University Physics I - Classical Mechanics, 2019

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162 CHAPTER 8. MOTION IN TWO DIMENSIONS y F θ F 180° − θ θ F y F x F x x Figure 8.1: The components of a vector that makes an angle θ with the positive x axis. Two examples are shown, for θ 0) and for 90 ◦
8.1. DEALING WITH FORCES IN TWO DIMENSIONS 163 eventually solve) the equations F net,x = ma x F net,y = ma y (8.2) We can show that Eqs. (8.2) must hold for any choice of orthogonal x and y axes, based on the fact that we know ⃗ F net = m⃗a holds along one particular direction, namely, the direction common to ⃗ F net and ⃗a, and the fact that we have defined the projection procedure to be the same for any kind of vector. Figure 8.2 shows how this works. Along the dashed line you just have the situation that is by now familiar to us from one-dimensional problems, where ⃗a lies along ⃗ F (assumed here to be the net force), and | ⃗ F | = m|⃗a|. However, in the figure I have chosen the axes to make an angle θ with this direction. Then, if you look at the projections of ⃗ F and ⃗a along the x axis, you will find a x = |⃗a| cos θ F x = | F ⃗ | cos θ = m|⃗a| cos θ = ma x (8.3) and similarly, F y = ma y . In words, each component of the force vector is responsible for only the corresponding component of the acceleration. A force in the x direction does not cause any acceleration in the y direction, and vice-versa. y F y a y a F θ a x F x x Figure 8.2: If you take the familiar, one-dimensional (see the black dashed line) form of ⃗ F = m⃗a, and project it onto orthogonal, rotated axes, you get the general two-dimensional case, showing that each orthogonal component of the acceleration is proportional, via the mass m, to only the corresponding component of the force (Eqs. (8.2)). In the rest of the chapter we shall see how to use Eqs. (8.2) in a number of examples. One thing I can anticipate is that, in general, we will try to choose our axes (unlike in Fig. 8.2 above) so that one of them does coincide with the direction of the acceleration, so the motion along the other direction is either nonexistent (v = 0) or trivial (constant velocity).
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University Physics I: Classical Mec
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ii
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iv CONTENTS 1.6 Problems . . . . .
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vi CONTENTS 5.1 Conservative intera
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viii CONTENTS 7.7 Examples . . . .
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x CONTENTS 10.4 Examples . . . . .
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xii CONTENTS 13.2.1 Temperature and
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xiv CONTENTS they have read. Then,
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Chapter 1 Reference frames, displac
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1.2. POSITION, DISPLACEMENT, VELOCI
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1.3. REFERENCE FRAME CHANGES AND RE
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1.3. REFERENCE FRAME CHANGES AND RE
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1.4. IN SUMMARY 19 with a velocity
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1.5. EXAMPLES 21 1.5 Examples 1.5.1
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1.5. EXAMPLES 23 (twice what it was
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1.5. EXAMPLES 25 Note that I have d
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1.6. PROBLEMS 27 1.6 Problems Probl
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1.6. PROBLEMS 29 Problem 5 You are
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Chapter 2 Acceleration 2.1 The law
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2.1. THE LAW OF INERTIA 33 This is
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2.2. ACCELERATION 35 2.2 Accelerati
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2.2. ACCELERATION 37 called “conc
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2.2. ACCELERATION 39 2.2.2 Motion w
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2.2. ACCELERATION 41 2.2.3 Accelera
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2.3. FREE FALL 43 proportional to t
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2.4. IN SUMMARY 45 4. Changes in ve
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2.5. EXAMPLES 47 which the rocket r
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2.6. PROBLEMS 49 (a) How far did yo
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Chapter 3 Momentum and Inertia 3.1
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3.1. INERTIA 53 reliable, repeatabl
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3.1. INERTIA 55 3.1.2 Inertial mass
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3.2. MOMENTUM 57 the system is p i
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3.3. EXTENDED SYSTEMS AND CENTER OF
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3.4. IN SUMMARY 61 3.3.2 Recoil and
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3.5. EXAMPLES 63 3.5 Examples 3.5.1
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3.5. EXAMPLES 65 3.5.2 Collision in
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3.6. PROBLEMS 67 3.6 Problems Probl
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Chapter 4 Kinetic Energy 4.1 Kineti
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4.1. KINETIC ENERGY 71 v 2i =0,v 1f
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4.1. KINETIC ENERGY 73 special case
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4.1. KINETIC ENERGY 75 This immedia
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4.2. “CONVERTIBLE” AND “TRANS
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4.2. “CONVERTIBLE” AND “TRANS
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4.3. IN SUMMARY 81 7. Besides the c
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4.4. EXAMPLES 83 On the other hand,
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4.4. EXAMPLES 85 K cm ′ = 1 2 (m
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4.5. PROBLEMS 87 (a) What is the in
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Chapter 5 Interactions and energy 5
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5.1. CONSERVATIVE INTERACTIONS 91 T
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5.1. CONSERVATIVE INTERACTIONS 93 A
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5.1. CONSERVATIVE INTERACTIONS 95 d
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5.2. DISSIPATION OF ENERGY AND THER
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5.4. CONSERVATION OF ENERGY 99 leve
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5.5. IN SUMMARY 101 gravitational p
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5.6. EXAMPLES 103 5.6 Examples 5.6.
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5.6. EXAMPLES 105 problem involves
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5.7. ADVANCED TOPICS 107 where the
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5.8. PROBLEMS 109 5.8 Problems Prob
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Page 139 and 140: 6.3. FORCES NOT DERIVED FROM A POTE
Page 145 and 146: 6.5. IN SUMMARY 127 F s,1 n a F s,1
Page 147 and 148: 6.6. EXAMPLES 129 6.6 Examples 6.6.
Page 149 and 150: 6.6. EXAMPLES 131 pushes on the bra
Page 151 and 152: 6.6. EXAMPLES 133 This is a huge di
Page 153 and 154: 6.7. PROBLEMS 135 Problem 6 Draw a
Page 155 and 156: Chapter 7 Impulse, Work and Power 7
Page 157 and 158: 7.2. WORK ON A SINGLE PARTICLE 139
Page 159 and 160: 7.3. THE “CENTER OF MASS WORK”
Page 161 and 162: 7.4. WORK DONE ON A SYSTEM BY ALL T
Page 169 and 170: 7.6. IN SUMMARY 151 which is equal
Page 171 and 172: 7.7. EXAMPLES 153 7.7 Examples 7.7.
Page 173 and 174: 7.7. EXAMPLES 155 To plot all this
Page 175 and 176: 7.7. EXAMPLES 157 (a) The external
Page 177 and 178: 7.8. PROBLEMS 159 Problem 5 A block
Page 179: Chapter 8 Motion in two dimensions
Page 183 and 184: 8.2. PROJECTILE MOTION 165 The over
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Page 187 and 188: 8.4. MOTION ON A CIRCLE (OR PART OF
Page 193 and 194: 8.5. IN SUMMARY 175 As we shall see
Page 195 and 196: 8.6. EXAMPLES 177 8.6 Examples You
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Page 201 and 202: 8.7. ADVANCED TOPICS 183 Note that
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Page 211 and 212: 9.2. ANGULAR MOMENTUM 193 9.2 Angul
Page 213 and 214: 9.2. ANGULAR MOMENTUM 195 thing is
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Page 219 and 220: 9.4. TORQUE 201 momentum involves t
Page 221 and 222: 9.4. TORQUE 203 wrenches). It is al
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9.8. EXAMPLES 213 9.8 Examples This
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9.8. EXAMPLES 215 Solution The figu
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9.9. PROBLEMS 217 9.9 Problems Prob
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9.9. PROBLEMS 219 A 20-kg plank of
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Chapter 10 Gravity 10.1 The inverse
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10.1. THE INVERSE-SQUARE LAW 223 ne
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10.1. THE INVERSE-SQUARE LAW 225 it
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10.1. THE INVERSE-SQUARE LAW 227 in
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10.1. THE INVERSE-SQUARE LAW 229 an
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10.1. THE INVERSE-SQUARE LAW 231 el
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10.1. THE INVERSE-SQUARE LAW 233 10
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10.1. THE INVERSE-SQUARE LAW 235 Fr
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10.2. WEIGHT, ACCELERATION, AND THE
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10.2. WEIGHT, ACCELERATION, AND THE
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10.3. IN SUMMARY 241 shifted and/or
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10.4. EXAMPLES 243 10.4 Examples 10
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10.4. EXAMPLES 245 The diagram of t
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10.5. ADVANCED TOPICS 247 10.5 Adva
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10.6. PROBLEMS 249 10.6 Problems Pr
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10.6. PROBLEMS 251 an orbit around
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Chapter 11 Simple harmonic motion 1
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11.2. SIMPLE HARMONIC MOTION 255 is
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11.2. SIMPLE HARMONIC MOTION 257 Th
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11.2. SIMPLE HARMONIC MOTION 259 If
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11.2. SIMPLE HARMONIC MOTION 261 Th
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11.3. PENDULUMS 263 o θ l F t F G
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11.3. PENDULUMS 265 11.3.2 The “p
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11.4. IN SUMMARY 267 9. A simple pe
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11.5. EXAMPLES 269 (b) The amplitud
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11.5. EXAMPLES 271 AsshowninSection
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11.6. ADVANCED TOPICS 273 Figure 11
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11.6. ADVANCED TOPICS 275 the oscil
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11.7. PROBLEMS 277 (that is, with r
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Chapter 12 Waves in one dimension 1
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12.1. TRAVELING WAVES 281 Perhaps t
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12.1. TRAVELING WAVES 283 ξ 1 0.5
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12.1. TRAVELING WAVES 285 Comparing
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12.1. TRAVELING WAVES 287 waves, th
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12.2. STANDING WAVES AND RESONANCE
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12.2. STANDING WAVES AND RESONANCE
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12.3. CONCLUSION, AND FURTHER RESOU
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12.5. EXAMPLES 297 However, if you
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12.6. ADVANCED TOPICS 299 12.6 Adva
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12.6. ADVANCED TOPICS 301 In the lo
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Chapter 13 Thermodynamics 13.1 Intr
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13.2. INTRODUCING TEMPERATURE 305 a
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13.2. INTRODUCING TEMPERATURE 307 m
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13.3. HEAT AND THE FIRST LAW 309 th
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13.3. HEAT AND THE FIRST LAW 311 ca
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13.4. THE SECOND LAW AND ENTROPY 31
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13.5. IN SUMMARY 319 13.5 In summar
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13.7. PROBLEMS 323 13.7 Problems Pr
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University Physics I - Classical Mechanics, 2019

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