Now Ready. Globe 8vo. Price 2s. 6d. MECHANICS FOR BEGINNERS BY W. GALLATLY, M.A. FORMERLY SCHOLAR OF PEMBROKE COLLEGE, CAMBRIDGE Adapted to the Elementary Stage of the South Kensington Syllabus London MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY All rights reserved CONTENTS Attention is called to the following special characteristics of the book: 1. It contains the unprecedented number of eight hundred examples, of which one hundred and sixty are worked in full. 2. Great attention is given to Work, Power, and Energy. In Practical Mechanics these subjects are supremely important; and considerable stress is being laid on them in public examinations. 3. Problems of the same type are classed together in small sections, the method of dealing with each section being clearly explained by a worked example. The work will be found suitable for the following examinations: 1. London University Matriculation. 2. Science and Art: Stage I. 3. Cambridge Local: Junior. 4. Oxford Local: Junior. 5. College of Preceptors: Professional Preliminary and Certificate. 6. Woolwich: Obligatory. 7. Royal University of Ireland: Matriculation. 8. Glasgow University: Preliminary. 9. Glasgow Technical College Preliminary. 10. St. Andrews University: Preliminary. 11. Victoria University: Preliminary. 12. Owens College: First Year. 13. Birmingham University; Junior Physics. 14. Firth College: Elementary Mechanics. PRESS OPINIONS NATURE" The student who uses the treatise as a text-book will be equipped for almost any examination in elementary theoretical mechanics.' GUARDIAN.-"There is much to praise in the book. The principles of Work and Energy have never been so simply and clearly stated in an elementary treatise." SCHOOLMASTER.-"Here everything possible seems to have been done to make the path a pleasant and easy one for the beginner. We feel certain the book will be received with marked favour." Now Ready. Crown 8vo. Price 4s. 6d. A TREATISE ON ELEMENTARY DYNAMICS DEALING WITH RELATIVE MOTION MAINLY IN TWO DIMENSIONS BY H. A. ROBERTS, M.A. London MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY All rights reserved This text-book assumes on the part of the student a knowledge of Trigonometry up to the Solution of Triangles, and of the simpler parts of Analytical Geometry. For the benefit of students who may be beginning to read the Differential Calculus, occasional reference is made, in the smaller print, to the notation of that subject. The Kinematical Section is fuller than has hitherto been usual in books of this class; the treatment of independent velocities occupies more space, that of acceleration is carried further, and such problems as Simple Harmonic Motion occur in their logical place. Mass is discussed from the point of view of inverse acceleration-ratios. Newton's formulation of the Laws of Motion is, as far as possible, retained: but an attempt is made, following modern authorities, to interpret the Laws so as to keep in view the Relativity of Motion. An entire chapter is devoted to Work and Energy. The Velocity diagram and the Hodograph are constantly employed; for example, the former is often of use in discussing the Impact of Particles, and the latter in discussing the Motion of Projectiles. THE LAWS OF MOTION. 73 If now we choose a certain particle A as being of unit mass, a particle B whose mass is m is such that magnitude of acceleration of A due to B m magnitude of acceleration of B due to A1 Thus the mass-ratio of the particle B to the unit particle A is a number m, which we may call the mass of B. Similarly, in the case of another particle C, the mass-ratio of C to A is a number m', which we may call the mass of C. If now the particles B, C be tested by experiment, we find, as previously remarked, that their mass-ratio is a constant number. Moreover, a comparison of experimental results shows that this number is equal to m m' The same is, of course, true of any other particles. Thus there is associated with each particle a certain number called its mass, such that the ratio of the masses of any two particles is equal to their mass-ratio as above defined. When two or more particles are rigidly connected so as to form a single particle, the mass of the single particle is found to be equal to the arithmetic sum of the masses of the separate particles composing it. For instance, the mass of a leaden bullet A, formed by melting together two other leaden bullets B and C, is the sum of the masses of B and C. Two homogeneous particles of the same material and volume are found to have equal masses. Hence the masses of homogeneous particles of the same material are proportional to their volumes, a result in accordance with observed facts. The mass of a particle, as is seen from the above discussion, is a constant scalar quantity. The constancy of mass has usually been assumed owing to Newton's definition of it. Units of Mass. The British unit of mass, i.e. the body regarded as a particle whose mass is arbitrarily assumed to be unity, is a lump of platinum kept at Westminster, and called the Imperial Standard Pound. The C.G.S. unit of mass is 100 part of the mass of a lump of platiniridium kept at Paris, and called the International Prototype Kilogram. This unit is called a Gram. The Kilogram was designed to represent, and approximately does so, the mass of 1000 cubic centimetres of distilled water at 4° C. and 760 millimetres barometric pressure. Our definition of mass is now complete as far as Galileo's axes are concerned; we shall presently extend it. As we shall give an independent proof of these important properties in the next article, we leave the above deductions as an exercise to the student. 157. To find from the hodograph the range of a projectile on a given inclined plane through the point of projection at right angles to the vertical plane of the path, and to discuss the circumstances of projection. (1) Range. Let QQ' be the range on the inclined plane, qq' the hodograph for the portion of the path QQ, o the pole of the hodograph, the middle point of qq', t the time from Q to Q'. Then (§ 40, ii., Cor.) ol is the average velocity from Q to Q'. Therefore the range=QQ'=ol. t. 2ql where u, v are the mag g nitudes of the (oblique) components of the initial velocity parallel to the plane and the vertical. This form of the expression for the range, which is convenient to remember, may be at once reduced to that of the last article. We have, in fact, oq = V, ol=u, lq=v, angle qol=a-i, angle oqla, angle olq=+i. Now Ready. Globe 8vo. Price 2s. 6d. ELEMENTARY MECHANICS OF SOLIDS BY W. T. A. EMTAGE, M.A. (OXON.) DIRECTOR OF PUBLIC INSTRUCTION IN MAURITIUS; LATE PRINCIPAL OF THE TECHNICAL IN UNIVERSITY COLLEGE, NOTTINGHAM, AND EXAMINER IN THE FINAL London MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY All rights reserved 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. 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 Acclerations; Uniform Circular Motion. Chapter XX. Simple Harmonic Motion; Pendulums. |