University Physics I - Classical Mechanics, 2019

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64 CHAPTER 3. MOMENTUM AND INERTIA for suitable intervals. In this way one gets v 1i = −1 m s v 2i =0.5 m s (b) Similarly, one gets v 1f =1 m s v 2f = −0.5 m s (c) Use this equation, or equivalent (conservation of momentum is OK) m 2 = − Δv 1 m 1 Δv 2 m 2 = − 1 − (−1) m 1 −0.5 − 0.5 =2 so the mass of the second cart is 2 kg. (d) Yes, the track appears to be level because the carts do not show any evidence of acceleration outside of the collision region (the position vs. time curves are straight lines outside of the region approximately given by 4.5s
3.5. EXAMPLES 65 3.5.2 Collision in different reference frames, center of mass, and recoil An 80-kg hockey player (call him player 1), moving at 3 m/s to the right, collides with a 90-kg player (player 2) who was moving at 2 m/s to the left. For a brief moment, they are stuck sliding together as they grab at each other. (a) What is their joint velocity as they slide together? (b) What was the velocity of their center of mass before and after the collision? (c) What does the collision look like to another player that was skating initially at 1.5m/s to the right? Give all the initial and final velocities as seen by this player, and show explicitly that momentum is also conserved in this player’s frame of reference. (d) Eventually, the 90-kg player manages to push the other one back, in such a way that player 1 (the 80-kg player) ends up moving at 1 m/s to the left relative to player 2. What are their final velocities in the earth frame of reference? Solution (a) Call the initial velocities v 1i and v 2i , the joint final velocity v f , and assume the two players are an isolated system for practical purposes. Then conservation of momentum reads Solving for the final velocity, we get Substituting the values given, we get m 1 v 1i + m 2 v 2i =(m 1 + m 2 )v f (3.13) v f = m 1v 1i + m 2 v 2i m 1 + m 2 (3.14) v f = 80 × 3 − 90 × 2 170 =0.353 m s (3.15) (b)AccordingtoEq.(3.10), the velocity of the center of mass, v cm , is just the same as what we just calculated (Eq. (3.14) above). This makes sense: after the collision, if the players are moving together, their system’s center of mass has to be moving with them. Also, if the system is isolated, the center of mass velocity should be the same before and after the collision. So the answer is v cm = v f =0.353 m/s (c) Let me refer to this third player as “player 3,” and introduce a subscript “3” to refer to the quantities as seen in his frame of reference. Let also the subscript “E” denote the original, “Earth” reference frame. From Eq. (1.19), we have then (for player 1, for instance) v 31 = v 3E + v E1 = v E1 − v E3 (3.16) because v 3E , the “velocity of the Earth in player 3’s reference frame,” is clearly equal to −v E3 ,the negative of the velocity of player 3 relative to the Earth. Basically, what Eq. (3.16) issayingis
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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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Page 37 and 38: 1.4. IN SUMMARY 19 with a velocity
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Page 43 and 44: 1.5. EXAMPLES 25 Note that I have d
Page 45 and 46: 1.6. PROBLEMS 27 1.6 Problems Probl
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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 61 and 62: 2.3. FREE FALL 43 proportional to t
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
Page 79 and 80: 3.4. IN SUMMARY 61 3.3.2 Recoil and
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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
Page 93 and 94: 4.1. KINETIC ENERGY 75 This immedia
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Page 99 and 100: 4.3. IN SUMMARY 81 7. Besides the c
Page 101 and 102: 4.4. EXAMPLES 83 On the other hand,
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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
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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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6.1. FORCE 115 however, somewhat mo
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6.2. FORCES AND POTENTIAL ENERGY 11
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6.2. FORCES AND POTENTIAL ENERGY 11
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6.3. FORCES NOT DERIVED FROM A POTE
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6.5. IN SUMMARY 127 F s,1 n a F s,1
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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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