University Physics I - Classical Mechanics, 2019

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80 CHAPTER 4. KINETIC ENERGY another we just add or subtract from all the velocities the relative velocity of the two frames. This operation, however, will not change any of the relative velocities of the parts of the system, since these are all differences to begin with. Mathematically, regardless of the value of v ′ . (v 2 + v ′ ) − (v 1 + v ′ )=v 2 − v 1 So there something we might call absolute (as opposed to “relative”) about the convertible kinetic energy: it is the same, it will have the same value, for any observer, regardless of how fast or in what direction that observer may be moving relative to the system as a whole. We may think of it as an intrinsic (meaning, observer-independent) property of the system. 4.3 In summary 1. The kinetic energy of a particle of mass m moving with velocity v is defined as K = 1 2 mv2 .It is a scalar quantity, and it is always positive. For a system of particles or an extended object, we define K sys as the sum of the kinetic energies of all the particles making up the system. 2. For any system, the total kinetic energy can be written as the sum of the translational (or center of mass) kinetic energy, K cm , and another term that involves the motion of the parts of the system relative to each other. (See Eq. (4.11) above.) The translational kinetic energy is constant for an isolated system, and is always given by K cm = 1 2 Mv2 cm . 3. The kinetic energy of relative motion (which, in the context of collisions, is called the convertible energy) is given, for the special case of a system consisting of two particles (or two non-rotating extended objects), by K conv = 1 2 μv2 12 ,whereμ = m 1m 2 /(m 1 +m 2 ) is the reduced mass, and v 12 = v 2 − v 1 is the relative velocity of the two objects. 4. In a one-dimensional collision between two objects that do not pass through each other, the convertible energy always drops to zero at some point, as a result of the interaction; that is, it is converted entirely into some other form of energy. At the end of the interaction, all the convertible energy may be recovered (elastic collision), or only part of it (inelastic collision), or none of it (completely inelastic collision). 5. In terms of the coefficient of restitution e, defined as e = −v 12,f /v 12,i , elastic collisions have e = 1, totally inelastic collisions have e = 0, and inelastic collisions 0
4.3. IN SUMMARY 81 7. Besides the cases considered above, one may have collisions where the objects pass through each other, giving e1. In these latter collisions some internal source of energy is converted into additional kinetic energy when the objects interact. The extreme case of this is an explosive separation, which is the reverse of a totally inelastic collision—two objects initially moving together fly apart, with a net increase in the system’s kinetic energy. 8. The translational kinetic energy of a system will, in general, have different values for observers moving with different velocities. The convertible kinetic energy, on the other hand, is seen by all observers to have the same value, regardless of their relative state of motion.
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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
Page 47 and 48: 1.6. PROBLEMS 29 Problem 5 You are
Page 49 and 50: Chapter 2 Acceleration 2.1 The law
Page 51 and 52: 2.1. THE LAW OF INERTIA 33 This is
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Page 57 and 58: 2.2. ACCELERATION 39 2.2.2 Motion w
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Page 63 and 64: 2.4. IN SUMMARY 45 4. Changes in ve
Page 65 and 66: 2.5. EXAMPLES 47 which the rocket r
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Page 69 and 70: Chapter 3 Momentum and Inertia 3.1
Page 71 and 72: 3.1. INERTIA 53 reliable, repeatabl
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Page 75 and 76: 3.2. MOMENTUM 57 the system is p i
Page 77 and 78: 3.3. EXTENDED SYSTEMS AND CENTER OF
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Page 81 and 82: 3.5. EXAMPLES 63 3.5 Examples 3.5.1
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Page 87 and 88: Chapter 4 Kinetic Energy 4.1 Kineti
Page 89 and 90: 4.1. KINETIC ENERGY 71 v 2i =0,v 1f
Page 91 and 92: 4.1. KINETIC ENERGY 73 special case
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Page 95 and 96: 4.2. “CONVERTIBLE” AND “TRANS
Page 97: 4.2. “CONVERTIBLE” AND “TRANS
Page 101 and 102: 4.4. EXAMPLES 83 On the other hand,
Page 103 and 104: 4.4. EXAMPLES 85 K cm ′ = 1 2 (m
Page 105 and 106: 4.5. PROBLEMS 87 (a) What is the in
Page 107 and 108: Chapter 5 Interactions and energy 5
Page 109 and 110: 5.1. CONSERVATIVE INTERACTIONS 91 T
Page 111 and 112: 5.1. CONSERVATIVE INTERACTIONS 93 A
Page 113 and 114: 5.1. CONSERVATIVE INTERACTIONS 95 d
Page 115 and 116: 5.2. DISSIPATION OF ENERGY AND THER
Page 117 and 118: 5.4. CONSERVATION OF ENERGY 99 leve
Page 119 and 120: 5.5. IN SUMMARY 101 gravitational p
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Page 131 and 132: Chapter 6 Interactions, part 2: For
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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.
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6.6. EXAMPLES 131 pushes on the bra
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6.6. EXAMPLES 133 This is a huge di
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6.7. PROBLEMS 135 Problem 6 Draw a
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Chapter 7 Impulse, Work and Power 7
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7.2. WORK ON A SINGLE PARTICLE 139
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7.3. THE “CENTER OF MASS WORK”
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7.4. WORK DONE ON A SYSTEM BY ALL T
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7.6. IN SUMMARY 151 which is equal
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7.7. EXAMPLES 153 7.7 Examples 7.7.
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7.7. EXAMPLES 155 To plot all this
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7.7. EXAMPLES 157 (a) The external
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7.8. PROBLEMS 159 Problem 5 A block
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Chapter 8 Motion in two dimensions
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8.1. DEALING WITH FORCES IN TWO DIM
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8.2. PROJECTILE MOTION 165 The over
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8.3. INCLINED PLANES 167 8.3 Inclin
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8.4. MOTION ON A CIRCLE (OR PART OF
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8.5. IN SUMMARY 175 As we shall see
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8.6. EXAMPLES 177 8.6 Examples You
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8.7. ADVANCED TOPICS 179 8.7 Advanc
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8.7. ADVANCED TOPICS 181 Now we jus
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8.7. ADVANCED TOPICS 183 Note that
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8.7. ADVANCED TOPICS 185 The cyan a
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8.8. PROBLEMS 187 (b) What is the w
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8.8. PROBLEMS 189 (e) The power is
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Chapter 9 Rotational dynamics Rotat
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9.2. ANGULAR MOMENTUM 193 9.2 Angul
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9.2. ANGULAR MOMENTUM 195 thing is
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9.3. THE CROSS PRODUCT AND ROTATION
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9.3. THE CROSS PRODUCT AND ROTATION
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9.4. TORQUE 201 momentum involves t
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9.4. TORQUE 203 wrenches). It is al
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9.5. STATICS 205 important in engin
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9.6. ROLLING MOTION 207 9.6 Rolling
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9.6. ROLLING MOTION 209 contact wit
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9.7. IN SUMMARY 211 (Note how I hav
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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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