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The Pendulum Plain and Puzzling The Pendulum Plain and Puzzling Chris Sangwin School of Mathematics University of Edinburgh April 2017 Chris Sangwin (University of Edinburgh) Pendulum April 2017 1 / 38 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 2017 2 / 38 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!) 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 2017 3 / 38 3 Contemporary examples relevant to A-level mathematics/FM. 4 Physics apparatus Illustrated by examples. (Equations on request!) 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 2017 3 / 38 4 Physics apparatus Illustrated by examples. (Equations on request!) 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 2017 3 / 38 Illustrated by examples. (Equations on request!) 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 2017 3 / 38 (Equations on request!) 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 2017 3 / 38 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 2017 3 / 38 Mechanics within mathematics Chris Sangwin (University of Edinburgh) Pendulum April 2017 4 / 38 y¨ = −k 2y? So y(t) = A cos(kt + ρ): Where is the x-coordinate in the model? The elastic pendulum What happens when I displace vertically and release from rest? Chris Sangwin (University of Edinburgh) Pendulum April 2017 5 / 38 So y(t) = A cos(kt + ρ): Where is the x-coordinate in the model? The elastic pendulum What happens when I displace vertically and release from rest? y¨ = −k 2y? Chris Sangwin (University of Edinburgh) Pendulum April 2017 5 / 38 Where is the x-coordinate in the model? The elastic pendulum What happens when I displace vertically and release from rest? y¨ = −k 2y? So y(t) = A cos(kt + ρ): Chris Sangwin (University of Edinburgh) Pendulum April 2017 5 / 38 The elastic pendulum What happens when I displace vertically and release from rest? y¨ = −k 2y? So y(t) = A cos(kt + ρ): Where is the x-coordinate in the model? Chris Sangwin (University of Edinburgh) Pendulum April 2017 5 / 38 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 2017 6 / 38 Pre-Newtonian mechanics Chris Sangwin (University of Edinburgh) Pendulum April 2017 7 / 38 Chris Sangwin (University of Edinburgh) Pendulum April 2017 8 / 38 Randomness Chris Sangwin (University of Edinburgh) Pendulum April 2017 9 / 38 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 2017 10 / 38 By conservation of energy, the kinetic energy at O equals the potential energy at P0. 1 gy = 2gl sin2(θ) = v 2: 2 g g v 2 = 4gl sin2(θ) = (2l sin(θ))2 = c2: l l q g Hence v = l c. MEI AS FM 6.02e y = l(1 − cos(2θ)) = 2l sin2(θ): Chris Sangwin (University of Edinburgh) Pendulum April 2017 11 / 38 1 gy = 2gl sin2(θ) = v 2: 2 g g v 2 = 4gl sin2(θ) = (2l sin(θ))2 = c2: l l q g Hence v = l c. MEI AS FM 6.02e y = l(1 − cos(2θ)) = 2l sin2(θ): By conservation of energy, the kinetic energy at O equals the potential energy at P0. Chris Sangwin (University of Edinburgh) Pendulum April 2017 11 / 38 q g Hence v = l c. MEI AS FM 6.02e y = l(1 − cos(2θ)) = 2l sin2(θ): By conservation of energy, the kinetic energy at O equals the potential energy at P0. 1 gy = 2gl sin2(θ) = v 2: 2 g g v 2 = 4gl sin2(θ) = (2l sin(θ))2 = c2: l l Chris Sangwin (University of Edinburgh) Pendulum April 2017 11 / 38 MEI AS FM 6.02e y = l(1 − cos(2θ)) = 2l sin2(θ): By conservation of energy, the kinetic energy at O equals the potential energy at P0. 1 gy = 2gl sin2(θ) = v 2: 2 g g v 2 = 4gl sin2(θ) = (2l sin(θ))2 = c2: l l q g Hence v = l c. Chris Sangwin (University of Edinburgh) Pendulum April 2017 11 / 38 y = l(1 − cos(2θ)) = 2l sin2(θ): By conservation of energy, the kinetic energy at O equals the potential energy at P0. 1 gy = 2gl sin2(θ) = v 2: 2 g g v 2 = 4gl sin2(θ) = (2l sin(θ))2 = c2: l l q g Hence v = l c. MEI AS FM 6.02e Chris Sangwin (University of Edinburgh) Pendulum April 2017 11 / 38 The pendulum as experimental apparatus MEI FM 6.03e Chris Sangwin (University of Edinburgh) Pendulum April 2017 12 / 38 The pendulum as experimental apparatus MEI FM 6.03e Chris Sangwin (University of Edinburgh) Pendulum April 2017 12 / 38 Issac Newton 1643 – 1727 Chris Sangwin (University of Edinburgh) Pendulum April 2017 13 / 38 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 2017 14 / 38 For small θ assume sin(θ) ≈ θ. mlθ¨ = −mgθ: Then the solution is θ(t) = A cos(!t + ρ): where !2 = g=l Simple harmonic motion Point at P. Note s = lθ and a = lθ¨. mlθ¨ = −mg sin(θ); θ(0) = θ0; θ_(0) = θ_0: Chris Sangwin (University of Edinburgh) Pendulum April 2017 15 / 38 mlθ¨ = −mgθ: Then the solution is θ(t) = A cos(!t + ρ): where !2 = g=l Simple harmonic motion Point at P. Note s = lθ and a = lθ¨. mlθ¨ = −mg sin(θ); θ(0) = θ0; θ_(0) = θ_0: For small θ assume sin(θ) ≈ θ. Chris Sangwin (University of Edinburgh) Pendulum April 2017 15 / 38 Then the solution is θ(t) = A cos(!t + ρ): where !2 = g=l Simple harmonic motion Point at P. Note s = lθ and a = lθ¨. mlθ¨ = −mg sin(θ); θ(0) = θ0; θ_(0) = θ_0: For small θ assume sin(θ) ≈ θ. mlθ¨ = −mgθ: Chris Sangwin (University of Edinburgh) Pendulum April 2017 15 / 38 Simple harmonic motion Point at P. Note s = lθ and a = lθ¨. mlθ¨ = −mg sin(θ); θ(0) = θ0; θ_(0) = θ_0: For small θ assume sin(θ) ≈ θ. mlθ¨ = −mgθ: Then the solution is θ(t) = A cos(!t + ρ): where !2 = g=l Chris Sangwin (University of Edinburgh) Pendulum April 2017 15 / 38 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 Notes on SHM Many assumptions: e.g. ignore friction. Chris Sangwin (University of Edinburgh) Pendulum April 2017 16 / 38 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.
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