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The principles of mechanics - Crew
THE PRINCIPLES OF MECHANICS
For students of physics and engineering.
BY HENRY CREW
LONGMANS, GREEN, AND CO., NEW YORK, 1908
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PREFACE
The following pages represent a lecture course which during several years past has been given to second-year students in physics at Northwestern University.
The prerequisites have been a course in general physics and a course, either concurrent or antecedent, in the calculus.
The author's efforts have been:
(1) To lead the student to clear dynamical views in the shortest possible time, without sacrificing him upon the altar of logic, yet pursuing a route which he can afterwards follow with safety.
(2) To build the discussion upon a few simple experiments and upon definitions which convey at once the physical meaning of the quantities defined. Thus, torque is introduced, not as the vector product of force and distance, but as the time rate of variation of angular momentum. Likewise moment of inertia is presented at the outset as the rotational inertia of a rigid body, and not as the integral of the second moment of the mass.
(3) To follow the example of Foppl in using vector analysis merely to present a clear, simple, and accurate picture of the facts, reserving the Cartesian analysis for purposes of computation.
(4) To confine the treatment to that part of mechanics which is common ground for the physicist and the engineer.
(5) To reduce the inherent difficulties of the subject to a minimum by treating dynamics in two analogous parts - rotational and translational - such that if either one is given the other may be immediately deduced. This point of view - this parallel treatment - which carries with it great economy of thought is already very old - dating at least from the introduction of generalized coordinates - but seems to have been fully utilized in few of our modern textbooks.
(6) To employ only two systems of units, the absolute C. G. S. and the "British Engineers."
All teachers of dynamics, it may fairly be supposed, hope to convince the student that the entire science of mechanics is practically nothing else than the application of Newton's Three Laws of Motion and the Principle of the Conservation of Energy to certain special systems, subject to certain boundary conditions. Since each of these laws is subject to mathematical expression, the science is quantitative throughout. From Galileo to Hertz, geometry and analysis have shown themselves indispensable to mechanics.
If this be true, the all-important matter for the engineer and the physicist is to see that the subject does not degenerate into a mere set of problems in illustration of the integral calculus and to see that equations are used only to quantify certain experimental results and to predict certain facts of nature. Which of these viewpoints shall dominate the classroom depends very little upon the text employed, very largely upon the attitude of the instructor.
In conclusion, the author wishes to express his obligations to Professor Malcolm McNeill of Lake Forest
The prerequisites have been a course in general physics and a course, either concurrent or antecedent, in the calculus.
The author's efforts have been:
(1) To lead the student to clear dynamical views in the shortest possible time, without sacrificing him upon the altar of logic, yet pursuing a route which he can afterwards follow with safety.
(2) To build the discussion upon a few simple experiments and upon definitions which convey at once the physical meaning of the quantities defined. Thus, torque is introduced, not as the vector product of force and distance, but as the time rate of variation of angular momentum. Likewise moment of inertia is presented at the outset as the rotational inertia of a rigid body, and not as the integral of the second moment of the mass.
(3) To follow the example of Foppl in using vector analysis merely to present a clear, simple, and accurate picture of the facts, reserving the Cartesian analysis for purposes of computation.
(4) To confine the treatment to that part of mechanics which is common ground for the physicist and the engineer.
(5) To reduce the inherent difficulties of the subject to a minimum by treating dynamics in two analogous parts - rotational and translational - such that if either one is given the other may be immediately deduced. This point of view - this parallel treatment - which carries with it great economy of thought is already very old - dating at least from the introduction of generalized coordinates - but seems to have been fully utilized in few of our modern textbooks.
(6) To employ only two systems of units, the absolute C. G. S. and the "British Engineers."
All teachers of dynamics, it may fairly be supposed, hope to convince the student that the entire science of mechanics is practically nothing else than the application of Newton's Three Laws of Motion and the Principle of the Conservation of Energy to certain special systems, subject to certain boundary conditions. Since each of these laws is subject to mathematical expression, the science is quantitative throughout. From Galileo to Hertz, geometry and analysis have shown themselves indispensable to mechanics.
If this be true, the all-important matter for the engineer and the physicist is to see that the subject does not degenerate into a mere set of problems in illustration of the integral calculus and to see that equations are used only to quantify certain experimental results and to predict certain facts of nature. Which of these viewpoints shall dominate the classroom depends very little upon the text employed, very largely upon the attitude of the instructor.
In conclusion, the author wishes to express his obligations to Professor Malcolm McNeill of Lake Forest
TABLE OF CONTENTS
KINEMATICS
Introductory
Position of a Particle
Vectors
Position of a Body
Change of Position; Linear Displacement
Analogue. Angular Displacement
Screw Motion
Rate of Change of Position; Velocity
Digression on Vector Algebra
Analogue. Rate of Change of Angular Displacement
Rate of Change of Velocity; Acceleration
Three Important Special Cases
Rectilinear Motion
Uniform Circular Motion
Simple Harmonic Motion
Analogue. Angular Acceleration
Three Important Special Cases
Rotation about a Fixed Axis
Spin Constant: Direction Varying
Simple Harmonic Oscillation
KINETICS
The Idea of Mass
Center of Mass
The Idea of Momentum
Analogue. The Idea of Rotational Inertia
Analogue. The Idea op Angular Momentum
Force and its Analogue, Torque
Newton's Laws of Motion
General Equations of Motion for a Particle
Rigorous Definition of Rotational Inertia
Digression on the Computation of Moments of Inertia
General Equations of Motion for Rigid Body
Cross-over Equations of Motion
Energy
Digression on Equilibrium
Power
Units of Kinematics and Dynamics
SOME APPLICATIONS OF GENERAL. PRINCIPLES TO SPECIAL PROBLEMS
Robin's Ballistic Pendulum
The Simple Pendulum
The Physical Pendulum
The Reversible Pendulum
Equivalent Mass of Vibrating Spiral Spring
Rigid Body set into Rotation by an Impulse
Center of Percussion of Slender Rod
Rotation of a Rigid Body (continued)
Rotation of a Rigid Body; More General Case
Centrifugal Couple, Numerical Illustration of
Rotation of a Rigid Body. No External Force
FRICTION
Definition of Friction
Angle of Repose. Friction Cone
Friction at Bearings. Friction Circle
Work of Friction
Fluid Friction
Free Oscillations of Loaded Spiral Spring. Theory of Damping
Digression on Centrifugal Force and its Analogue, Precessional Couple
Digression on Inertia Skeletons and Products op Inertia
DYNAMICS OF ELASTIC BODIES
Rough Definitions of Stress and Strain
Distinction between Solids and Fluids
Kinematics of Strain
Simple Extension
Simple Shear
Homogeneous Strain
Cubical Dilatation
Pure Shear in Terms of Extension and Contraction
General Shear in Terms of Extension and Contraction
Heterogeneous Strains
Analysis op Stress
Simple Longitudinal Stress
Simple Shearing Stress
Most General Stress at a Point
Fluid Pressure
Pure Shearing Stress in Terms of Simple Push and Pull
General Shearing Stress in Terms of Push and Pull
Heterogeneous Stress
Hooke's Law
Elastic Moduli
Bulk Modulus
Rigidity Modulus
Young's Modulus
Poisson's Ratio
Experimental Determination of Elastic Constants
FLUID MOTION,
Hydrostatics
Equations of Equilibrium
Theory of Barometer
Center of Pressure
Center of Buoyancy
Stability of a Floating Body
Hydrokinetics, General Case
Case of Very Slow Motions
Case of Steady Motion
Bernoulli's Theorem
Torricelli's Theorem
Form of Free Surface in Vessel of Rotating Liquid
INTRODUCTORY.
1. In general, the pursuit of any particular study presumes a certain amount of previous training. In the case of Dynamics, it is necessary to assume that the student is more or less familiar with the language of mathematical analysis; and in particular that he has at least a slight acquaintance with Analytical Geometry and with the elements of the Infinitesimal Calculus.
In order to take up simpler matters first, and to make sure that reader and writer are each using the same vocabulary, a brief resume of the geometry of motion - generally called Kinematics - is here given. The necessity for introducing Kinematics at this point lies also in the fact that Dynamics is an attempt to draw a picture of the physical universe in the briefest and most accurate terms; and many of the best terms for this purpose are those of Kinematics.
The masses, forces, and energies which are studied under the head of Dynamics proper, do not here come up for consideration; for Kinematics deals only with the motions of mathematical points and of geometrical figures. Kinematics is not, therefore, a branch of physical science, but belongs rather to mathematics. It is a science which was described by Lagrange as the geometry of four dimensions, because it deals with four, and only four, independent variables, namely, time and the three space coordinates which determine the position of a point. The name, Kinematics, is due to Ampere.
2. A body is a limited portion of matter; but, when we have to deal with a body which is so small that its dimensions may be neglected in comparison with other distances involved, we call this body a material particle. Thus the radius of the earth is so small in comparison with the radius of its orbit about the sun, that the astronomer ordinarily treats the earth as a particle; while for the geologist the earth is a large and important body. The shot of a 13-inch rifle is treated as a particle by one who is computing its range; but for the man who is shaping the shot on the lathe, or for the sailor who is loading it upon the ammunition-hoist, it is a body of very considerable dimensions.
In what immediately follows we shall have frequent occasion to speak of particles and of bodies; but during the entire treatment of Kinematics we shall consider them both as massless; in other words, we shall for the present employ the word “particle” to denote a mathematical point, assigning to it only the property of position; in like manner the word "body'' will be used to denote a geometrical figure. Later, when we reach the subject of Dynamics proper, we shall see that the mass of a body is an all- important factor.
DOWNLOAD FREE BOOK: The principles of mechanics
In order to take up simpler matters first, and to make sure that reader and writer are each using the same vocabulary, a brief resume of the geometry of motion - generally called Kinematics - is here given. The necessity for introducing Kinematics at this point lies also in the fact that Dynamics is an attempt to draw a picture of the physical universe in the briefest and most accurate terms; and many of the best terms for this purpose are those of Kinematics.
The masses, forces, and energies which are studied under the head of Dynamics proper, do not here come up for consideration; for Kinematics deals only with the motions of mathematical points and of geometrical figures. Kinematics is not, therefore, a branch of physical science, but belongs rather to mathematics. It is a science which was described by Lagrange as the geometry of four dimensions, because it deals with four, and only four, independent variables, namely, time and the three space coordinates which determine the position of a point. The name, Kinematics, is due to Ampere.
2. A body is a limited portion of matter; but, when we have to deal with a body which is so small that its dimensions may be neglected in comparison with other distances involved, we call this body a material particle. Thus the radius of the earth is so small in comparison with the radius of its orbit about the sun, that the astronomer ordinarily treats the earth as a particle; while for the geologist the earth is a large and important body. The shot of a 13-inch rifle is treated as a particle by one who is computing its range; but for the man who is shaping the shot on the lathe, or for the sailor who is loading it upon the ammunition-hoist, it is a body of very considerable dimensions.
In what immediately follows we shall have frequent occasion to speak of particles and of bodies; but during the entire treatment of Kinematics we shall consider them both as massless; in other words, we shall for the present employ the word “particle” to denote a mathematical point, assigning to it only the property of position; in like manner the word "body'' will be used to denote a geometrical figure. Later, when we reach the subject of Dynamics proper, we shall see that the mass of a body is an all- important factor.
DOWNLOAD FREE BOOK: The principles of mechanics
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