University Physics I - Classical Mechanics, 2019

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272 CHAPTER 11. SIMPLE HARMONIC MOTION 11.6 Advanced Topics 11.6.1 Mass on a spring damped by friction with a surface Consider the system depicted in Figure 11.2 in the presence of friction between the block and the surface. Let the coefficient of kinetic friction be μ k and the coefficient of static friction be μ s . As usual, we will assume that μ s ≥ μ k . As the mass oscillates, it will experience a kinetic friction force of magnitude F k = μ k mg, inthe direction opposite the direction of motion; that is to say, a force that changes direction every half period. As shown in section 11.2.2, this force does not change the frequency of the motion, but it displaces the equilibrium position in the direction of the force, which is to say, closer to the starting point for each half-swing. As a result of that, the amplitude for each half-swing is reduced from the previous one. Let the original equilibrium position (in the absence of friction) be x 0 = 0. Suppose we displace the mass a distance A to the right (call this position, the starting point for the first half-swing, x 1 = A), and let go. In the presence of friction, the equilibrium position for this first half-swing becomes the point x ′ 0 = F k /k = μ k mg, so the real amplitude of this first half-oscillation will be A 1 = x 1 − x ′ 0 = A − x′ 0 , and the resulting motion will be x(t) =x ′ 0 + A 1 cos(ωt) (first half-period, 0 ≤ t ≤ π/ω) (11.32) The mass then stops, momentarily, at t = π/ω, atthepositionx 2 = x ′ 0 − A 1, and turns around for the second half-swing. However, now the external force has reversed direction, so the new equilibrium position is at −x ′ 0 , and the new amplitude is A 2 = −x ′ 0 − x 2 = A 1 − 2x ′ 0 = A − 3x′ 0 . The motion for this next half-period is then x(t) =−x ′ 0 + A 2 cos(ωt) (second half-period, π/ω ≤ t ≤ 2π/ω) (11.33) Continuing the process, we see that A 1 = A−x ′ 0 , A 2 = A−3x ′ 0 , A 3 = A−5x ′ 0 ...A n = A−(2n−1)x ′ 0 . Of course, this can’t go on forever, since we require the amplitude to be a positive quantity; so the motion will consist of only n half-periods, where n is an integer such that A − (2n − 1)x ′ 0 > 0 but A − (2n +1)x ′ 0 < 0. (That is to say, n is equal to the integer part of (A/x′ 0 +1)/2.) The figure shows an example of how this would go, for the following choice of parameters: period T =1s,μ k =0.1, and A =0.18 m. Note that, since x ′ 0 depends only on the ratio m/k =1/ω2 ,there is no need to specify m and k separately. We get ω =2π/T =2π rad/s, x ′ 0 = μ kg/ω 2 =0.0248 m, and (A/x ′ 0 +1)/2 =4.13, which means that the motion will go on for 4 half-periods before stopping.
11.6. ADVANCED TOPICS 273 Figure 11.9: Damped oscillations. Note that, in general, the oscillator does not stop at the equilibrium position. Rather, its final position will be at the end of the last half-swing, which is either x ′ 0 − A n (if the number n of half-periods is odd), or −x ′ 0 + A n,ifthenumbern is even. Either way, at that point the spring will be exerting a force of magnitude F spr = k|x ′ 0 − A n | = k|x ′ 0 − A +(2n − 1)x ′ 0| = k|A − 2nx ′ 0|
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University Physics I: Classical Mec
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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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6.3. 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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7.6. 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 219 A 20-kg plank of
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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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