remains at rest, and if in motion, the mass continues to move with uniform velocity.
III. When the force applied to a certain mass is doubled or trebled, etc., then the acceleration produced is doubled or trebled, etc
IV. When the mass to which a certain force is applied is doubled, or trebled, etc., then the acceleration is halved, or divided by three, etc.
V. Every force applied to a mass produces in that mass its proper acceleration in its own direction, independently of any other motion which the mass may have.
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