University Physics I - Classical Mechanics, 2019

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276 CHAPTER 11. SIMPLE HARMONIC MOTION 11.7 Problems Problem 1 Ablockofmassm is sliding on a frictionless, horizontal surface, with a velocity v i . It hits an ideal spring, of spring constant k, which is attached to the wall. The spring compresses until the block momentarily stops, and then starts expanding again, so the block ultimately bounces off (see Example 5.6.2). (a) Write down an equation of motion (a function x(t)) for the block, which is valid for as long as it is in contact with the spring. For simplicity, assume the block is initially moving to the right, take the time when it first makes contact with the spring to be t = 0, and let the position of the block at that time to be x = 0. Make sure that you express any constants in your equation (such as A or ω) in terms of the given data, namely, m, v i ,andk. (b) Sketch the function x(t) for the relevant time interval. Problem 2 For this problem, imagine that you are on a ship that is oscillating up and down on a rough sea. Assume for simplicity that this is simple harmonic motion (in the vertical direction) with amplitude 5 cm and frequency 2 Hz. There is a box on the floor with mass m =1kg. (a) Assuming the box remains in contact with the floor throughout, find the maximum and minimum values of the normal force exerted on it by the floor over an oscillation cycle. (b) How large would the amplitude of the oscillations have to become for the box to lose contact with the floor, assuming the frequency remains constant? (Hint: what is the value of the normal force at the moment the box loses contact with the floor?) Problem 3 Imagine a simple pendulum swinging in an elevator. If the cable holding the elevator up was to snap, allowing the elevator to go into free fall, what would happen to the frequency of oscillation of the pendulum? Justify your answer. Problem 4 Consider a block of mass m attached to two springs, one on the left with spring constant k 1 and one on the right with spring constant k 2 . Each spring is attached on the other side to a wall, and the block slides without friction on a horizontal surface. When the block is sitting at x =0,both springs are relaxed. Write Newton’s second law, F = ma, as a differential equation for an arbitrary position x of the block. What is the period of oscillation of this system? Problem 5 Consider the block hanging from a spring shown in Figure 11.6. Suppose the mass of the block is 1.5 kg and the system is at rest when the spring has been stretched 2 cm from its original length
11.7. PROBLEMS 277 (that is, with reference to the figure, y 0 − y 0 ′ =0.02 m). (a) What is the value of the spring constant k? (b) If you stretch the spring by an additional 2 cm downward from this equilibrium position, and release it, what will be the frequency of the oscillations? (c) Now consider the system formed by the spring, the block, and the earth. Take the “zero” of gravitational potential energy to be at the height y 0 ′ (the equilibrium point; you may also use this as the origin for the vertical coordinate!), and calculate all the energies in the system (kinetic, spring/elastic, and gravitational) at the highest point in the oscillation, the equilibrium point, and the lowest point. Verify that the sum is indeed constant.
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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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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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University Physics I - Classical Mechanics, 2019

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