The Pendulum Plain and Puzzling
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1 The Pendulum Plain and Puzzling Chris Sangwin School of Mathematics University of Edinburgh April 2017 Chris Sangwin (University of Edinburgh) Pendulum April / 38
2 Outline 1 Introduction and motivation 2 Geometrical physics 3 Classical mechanics 4 Driven pendulum 5 The elastic pendulum 6 Chaos 7 Conclusion Chris Sangwin (University of Edinburgh) Pendulum April / 38
3 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. Chris Sangwin (University of Edinburgh) Pendulum April / 38
4 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. 1 Divinely ordered world of classical mechanics 2 Catastrophe theory 3 Chaos Chris Sangwin (University of Edinburgh) Pendulum April / 38
5 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. 1 Divinely ordered world of classical mechanics 2 Catastrophe theory 3 Chaos 3 Contemporary examples relevant to A-level mathematics/fm. Chris Sangwin (University of Edinburgh) Pendulum April / 38
6 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. 1 Divinely ordered world of classical mechanics 2 Catastrophe theory 3 Chaos 3 Contemporary examples relevant to A-level mathematics/fm. 4 Physics apparatus Chris Sangwin (University of Edinburgh) Pendulum April / 38
7 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. 1 Divinely ordered world of classical mechanics 2 Catastrophe theory 3 Chaos 3 Contemporary examples relevant to A-level mathematics/fm. 4 Physics apparatus Illustrated by examples. Chris Sangwin (University of Edinburgh) Pendulum April / 38
8 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. 1 Divinely ordered world of classical mechanics 2 Catastrophe theory 3 Chaos 3 Contemporary examples relevant to A-level mathematics/fm. 4 Physics apparatus Illustrated by examples. (Equations on request!) Chris Sangwin (University of Edinburgh) Pendulum April / 38
9 Mechanics within mathematics Chris Sangwin (University of Edinburgh) Pendulum April / 38
10 The elastic pendulum What happens when I displace vertically and release from rest? Chris Sangwin (University of Edinburgh) Pendulum April / 38
11 The elastic pendulum What happens when I displace vertically and release from rest? ÿ = k 2 y? Chris Sangwin (University of Edinburgh) Pendulum April / 38
12 The elastic pendulum What happens when I displace vertically and release from rest? ÿ = k 2 y? So y(t) = A cos(kt + ρ). Chris Sangwin (University of Edinburgh) Pendulum April / 38
13 The elastic pendulum What happens when I displace vertically and release from rest? ÿ = k 2 y? So y(t) = A cos(kt + ρ). Where is the x-coordinate in the model? Chris Sangwin (University of Edinburgh) Pendulum April / 38
14 Modelling I hope I shall shock a few people in asserting that the most important single task of mathematical instruction in the secondary school is to teach the setting up of equations to solve word problems. [...] And so the future engineer, when he learns in the secondary school to set up equations to solve word problems" has a first taste of, and has an opportunity to acquire the attitude essential to, his principal professional use of mathematics. Polya (1962) Chris Sangwin (University of Edinburgh) Pendulum April / 38
15 Pre-Newtonian mechanics Chris Sangwin (University of Edinburgh) Pendulum April / 38
16 Chris Sangwin (University of Edinburgh) Pendulum April / 38
17 Randomness Chris Sangwin (University of Edinburgh) Pendulum April / 38
18 A proposition well known to geometers... it is a proposition well known to geometers, that the velocity of a pendulous body in the lowest point is as the chord of the arc which it has described in its descent. (Newton, Principia, (I), pg 25) Chris Sangwin (University of Edinburgh) Pendulum April / 38
19 y = l(1 cos(2θ)) = 2l sin 2 (θ). Chris Sangwin (University of Edinburgh) Pendulum April / 38
20 y = l(1 cos(2θ)) = 2l sin 2 (θ). By conservation of energy, the kinetic energy at O equals the potential energy at P 0. Chris Sangwin (University of Edinburgh) Pendulum April / 38
21 y = l(1 cos(2θ)) = 2l sin 2 (θ). By conservation of energy, the kinetic energy at O equals the potential energy at P 0. gy = 2gl sin 2 (θ) = 1 2 v 2. v 2 = 4gl sin 2 (θ) = g l (2l sin(θ))2 = g l c2. Chris Sangwin (University of Edinburgh) Pendulum April / 38
22 y = l(1 cos(2θ)) = 2l sin 2 (θ). By conservation of energy, the kinetic energy at O equals the potential energy at P 0. gy = 2gl sin 2 (θ) = 1 2 v 2. Hence v = g l c. v 2 = 4gl sin 2 (θ) = g l (2l sin(θ))2 = g l c2. Chris Sangwin (University of Edinburgh) Pendulum April / 38
23 y = l(1 cos(2θ)) = 2l sin 2 (θ). By conservation of energy, the kinetic energy at O equals the potential energy at P 0. gy = 2gl sin 2 (θ) = 1 2 v 2. Hence v = v 2 = 4gl sin 2 (θ) = g l (2l sin(θ))2 = g l c2. g l c. MEI AS FM 6.02e Chris Sangwin (University of Edinburgh) Pendulum April / 38
24 The pendulum as experimental apparatus MEI FM 6.03e Chris Sangwin (University of Edinburgh) Pendulum April / 38
25 The pendulum as experimental apparatus MEI FM 6.03e Chris Sangwin (University of Edinburgh) Pendulum April / 38
26 Issac Newton Chris Sangwin (University of Edinburgh) Pendulum April / 38
27 Newton s Laws of Motion Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. The relationship between an object s mass m, its acceleration a, and the applied force F is F = ma. For every action there is an equal and opposite reaction. AS/A level core Chris Sangwin (University of Edinburgh) Pendulum April / 38
28 Simple harmonic motion Point at P. Note s = lθ and a = l θ. ml θ = mg sin(θ), θ(0) = θ 0, θ(0) = θ 0. Chris Sangwin (University of Edinburgh) Pendulum April / 38
29 Simple harmonic motion Point at P. Note s = lθ and a = l θ. ml θ = mg sin(θ), θ(0) = θ 0, θ(0) = θ 0. For small θ assume sin(θ) θ. Chris Sangwin (University of Edinburgh) Pendulum April / 38
30 Simple harmonic motion Point at P. Note s = lθ and a = l θ. ml θ = mg sin(θ), θ(0) = θ 0, θ(0) = θ 0. For small θ assume sin(θ) θ. ml θ = mgθ. Chris Sangwin (University of Edinburgh) Pendulum April / 38
31 Simple harmonic motion Point at P. Note s = lθ and a = l θ. ml θ = mg sin(θ), θ(0) = θ 0, θ(0) = θ 0. For small θ assume sin(θ) θ. ml θ = mgθ. Then the solution is where ω 2 = g/l θ(t) = A cos(ωt + ρ). Chris Sangwin (University of Edinburgh) Pendulum April / 38
32 Notes on SHM Many assumptions: e.g. ignore friction. Chris Sangwin (University of Edinburgh) Pendulum April / 38
33 Notes on SHM Many assumptions: e.g. ignore friction. sin(θ) θ makes it easier. Chris Sangwin (University of Edinburgh) Pendulum April / 38
34 Notes on SHM Many assumptions: e.g. ignore friction. sin(θ) θ makes it easier. Physics experiment to establish value of g. Chris Sangwin (University of Edinburgh) Pendulum April / 38
35 Notes on SHM Many assumptions: e.g. ignore friction. sin(θ) θ makes it easier. Physics experiment to establish value of g. For SHM the period does not depend on amplitude. Chris Sangwin (University of Edinburgh) Pendulum April / 38
36 Notes on SHM Many assumptions: e.g. ignore friction. sin(θ) θ makes it easier. Physics experiment to establish value of g. For SHM the period does not depend on amplitude. For a real pendulum it does. Chris Sangwin (University of Edinburgh) Pendulum April / 38
37 Notes on SHM Many assumptions: e.g. ignore friction. sin(θ) θ makes it easier. Physics experiment to establish value of g. For SHM the period does not depend on amplitude. For a real pendulum it does. The Hooke s law spring/mass gives SHM. MEI AS FM 4.10f Chris Sangwin (University of Edinburgh) Pendulum April / 38
38 Christiaan Huygens Chris Sangwin (University of Edinburgh) Pendulum April / 38
39 Christiaan Huygens Horologium oscillatorium, (1673) The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula As Applied to Clocks Chris Sangwin (University of Edinburgh) Pendulum April / 38
40 Christiaan Huygens Horologium oscillatorium, (1673) The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula As Applied to Clocks A pendulum which changes its length (cf elastic pendulum) Chris Sangwin (University of Edinburgh) Pendulum April / 38
41 Energy considerations (simple pendulum) ml θ = mg sin(θ), θ(0) = θ 0, θ(0) = θ0. Chris Sangwin (University of Edinburgh) Pendulum April / 38
42 Energy considerations (simple pendulum) Multiply by l θ so that, ml θ = mg sin(θ), θ(0) = θ 0, θ(0) = θ0. ml 2 θ θ + mgl θ sin(θ) = 0. Chris Sangwin (University of Edinburgh) Pendulum April / 38
43 Energy considerations (simple pendulum) ml θ = mg sin(θ), θ(0) = θ 0, θ(0) = θ0. Multiply by l θ so that, ml 2 θ θ + mgl θ sin(θ) = 0. Integrating this gives or 1 2 ml2 θ 2 mgl cos(θ) = E 0, 1 2 ml2 θ 2 + mgl(1 cos(θ)) = E 1. Chris Sangwin (University of Edinburgh) Pendulum April / 38
44 Energy considerations (simple pendulum) Multiply by l θ so that, Integrating this gives or ml θ = mg sin(θ), θ(0) = θ 0, θ(0) = θ0. ml 2 θ θ + mgl θ sin(θ) = ml2 θ 2 mgl cos(θ) = E 0, 1 2 ml2 θ 2 + mgl(1 cos(θ)) = E 1. Kinetic and potential energy. MEI FM 6.02 Chris Sangwin (University of Edinburgh) Pendulum April / 38
45 Energy considerations 1 2 ml2 θ2 + mgl(1 cos(θ)) = E 1. Conservation equation: E 1, is constant Chris Sangwin (University of Edinburgh) Pendulum April / 38
46 Energy considerations 1 2 ml2 θ2 + mgl(1 cos(θ)) = E 1. Conservation equation: E 1, is constant Writing this as θ 2 = 2g (cos(θ) 1) + 2E 1 l ml 2, The form y 2 = α(cos(x) 1) + β where α, β > 0. Chris Sangwin (University of Edinburgh) Pendulum April / 38
47 Phase plane y 2 = α(cos(x) 1) + β Chris Sangwin (University of Edinburgh) Pendulum April / 38
48 Blackburn Y pendulum Chris Sangwin (University of Edinburgh) Pendulum April / 38
49 Blackburn Y pendulum For small oscillations x 1 (t) = A 1 cos(ω 1 t + ρ 1 ). x 2 (t) = A 2 cos(ω 2 t + ρ 2 ). Chris Sangwin (University of Edinburgh) Pendulum April / 38
50 Blackburn Y pendulum For small oscillations If ω 1 ω 2 is rational... x 1 (t) = A 1 cos(ω 1 t + ρ 1 ). x 2 (t) = A 2 cos(ω 2 t + ρ 2 ). Chris Sangwin (University of Edinburgh) Pendulum April / 38
51 Blackburn Y pendulum For small oscillations x 1 (t) = A 1 cos(ω 1 t + ρ 1 ). x 2 (t) = A 2 cos(ω 2 t + ρ 2 ). If ω 1 ω 2 is rational... Lissajous figure Chris Sangwin (University of Edinburgh) Pendulum April / 38
52 Add in damping Chris Sangwin (University of Edinburgh) Pendulum April / 38
53 Coupled oscillators Chris Sangwin (University of Edinburgh) Pendulum April / 38
54 Coupled oscillators Wilberforce s spring (1894) Chris Sangwin (University of Edinburgh) Pendulum April / 38
55 Coupled oscillators Wilberforce s spring (1894) mÿ = ky 1 2 ɛφ, I φ = τφ 1 2 ɛy. These are weakly coupled linear oscillators. Chris Sangwin (University of Edinburgh) Pendulum April / 38
56 Coupled oscillators Wilberforce s spring (1894) mÿ = ky 1 2 ɛφ, I φ = τφ 1 2 ɛy. These are weakly coupled linear oscillators. ω 2 s = k m, ω2 t = τ I. Chris Sangwin (University of Edinburgh) Pendulum April / 38
57 Techniques Many methods: Direct substitution of candidate y(t) = e ωt. Set up whole system as ẋ = Ax. Laplace transforms Chris Sangwin (University of Edinburgh) Pendulum April / 38
58 Solution After some work... and 2ω 2 = ( ) (ω ωs 2 + ωt 2 ) ± 2 s ωt ɛ 2 /(mi). (1) φ(t) = y(t) = ɛy 0 2I(ωF 2 ω2 S ) (cos(ω F t) cos(ω S t)), y ( ) 0 I(ωF 2 ω2 S ) (IωF 2 τ) cos(ω F t) (IωS 2 τ) cos(ω St). Chris Sangwin (University of Edinburgh) Pendulum April / 38
59 Solution After some work... and φ(t) = y(t) = 2ω 2 = ( ) (ω ωs 2 + ωt 2 ) ± 2 s ωt ɛ 2 /(mi). (1) ɛy 0 2I(ωF 2 ω2 S ) (cos(ω F t) cos(ω S t)), y ( ) 0 I(ωF 2 ω2 S ) (IωF 2 τ) cos(ω F t) (IωS 2 τ) cos(ω St). ( ) α + β cos(α) cos(β) = 2 sin sin 2 ( β α 2 ). Chris Sangwin (University of Edinburgh) Pendulum April / 38
60 ( ) α + β cos(α) cos(β) = 2 sin sin 2 ( β α 2 ). Chris Sangwin (University of Edinburgh) Pendulum April / 38
61 ( ) α + β cos(α) cos(β) = 2 sin sin 2 φ(t) = ( β α ). 2 ( ) ( ) ɛy 0 I(ωF 2 ω2 S ) sin ωf + ω S ωf ω t S t 2 2 which is the product of two sinusoids. Chris Sangwin (University of Edinburgh) Pendulum April / 38
62 ( ) α + β cos(α) cos(β) = 2 sin sin 2 φ(t) = ( β α ). 2 ( ) ( ) ɛy 0 I(ωF 2 ω2 S ) sin ωf + ω S ωf ω t S t 2 2 which is the product of two sinusoids. Special case (ω F = ω S ) when ω s = ω t complete energy transfer between torsion and vertical modes. Chris Sangwin (University of Edinburgh) Pendulum April / 38
63 ( ) α + β cos(α) cos(β) = 2 sin sin 2 φ(t) = ( β α ). 2 ( ) ( ) ɛy 0 I(ωF 2 ω2 S ) sin ωf + ω S ωf ω t S t 2 2 which is the product of two sinusoids. Special case (ω F = ω S ) when ω s = ω t complete energy transfer between torsion and vertical modes. Physics experiment to measure the value of τ. Chris Sangwin (University of Edinburgh) Pendulum April / 38
64 Elastic pendulum I first noticed the mode coupling in the swinging pendulum when I was a Teaching Assistant as a graduate student when (by chance) a student could not make the lab experiment work. This classical experiment, as you know, first measures the spring constant and then predicts the period. After stewing about this for a while I was able to work it out. (Olsson, M. 2008, private communication) See Olsson Chris Sangwin (University of Edinburgh) Pendulum April / 38
65 Deriving equations Kinetic energy Potential energy T = m 2 (ẋ 2 + ẏ 2 ), (2) V = mgy + k 2 (l l 0) 2, (3) l 2 = (l 1 y) 2 + x 2. Chris Sangwin (University of Edinburgh) Pendulum April / 38
66 Deriving equations Kinetic energy Potential energy T = m 2 (ẋ 2 + ẏ 2 ), (2) V = mgy + k 2 (l l 0) 2, (3) l 2 = (l 1 y) 2 + x 2. Approximate the Lagrangian L = T V as Chris Sangwin (University of Edinburgh) Pendulum April / 38
67 Approximate equations of motion ẍ = ω 2 px + λxy, ÿ = ω 2 sy + λ 2 x 2. Chris Sangwin (University of Edinburgh) Pendulum April / 38
68 Approximate equations of motion ẍ = ω 2 px + λxy, ÿ = ω 2 sy + λ 2 x 2. where ω 2 p = g l 1, ω 2 s = k m λ = ω 2 s l 0 l1 2 = k m l 0 l1 2. Chris Sangwin (University of Edinburgh) Pendulum April / 38
69 Numerical solutions Chris Sangwin (University of Edinburgh) Pendulum April / 38
70 Start with a vertical motion x << 1 y(t) a cos(ω s t). Chris Sangwin (University of Edinburgh) Pendulum April / 38
71 Start with a vertical motion x << 1 So y(t) a cos(ω s t). ẍ(t) + (ω 2 p + λa cos(ω s t))x(t) = 0. Chris Sangwin (University of Edinburgh) Pendulum April / 38
72 Start with a vertical motion x << 1 So Mathieu s equation ω s = 2ω p. y(t) a cos(ω s t). ẍ(t) + (ω 2 p + λa cos(ω s t))x(t) = 0. Chris Sangwin (University of Edinburgh) Pendulum April / 38
73 Mathieu s equation ẍ(t) + (α + β cos(t))x(t) = 0 Chris Sangwin (University of Edinburgh) Pendulum April / 38
74 Mathieu s equation ẍ(t) + (α + β cos(t))x(t) = 0 Instability of the elastic pendulum Stability of the driven inverted pendulum Chris Sangwin (University of Edinburgh) Pendulum April / 38
75 Driven Double Pendulum Chris Sangwin (University of Edinburgh) Pendulum April / 38
76 Chaos Simple systems: simple behavior. Chris Sangwin (University of Edinburgh) Pendulum April / 38
77 Chaos Simple systems: simple behavior. complex behavior! Chris Sangwin (University of Edinburgh) Pendulum April / 38
78 Chaos Simple systems: simple behavior. complex behavior! Double pendulum. Chris Sangwin (University of Edinburgh) Pendulum April / 38
79 Chaos Simple systems: simple behavior. complex behavior! Double pendulum. Complexity very different from irrational Lissajous. Chris Sangwin (University of Edinburgh) Pendulum April / 38
80 Magnetic pendulum Release the pendulum from rest. Which magnet does it end up above? Chris Sangwin (University of Edinburgh) Pendulum April / 38
81 Magnetic pendulum Release the pendulum from rest. Which magnet does it end up above? Chris Sangwin (University of Edinburgh) Pendulum April / 38
82 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. Chris Sangwin (University of Edinburgh) Pendulum April / 38
83 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. Chris Sangwin (University of Edinburgh) Pendulum April / 38
84 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. 3 Contemporary examples relevant to the new A-level. Chris Sangwin (University of Edinburgh) Pendulum April / 38
85 Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. 3 Contemporary examples relevant to the new A-level. 4 Physics apparatus Chris Sangwin (University of Edinburgh) Pendulum April / 38
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