Tài liệu Schaum_s outline of college physics 9th

SCHAUM'S OUTLINE OF THEORY AND PROBLEMS OF COLLEGE PHYSICS Ninth Edition . FREDERICK J. BUECHE, Ph.D. Distinguished Professor at Large University of Dayton EUGENE HECHT, Ph.D. Professor of Physics Adelphi University . SCHAUM'S OUTLINE SERIES McGRAW-HILL New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto abc McGraw-Hill Copyright © 1997, 1989, 1979, 1961, 1942, 1940, 1939, 1936 by The McGraw-Hill Companies, Inc All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-1367497 The material in this eBook also appears in the print version of this title: 0-07-008941-8. 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DOI: 10.1036/0071367497 Preface The introductory physics course, variously known as ``general physics'' or ``college physics,'' is usually a two-semester in-depth survey of classical topics capped o€ with some selected material from modern physics. Indeed the name ``college physics'' has become a euphemism for introductory physics without calculus. Schaum's Outline of College Physics was designed to uniquely complement just such a course, whether given in high school or in college. The needed mathematical knowledge includes basic algebra, some trigonometry, and a tiny bit of vector analysis. It is assumed that the reader already has a modest understanding of algebra. Appendix B is a general review of trigonometry that serves nicely. Even so, the necessary ideas are developed in place, as needed. And the same is true of the rudimentary vector analysis that's requiredÐit too is taught as the situation requires. In some ways learning physics is unlike learning most other disciplines. Physics has a special vocabulary that constitutes a language of its own, a language immediately transcribed into a symbolic form that is analyzed and extended with mathematical logic and precision. Words like energy, momentum, current, ¯ux, interference, capacitance, and so forth, have very speci®c scienti®c meanings. These must be learned promptly and accurately because the discipline builds layer upon layer; unless you know exactly what velocity is, you cannot know what acceleration or momentum are, and without them you cannot know what force is, and on and on. Each chapter in this book begins with a concise summary of the important ideas, de®nitions, relationships, laws, rules, and equations that are associated with the topic under discussion. All of this material constitutes the conceptual framework of the discourse, and its mastery is certainly challenging in and of itself, but there's more to physics than the mere recitation of its principles. Every physicist who has ever tried to teach this marvelous subject has heard the universal student lament, ``I understand everything; I just can't do the problems.'' Nonetheless most teachers believe that the ``doing'' of problems is the crucial culmination of the entire experience, it's the ultimate proof of understanding and competence. The conceptual machinery of de®nitions and rules and laws all come together in the process of problem solving as nowhere else. Moreover, insofar as the problems re¯ect the realities of our world, the student learns a skill of immense practical value. This is no easy task; carrying out the analysis of even a moderately complex problem requires extraordinary intellectual vigilance and un¯agging attention to detail above and beyond just ``knowing how to do it.'' Like playing a musical instrument, the student must learn the basics and then practice, practice, practice. A single missed note in a sonata is overlookable; a single error in a calculation, however, can propagate through the entire e€ort producing an answer that's completely wrong. Getting it right is what this book is all about. Although a selection of new problems has been added, the 9th-edition revision of this venerable text has concentrated on modernizing the work, and improving the pedagogy. To that end, the notation has been simpli®ed and made consistent throughout. For example, force is now symbolized by F and only F; thus centripetal force is FC, weight is FW, tension is FT, normal force is FN, friction is Ff, and so on. Work (W ) will never again be confused with weight (FW), and period iii Copyright 1997, 1989, 1979, 1961, 1942, 1940, 1939, 1936 The McGraw-Hill Companies, Inc. Click Here for Terms of Use. iv SIGNIFICANT FIGURES (T ) will never be mistaken for tension (FT). To better match what's usually written in the classroom, a vector is now indicated by a boldface symbol with a tiny arrow above it. The idea of signi®cant ®gures is introduced (see Appendix A) and scrupulously adhered to in every problem. Almost all the de®nitions have been revised to make them more precise or to re¯ect a more modern perspective. Every drawing has been redrawn so that they are now more accurate, realistic, and readable. If you have any comments about this edition, suggestions for the next edition, or favorite problems you'd like to share, send them to E. Hecht, Adelphi University, Physics Department, Garden City, NY 11530. Freeport, NY EUGENE HECHT Contents Chapter 1 INTRODUCTION TO VECTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 UNIFORMLY ACCELERATED MOTION . . . . . . . . . . . . . . . . . . . . . 13 Chapter 3 NEWTON'S LAWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Chapter 4 EQUILIBRIUM UNDER THE ACTION OF CONCURRENT FORCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 EQUILIBRIUM OF A RIGID BODY UNDER COPLANAR FORCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Scalar quantity. Vector quantity. Resultant. Graphical addition of vectors (polygon method). Parallelogram method. Subtraction of vectors. Trigonometric functions. Component of a vector. Component method for adding vectors. Unit vectors. Displacement. Speed. Velocity. Acceleration. Uniformly accelerated motion along a straight line. Direction is important. Instantaneous velocity. Graphical interpretations. Acceleration due to gravity. Velocity components. Projectile problems. Mass. Standard kilogram. Force. Net external force. The newton. Newton's First Law. Newton's Second Law. Newton's Third Law. Law of universal gravitation. Weight. Relation between mass and weight. Tensile force. Friction force. Normal force. Coecient of kinetic friction. Coecient of static friction. Dimensional analysis. Mathematical operations with units. Concurrent forces. An object is in equilibrium. First condition for equilibrium. Problem solution method (concurrent forces). Weight of an object. Tensile force. Friction force. Normal force. Chapter 5 Torque (or moment). Two conditions for equilibrium. Position of the axis is arbitrary. Center of gravity. Chapter 6 WORK, ENERGY, AND POWER. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Chapter 7 SIMPLE MACHINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Work. Unit of work. Energy. Kinetic energy. Gravitational potential energy. Work-energy theorem. Conservation of energy. Power. Kilowatt-hour. A machine. Principle of work. Mechanical advantage. Eciency. v Copyright 1997, 1989, 1979, 1961, 1942, 1940, 1939, 1936 The McGraw-Hill Companies, Inc. Click Here for Terms of Use. vi PREFACE Chapter 8 IMPULSE AND MOMENTUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Chapter 9 ANGULAR MOTION IN A PLANE . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Chapter 10 RIGID-BODY ROTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Chapter 11 SIMPLE HARMONIC MOTION AND SPRINGS. . . . . . . . . . . . . . . . 126 Chapter 12 DENSITY; ELASTICITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Chapter 13 FLUIDS AT REST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Chapter 14 FLUIDS IN MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Chapter 15 THERMAL EXPANSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Linear momentum. Impulse. Impulse causes change in momentum. Conservation of linear momentum. Collisions and explosions. Perfectly elastic collision. Coecient of restitution. Center of mass. Angular displacement. Angular speed. Angular acceleration. Equations for uniformly accelerated motion. Relations between angular and tangential quantities. Centripetal acceleration. Centripetal force. Torque (or moment). Moment of inertia. Torque and angular acceleration. Kinetic energy of rotation. Combined rotation and translation. Work. Power. Angular momentum. Angular impulse. Parallel-axis theorem. Analogous linear and angular quantities. Period. Frequency. Graph of a vibratory motion. Displacement. Restoring force. Simple harmonic motion. Hookean system. Elastic potential energy. Energy interchange. Speed in SHM. Acceleration in SHM. Reference circle. Period in SHM. Acceleration in terms of T. Simple pendulum. SHM. Mass density. Speci®c gravity. Elasticity. Stress. Strain. Young's modulus. Bulk modulus. Shear modulus. Elastic limit. Average pressure. Standard atmospheric pressure. Hydrostatic pressure. Pascal's principle. Archimedes' principle. Fluid ¯ow or discharge. Equation of continuity. Shear rate. Viscosity. Poiseuille's Law. Work done by a piston. Work done by a pressure. Bernoulli's equation. Torricelli's theorem. Reynolds number. Temperature. expansion. Linear expansion of solids. Area expansion. Volume vii SIGNIFICANT FIGURES Chapter 16 IDEAL GASES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Chapter 17 KINETIC THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Chapter 18 HEAT QUANTITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Chapter 19 TRANSFER OF HEAT ENERGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Chapter 20 FIRST LAW OF THERMODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . 198 Chapter 21 ENTROPY AND THE SECOND LAW . . . . . . . . . . . . . . . . . . . . . . . . 209 Chapter 22 WAVE MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Chapter 23 SOUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Ideal (or perfect) gas. One mole of a substance. Ideal Gas Law. Special cases. Absolute zero. Standard conditions or standard temperature and pressure (S.T.P.). Dalton's Law of partial pressures. Gas-law problems. Kinetic theory. Avogadro's number. Mass of a molecule. Average translational kinetic energy. Root mean square speed. Absolute temperature. Pressure. Mean free path. Thermal energy. Heat. Speci®c heat. Heat gained (or lost). Heat of fusion. Heat of vaporization. Heat of sublimation. Calorimetry problems. Absolute humidity. Relative humidity. Dew point. Energy can be transferred. Radiation. Conduction. Thermal resistance. Convection. Heat. Internal energy. Work done by a system. First Law of Thermodynamics. Isobaric process. Isovolumic process. Isothermal process. Adiabatic process. Speci®c heats of gases. Speci®c heat ratio. Work related to area. Eciency of a heat engine. Second Law of Thermodynamics. Most probable state. Entropy. Entropy is a measure of disorder. Propagating wave. Wave terminology. In-phase vibrations. Speed of a transverse wave. Standing waves. Conditions for resonance. Longitudinal (compressional) waves. Sound waves. Equations for sound speed. Speed of sound in air. Loudness. Intensity (or loudness) level. Beats. Doppler e€ect. Interference e€ects. Intensity. viii CONTENTS Chapter 24 COULOMB'S LAW AND ELECTRIC FIELDS . . . . . . . . . . . . . . . . . . 232 Chapter 25 POTENTIAL; CAPACITANCE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Chapter 26 CURRENT, RESISTANCE, AND OHM'S LAW . . . . . . . . . . . . . . . . . 256 Chapter 27 ELECTRICAL POWER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Chapter 28 EQUIVALENT RESISTANCE; SIMPLE CIRCUITS . . . . . . . . . . . . . . 270 Chapter 29 KIRCHHOFF'S LAWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Chapter 30 FORCES IN MAGNETIC FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Chapter 31 SOURCES OF MAGNETIC FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . 299 Chapter 32 INDUCED EMF; MAGNETIC FLUX . . . . . . . . . . . . . . . . . . . . . . . . . 305 Coulomb's Law. Charge quantized. Conservation of charge. Test-charge concept. Electric ®eld. Strength of the electric ®eld. Electric ®eld due to a point charge. Superposition principle. Potential di€erence. Absolute potential. Electrical potential energy. V related to E. Electron volt energy unit. Capacitor. Parallel-plate capacitor. Capacitors in parallel and series. Energy stored in a capacitor. Current. Battery. Resistance. Ohm's Law. Measurement of resistance by ammeter and voltmeter. Terminal potential di€erence. Resistivity. Resistance varies with temperature. Potential changes. Electrical work. Electrical power. Power loss in a resistor. generated in a resistor. Convenient conversions. Resistors in series. Thermal energy Resistors in parallel. Kirchho€'s node (or junction) rule. equations obtained. Kirchho€'s loop (or circuit) rule. Set of Magnetic ®eld. Magnetic ®eld lines. Magnet. Magnetic poles. Charge moving through a magnetic ®eld. Direction of the force. Magnitude of the force. Magnetic ®eld at a point. Force on a current in a magnetic ®eld. Torque on a ¯at coil. Magnetic ®elds are produced. Direction of the magnetic ®eld. Ferromagnetic materials. Magnetic moment. Magnetic ®eld of a current element. Magnetic e€ects of matter. Magnetic ®eld lines. Magnetic ¯ux. Faraday's Law for induced emf. Lenz's Law. Motional emf. Induced emf. ix SIGNIFICANT FIGURES Chapter 33 ELECTRIC GENERATORS AND MOTORS . . . . . . . . . . . . . . . . . . . 315 Chapter 34 INDUCTANCE; R-C AND R-L TIME CONSTANTS . . . . . . . . . . . . . 321 Chapter 35 ALTERNATING CURRENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Chapter 36 REFLECTION OF LIGHT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 Chapter 37 REFRACTION OF LIGHT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Chapter 38 THIN LENSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Chapter 39 OPTICAL INSTRUMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Chapter 40 INTERFERENCE AND DIFFRACTION OF LIGHT . . . . . . . . . . . . . . 366 Chapter 41 RELATIVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Electric generators. Electric motors. Self-inductance. Mutual inductance. Energy stored in an inductor. constant. R-L time constant. Exponential functions. R-C time Emf generated by a rotating coil. Meters. Thermal energy generated or power lost. Forms of Ohm's Law. Phase. Impedance. Phasors. Resonance. Power loss. Transformer. Nature of light. Law of re¯ection. Plane mirrors. Mirror equation. Size of the image. Speed of light. Index of refraction. total internal re¯ection. Prism. Refraction. Type of lenses. Object and image relation. Lenses in contact. Combination of thin lenses. Telescope. Spherical mirrors. Snell's Law. Critical angle for Lensmaker's equation. The eye. Magnifying glass. Lens power. Microscope. Coherent waves. Relative phase. Interference e€ects. Di€raction. Single-slit di€raction. Limit of resolution. Di€raction grating equation. Di€raction of X-rays. Optical path length. Reference frame. Special theory of relativity. Relativistic linear momentum. Limiting speed. Relativistic energy. Time dilation. Simultaneity. Length contraction. Velocity addition formula. x CONTENTS Chapter 42 QUANTUM PHYSICS AND WAVE MECHANICS. . . . . . . . . . . . . . . 382 Chapter 43 THE HYDROGEN ATOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Chapter 44 MULTIELECTRON ATOMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Chapter 45 NUCLEI AND RADIOACTIVITY. . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Chapter 46 APPLIED NUCLEAR PHYSICS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Appendix A SIGNIFICANT FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Appendix B TRIGONOMETRY NEEDED FOR COLLEGE PHYSICS . . . . . . . . . . 419 Appendix C EXPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Appendix D LOGARITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Appendix E PREFIXES FOR MULTIPLES OF SI UNITS; THE GREEK ALPHABET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Appendix F FACTORS FOR CONVERSIONS TO SI UNITS . . . . . . . . . . . . . . . . . 428 Appendix G PHYSICAL CONSTANTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Appendix H TABLE OF THE ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 INDEX ...................................................... 433 Quanta of radiation. Photoelectric e€ect. Momentum of a photon. Compton e€ect. De Broglie waves. Resonance of de Broglie waves. Quantized energies. Hydrogen atom. Electron orbits. Energy-level diagrams. Emission of light. Spectral lines. Origin of spectral series. Absorption of light. Neutral atom. Quantum numbers. Pauli exclusion principle. Nucleus. Nuclear charge and atomic number. Atomic mass unit. Mass number. Isotopes. Binding energies. Radioactivity. Nuclear equations. Nuclear binding energies. Fission reaction. Fusion reaction. Radiation dose. Radiation damage potential. E€ective radiation dose. High-energy accelerators. Momentum of a particle. Chapter 1 Introduction to Vectors A SCALAR QUANTITY, or scalar, is one that has nothing to do with spatial direction. Many physical concepts such as length, time, temperature, mass, density, charge, and volume are scalars; each has a scale or size, but no associated direction. The number of students in a class, the quantity of sugar in a jar, and the cost of a house are familiar scalar quantities. Scalars are speci®ed by ordinary numbers and add and subtract in the usual way. Two candies in one box plus seven in another give nine candies total. A VECTOR QUANTITY is one that can be speci®ed completely only if we provide both its magnitude (size) and direction. Many physical concepts such as displacement, velocity, acceleration, force, and momentum are vector quantities. For example, a vector displacement might be a change in position from a certain point to a second point 2 cm away and in the x-direction from the ®rst point. As another example, a cord pulling northward on a post gives rise to a vector force on the post of 20 newtons (N) northward. One newton is 0.225 pound (1.00 N ˆ 0:225 lb). Similarly, a car moving south at 40 km/h has a vector velocity of 40 km/h-SOUTH. A vector quantity can be represented by an arrow drawn to scale. The length of the arrow is proportional to the magnitude of the vector quantity (2 cm, 20 N, 40 km/h in the above examples). The direction of the arrow represents the direction of the vector quantity. In printed material, vectors are often represented by boldface type, such as F. When written by hand, the designations ~ F and are commonly used. A vector is not completely de®ned until we establish some rules for its behavior. THE RESULTANT, or sum, of a number of vectors of a particular type (force vectors, for example) is that single vector that would have the same e€ect as all the original vectors taken together. GRAPHICAL ADDITION OF VECTORS (POLYGON METHOD): This method for ®nding the resultant ~ R of several vectors (~ A, ~ B, and ~ C) consists in beginning at any convenient point and drawing (to scale and in the proper directions) each vector arrow in turn. They may be taken in any order of succession: ~ A ‡~ B‡~ Cˆ~ C‡~ A ‡~ Bˆ~ R. The tail end of each arrow is positioned at the tip end of the preceding one, as shown in Fig. 1-1. Fig. 1-1 1 Copyright 1997, 1989, 1979, 1961, 1942, 1940, 1939, 1936 The McGraw-Hill Companies, Inc. Click Here for Terms of Use. 2 INTRODUCTION TO VECTORS [CHAP. 1 The resultant is represented by an arrow with its tail end at the starting point and its tip end at the tip of the last vector added. If ~ R is the resultant, R ˆ j~ Rj is the size or magnitude of the resultant. PARALLELOGRAM METHOD for adding two vectors: The resultant of two vectors acting at any angle may be represented by the diagonal of a parallelogram. The two vectors are drawn as the sides of the parallelogram and the resultant is its diagonal, as shown in Fig. 1-2. The direction of the resultant is away from the origin of the two vectors. Fig. 1-2 SUBTRACTION OF VECTORS: To subtract a vector ~ B from a vector ~ A, reverse the direction ~ ~ ~ ~ ~ of B and add individually to vector A, that is, A B ˆ A ‡ … ~ B†: THE TRIGONOMETRIC FUNCTIONS are de®ned in relation to a right angle. For the right triangle shown in Fig. 1-3, by de®nition sin  ˆ opposite B ˆ ; hypotenuse C cos  ˆ adjacent A ˆ ; hypotenuse C tan  ˆ opposite B ˆ adjacent A We often use these in the forms B ˆ C sin  A ˆ C cos  B ˆ A tan  Fig. 1-3 A COMPONENT OF A VECTOR is its e€ective value in a given direction. For example, the xcomponent of a displacement is the displacement parallel to the x-axis caused by the given displacement. A vector in three dimensions may be considered as the resultant of its component vectors resolved along any three mutually perpendicular directions. Similarly, a vector in two dimensions CHAP. 1] INTRODUCTION TO VECTORS 3 Fig. 1-4 may be resolved into two component vectors acting along any two mutually perpendicular directions. Figure 1-4 shows the vector ~ R and its x and y vector components, ~ Rx and ~ Ry , which have magnitudes j~ Rx j ˆ j~ Rj cos  and j~ Ry j ˆ j~ Rj sin  Rx ˆ R cos  and Ry ˆ R sin  or equivalently COMPONENT METHOD FOR ADDING VECTORS: Each vector is resolved into its x-, y-, and z-components, with negatively directed components taken as negative. The scalar x-component R is the algebraic sum of all the scalar x-components. The scalar y- and zRx of the resultant ~ components of the resultant are found in a similar way. With the components known, the magnitude of the resultant is given by q R ˆ R2x ‡ R2y ‡ R2z In two dimensions, the angle of the resultant with the x-axis can be found from the relation tan  ˆ Ry Rx UNIT VECTORS have a magnitude of one and are represented by a boldface symbol topped ^ are assigned to the x-, y-, and z-axes, respecwith a caret. The special unit vectors ^i, ^j, and k ^ ^ represents a ®vetively. A vector 3i represents a three-unit vector in the ‡x-direction, while 5k unit vector in the z-direction. A vector ~ R that has scalar x-, y-, and z-components Rx , Ry , and ^ Rz , respectively, can be written as ~ R ˆ Rx^i ‡ Ry^j ‡ Rz k. THE DISPLACEMENT: When an object moves from one point in space to another the displacement is the vector from the initial location to the ®nal location. It is independent of the actual distance traveled. 4 INTRODUCTION TO VECTORS [CHAP. 1 Solved Problems 1.1 Using the graphical method, ®nd the resultant of the following two displacements: 2.0 m at 408 and 4.0 m at 1278, the angles being taken relative to the ‡x-axis, as is customary. Give your answer to two signi®cant ®gures. (See Appendix A on signi®cant ®gures.) Choose x- and y-axes as shown in Fig. 1-5 and lay out the displacements to scale, tip to tail from the origin. Notice that all angles are measured from the ‡x-axis. The resultant vector ~ R points from starting point to end point as shown. We measure its length on the scale diagram to ®nd its magnitude, 4.6 m. Using a protractor, we measure its angle  to be 1018. The resultant displacement is therefore 4.6 m at 1018: Fig. 1-5 1.2 Fig. 1-6 Find the x- and y-components of a 25.0-m displacement at an angle of 210:08: The vector displacement and its components are shown in Fig. 1-6. The scalar components are x-component ˆ y-component ˆ …25:0 m† cos 30:08 ˆ 21:7 m …25:0 m† sin 30:08 ˆ 12:5 m Notice in particular that each component points in the negative coordinate direction and must therefore be taken as negative. 1.3 Solve Problem 1.1 by use of rectangular components. We resolve each vector into rectangular components as shown in Fig. 1-7(a) and (b). (Place a crosshatch symbol on the original vector to show that it is replaced by its components.) The resultant has scalar components of Rx ˆ 1:53 m 2:41 m ˆ 0:88 m Ry ˆ 1:29 m ‡ 3:19 m ˆ 4:48 m Notice that components pointing in the negative direction must be assigned a negative value. The resultant is shown in Fig. 1.7(c); there, we see that q 4:48 m R ˆ …0:88 m†2 ‡ …4:48 m†2 ˆ 4:6 m tan  ˆ 0:88 m and  ˆ 798, from which  ˆ 1808  ˆ 1018. Hence ~ R ˆ 4:6 m Ð 1018 FROM ‡X-AXIS; remember vectors must have their directions stated explicitly. CHAP. 1] INTRODUCTION TO VECTORS 5 Fig. 1-7 1.4 Add the following two force vectors by use of the parallelogram method: 30 N at 308 and 20 N at 1408. Remember that numbers like 30 N and 20 N have two signi®cant ®gures. The force vectors are shown in Fig. 1-8(a). We construct a parallelogram using them as sides, as shown in Fig. 1-8(b). The resultant ~ R is then represented by the diagonal. By measurement, we ®nd that ~ R is 30 N at 728: Fig. 1-8 1.5 Four coplanar forces act on a body at point O as shown in Fig. 1-9(a). Find their resultant graphically. Starting from O, the four vectors are plotted in turn as shown in Fig. 1-9(b). We place the tail end of each vector at the tip end of the preceding one. The arrow from O to the tip of the last vector represents the resultant of the vectors. Fig. 1-9 6 INTRODUCTION TO VECTORS [CHAP. 1 We measure R from the scale drawing in Fig. 1-9(b) and ®nd it to be 119 N. Angle  is measured by protractor and is found to be 378. Hence the resultant makes an angle  ˆ 1808 378 ˆ 1438 with the positive x-axis. The resultant is 119 N at 1438: 1.6 The ®ve coplanar forces shown in Fig. 1-10(a) act on an object. Find their resultant. (1) First we ®nd the x- and y-components of each force. These components are as follows: Force 19.0 15.0 16.0 11.0 22.0 N N N N N Notice the ‡ and x-Component y-Component 19.0 N …15:0 N) cos 60:08 ˆ 7:50 N …16:0 N) cos 45:08 ˆ 11:3 N …11:0 N) cos 30:08 ˆ 9:53 N 0N 0N …15:0 N) sin 60:08 ˆ 13:0 N …16:0 N) sin 45:08 ˆ 11:3 N …11:0 N) sin 30:08 ˆ 5:50 N 22:0 N signs to indicate direction. (2) The resultant ~ R has components Rx ˆ  Fx and Ry ˆ  Fy , where we read  Fx as ``the sum of all the xforce components.'' We then have Rx ˆ 19:0 N ‡ 7:50 N 11:3 N 9:53 N ‡ 0 N ˆ ‡5:7 N Ry ˆ 0 N ‡ 13:0 N ‡ 11:3 N 5:50 N 22:0 N ˆ 3:2 N (3) The magnitude of the resultant is Rˆ (4) q R2x ‡ R2y ˆ 6:5 N Finally, we sketch the resultant as shown in Fig. 1-10(b) and ®nd its angle. We see that tan  ˆ from which  ˆ 298. Then  ˆ 3608 ~ R ˆ 6:5 N Ð 3318 FROM ‡X-AXIS. 3:2 N ˆ 0:56 5:7 N 298 ˆ 3318. The resultant is 6.5 N at 3318 (or Fig. 1-10 298) or CHAP. 1] 1.7 7 INTRODUCTION TO VECTORS Solve Problem 1.5 by use of the component method. Give your answer for the magnitude to two signi®cant ®gures. The forces and their components are: Force 80 100 110 160 N N N N x-Component y-Component 80 N (100 N) cos 45 8 ˆ 71 N …110 N) cos 308 ˆ 95 N …160 N) cos 20 8 ˆ 150 N 0 (100 N) sin 458 ˆ 71 N (110 N) sin 308 ˆ 55 N …160 N) sin 208 ˆ 55 N Notice the sign of each component. To ®nd the resultant, we have Rx ˆ  Fx ˆ 80 N ‡ 71 N 95 N 150 N ˆ 94 N Ry ˆ  Fy ˆ 0 ‡ 71 N ‡ 55 N 55 N ˆ 71 N The resultant is shown in Fig. 1-11; there, we see that q R ˆ …94 N†2 ‡ …71 N†2 ˆ 1:2  102 N Further, tan  ˆ …71 N†=…94 N†, from which  ˆ 378. Therefore the resultant is 118 N at 1808 R ˆ 118 N Ð 1438 FROM ‡X-AXIS. or ~ Fig. 1-11 1.8 378 ˆ 1438 Fig. 1-12 A force of 100 N makes an angle of  with the x-axis and has a scalar y-component of 30 N. Find both the scalar x-component of the force and the angle . (Remember that the number 100 N has three signi®cant ®gures whereas 30 N has only two.) The data are sketched roughly in Fig. 1-12. We wish to ®nd Fx and . We know that sin  ˆ 30 N ˆ 0:30 100 N  ˆ 17:468, and thus, to two signi®cant ®gures,  ˆ 178: Then, using the cos , we have Fx ˆ …100 N† cos 17:468 ˆ 95 N 1.9 A child pulls on a rope attached to a sled with a force of 60 N. The rope makes an angle of 408 to the ground. (a) Compute the e€ective value of the pull tending to move the sled along the ground. (b) Compute the force tending to lift the sled vertically. 8 INTRODUCTION TO VECTORS [CHAP. 1 As shown in Fig. 1-13, the components of the 60 N force are 39 N and 46 N. (a) The pull along the ground is the horizontal component, 46 N. (b) The lifting force is the vertical component, 39 N. Fig. 1-13 1.10 Fig. 1-14 A car whose weight is FW is on a ramp which makes an angle  to the horizontal. How large a perpendicular force must the ramp withstand if it is not to break under the car's weight? As shown in Fig. 1-14, the car's weight is a force ~ FW that pulls straight down on the car. We take components of ~ F along the incline and perpendicular to it. The ramp must balance the force component FW cos  if the car is not to crash through the ramp. 1.11 ^ Express the forces shown in Figs. 1-7(c), 1-10(b), 1-11, and 1-13 in the form ~ R ˆ Rx^i ‡ Ry^j ‡ Rz k (leave out the units). Remembering that plus and minus signs must be used to show direction along an axis, we can write For Fig. 1-7(c): For Fig. 1-10(b): For Fig. 1-11: For Fig. 1-13: 1.12 ~ R ˆ 0:88^i ‡ 4:48^j ~ R ˆ 5:7^i 3:2^j ~ R ˆ 94^i ‡ 71^j ~ R ˆ 46^i ‡ 39^j ^ N, Three forces that act on a particle are given by ~ F1 ˆ …20^i 36^j ‡ 73k† ^ N, and ~ ^ N. Find their resultant vector. Also ®nd the mag~ F2 ˆ … 17^i ‡ 21^j 46k† F3 ˆ … 12k† nitude of the resultant to two signi®cant ®gures. We know that Rx ˆ  Fx ˆ 20 N 17 N ‡ 0 N ˆ 3 N Ry ˆ  Fy ˆ 36 N ‡ 21 N ‡ 0 N ˆ 15 N Rz ˆ  Fz ˆ 73 N 46 N 12 N ˆ 15 N ^ we ®nd Since ~ R ˆ Rx^i ‡ Ry^j ‡ Rz k, ~ R ˆ 3^i ^ 15^j ‡ 15k To two signi®cant ®gures, the three-dimensional pythagorean theorem then gives q p R ˆ R2x ‡ R2y ‡ R2z ˆ 459 ˆ 21 N CHAP. 1] 1.13 9 INTRODUCTION TO VECTORS Perform graphically the following vector additions and subtractions, where ~ A, ~ B, and ~ C are the vectors shown in Fig. 1-15: (a) ~ A ‡~ B; (b) ~ A ‡~ B‡~ C; (c) ~ A ~ B; (d ) ~ A ‡~ B ~ C: See Fig. 1-15(a) through (d ). In (c), ~ A ~ Bˆ~ A‡… ~ B†; that is, to subtract ~ B from ~ A, reverse the direction of ~ B and add it vectorially to ~ A. Similarly, in (d ), ~ A‡~ B ~ Cˆ~ A ‡~ B‡… ~ C†, where ~ C is equal in magnitude but opposite in direction to ~ C: Fig. 1-15 1.14 If ~ Aˆ ^ and ~ 12^i ‡ 25^j ‡ 13k Bˆ ^ ®nd the resultant when ~ 3^j ‡ 7k, A is subtracted from ~ B: From a purely mathematical approach, we have ~ B Notice that 12^i added it to ~ B. 1.15 1.16 25^j ^ ^ ~ A ˆ … 3^j ‡ 7k† … 12^i ‡ 25^j ‡ 13k† ^ ‡ 12^i 25^j 13k ^ ˆ 12^i ˆ 3^j ‡ 7k 28^j ^ 6k ^ is simply ~ 13k A reversed in direction. Therefore we have, in essence, reversed ~ A and A boat can travel at a speed of 8 km/h in still water on a lake. In the ¯owing water of a stream, it can move at 8 km/h relative to the water in the stream. If the stream speed is 3 km/h, how fast can the boat move past a tree on the shore when it is traveling (a) upstream and (b) downstream? (a) If the water was standing still, the boat's speed past the tree would be 8 km/h. But the stream is carrying it in the opposite direction at 3 km/h. Therefore the boat's speed relative to the tree is 8 km=h 3 km=h ˆ 5 km=h: (b) In this case, the stream is carrying the boat in the same direction the boat is trying to move. Hence its speed past the tree is 8 km=h ‡ 3 km=h ˆ 11 km=h: A plane is traveling eastward at an airspeed of 500 km/h. But a 90 km/h wind is blowing southward. What are the direction and speed of the plane relative to the ground? The plane's resultant velocity is the sum of two velocities, 500 km/h Ð EAST and 90 km/h Ð SOUTH. These component velocities are shown in Fig. 1-16. The plane's resultant velocity is then q R ˆ …500 km=h†2 ‡ …90 km=h†2 ˆ 508 km=h