University Physics I - Classical Mechanics, 2019

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218 CHAPTER 9. ROTATIONAL DYNAMICS (c) If the worker were to let go of the plank, what would its angular acceleration be as it starts swinging down? The moment of inertia is I = 1 3 Ml2 . (Note: assume the toolbox stops pressing down on the plank immediately. This is a good approximation, as you shall see below.) (d) Consider a point on the plank located immediately below the toolbox. As the plank swings, this point moves in a circle of radius 1.5m. What is its linear (tangential) acceleration as it starts going down, and how does it compare to the acceleration of gravity? Problem 4 A solid sphere of radius 5 cm and mass 0.2 kg is rolling without slipping, on level ground, at a constant speed of 0.5 m/s. It is heading toward a ramp that makes an angle of 30 ◦ with the horizontal. (a) What is the angular velocity of the sphere, in radians per second? (b) If the moment of inertia of a solid, homogeneous sphere is I = 2 5 mR2 , what is the initial angular momentum of this sphere? (c) What is the translational kinetic energy of the sphere? (d) What is its rotational kinetic energy? (e) How high (vertically) will the sphere rise as it goes up the ramp, still rolling without slipping, before it stops and turns around? (f) Draw an extended free-body diagram for the sphere as it rolls up the ramp. On the diagram, indicate the direction of rotation, and the direction of the acceleration of the center of mass. (g) Referring to the extended free-body diagram in the previous question, which force is responsible for the change in the angular momentum of the sphere as it rolls up the ramp? Explain briefly: Why that force and not another one? Does the direction you have assumed for this force agree with the torque it has to provide? (If not, you’d better go back and fix that right now!) (h) Calculate the acceleration of the sphere as it rolls up the ramp. Problem 5 A very light, inextensible string is wrapped around a cylindrical spool. The end of the string is held fixed, and the spool is released so it starts falling, as the string unwinds. Because the spool is basically hollow, you can take its moment of inertia to be I = mR 2 . (a) Draw an extended free-body diagram for the spool as it unrolls. (b) Find the linear acceleration of the spool as it accelerates toward the ground. (c) Let the mass of the spool be 0.1 kg. What is its translational kinetic energy after it has fallen for 0.5s? (d) What is its rotational kinetic energy at that time? (e) What is the tension in the string? Does this change as the spool falls? (Remember the mass of the string is negligible.) (f) If the radius of the spool is 3 cm, what is the magnitude of the torque (around the center of mass) exerted by the tension? Problem 6
9.9. PROBLEMS 219 A 20-kg plank of length l = 4 m is supported at both ends as shown in the figure. A 60-kg man is standing a distance l/3 from the right end of the plank. 4 m 1.33 m (a) Draw an extended free-body diagram for the plank. Try to get the scale of the forces at least qualitatively right. (b) Find the upwards force on the plank exerted by each of the two supports.
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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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5.8. PROBLEMS 111 here? Explain. (f
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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 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 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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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 295 12.5 Examples 12
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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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13.6. EXAMPLES 321 13.6 Examples 13
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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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