IntroductionModeling Poe's Pendulum with Newtonian MechanicsModeling Poe's Pendulum Revisited Using Lagrangian MechanicsA System of EquationsA Change of VariablesFinding Initial Conditions for our Bessel Function from and TimeComputing and analyzing the BesselConclusionAcknowledgementsAppendixIntroductionModeling Poe’s . . .Modeling Poe’s . . .A System of EquationsA Change of VariablesFinding Initial . . .Computing and . . .ConclusionAcknowledgementsAppendixHome PageTitle PageJJ IIJ IPage 1 of 36Go BackFull ScreenCloseQuitThe Pit and the PendulumAaron Kavanaugh and Travis MoeMay 12, 2005AbstractEdgar Allen Poe wrote some of the most original and terrifying hor-ror stories of the 19th century. Poe’s protagonists often met with grizzlydeaths. His classic “The Pit and the Pendulum” is no exception to this.In Poe’s story, a man is strapped to a table and forced to watch as a everlengthening pendulum tippe d w ith a scythe swings closer and closer to hischest. In this paper, we investigate not the psychological, but the physicalaspects of a lengthening pendulum. We describe the lengthening pendulumin terms of Newtonian and Lagrangian mechanics and derive a differentialequation for the pendulum’s motion in terms of its angle. We then examinenumerical methods of solving this differential equation and transform ourdifferential e quation into a Bessel equation to discover an analytical solu-tion. Finally, we examine the motion of such a pe ndulum in reference toPoe’s story. While our paper may not be as terrifying as a Poe tale (unlessyou regard intense applications of the chain rule terrifying), it is sure to bemore informative.IntroductionModeling Poe’s . . .Modeling Poe’s . . .A System of EquationsA Change of VariablesFinding Initial . . .Computing and . . .ConclusionAcknowledgementsAppendixHome PageTitle PageJJ IIJ IPage 2 of 36Go BackFull ScreenCloseQuit1. Introduction“Looking upward, I surveyed the ceiling of my prison. It was somethirty or forty feet overhead, and constructed much as the side walls.In one of its panels a very singular figure riveted my whole attention .It was the painted figure of Time as he is commonly represented, savethat in lieu of a scythe he held what at a casual glance I supposedto be the pictured image of a huge pendulum, such as we see onantique clocks. There was something, however, in the appearance ofthis machine which caused me to regard it more attentively. WhileI gazed directly upward at it (for its position was immediately overmy own), I fancied that I saw it in motion. In an instant afterwardthe fancy was confirmed. Its sweep was brief, and of course slow. Iwatched it for some minutes, somewhat in fear but more in wonder [...]It might have been half-an-hour, perhaps even an hour (for I could takebut imperfect note of time) before I again cast my eyes upward. WhatI then saw confounded and amazed me. The sweep of the pendulumhad increased in extent by nearly a yard. As a natural consequence,its velocity was also much greater. But what mainly disturbed me wasthe idea that it had p erceptibly DESCENDED. I now observed, withwhat horror it is needless to say, that its nether extremity was formedof a crescent of glittering steel, about a foot in length from horn tohorn; the horns upward, and the under edge evidently as keen as thatof a razor. Like a razor also it seemed massy and heavy, tapering fromthe edge into a solid and broad structure above. It was appended to aweighty rod of brass, and the whole HISSED as it swung through theIntroductionModeling Poe’s . . .Modeling Poe’s . . .A System of EquationsA Change of VariablesFinding Initial . . .Computing and . . .ConclusionAcknowledgementsAppendixHome PageTitle PageJJ IIJ IPage 3 of 36Go BackFull ScreenCloseQuitair [...] What boots it to tell of the long, long hours of horror morethan mortal, during which I counted the rushing oscillations of thesteel! Inch by inch – line by line – with a descent only appreciable atintervals that seemed ages – down and still down it came! Days passed– it might have been that many days passed – ere it swept so closelyover me as to fan me with its acrid breath. [...] The vibration of thependulum was at right angles to my length. I saw that the crescentwas designed to cross the region of the heart. It would fray the sergeof my robe; it would return and repeat its operations – again – andagain. Notwithstanding its terrifically wide sweep (some thirty feet ormore) and the hissing vigour of its descent, sufficient to s under thesevery walls of iron, still the fraying of my robe would be all that, forseveral minutes, it would accomplish; and at this thought I paused. Idared not go farther than this reflection.”This is the descending pendulum as Poe describes it. An ingeniously crueltorture device which starts with a sweep that is “brief, and [...]slow” but, whichby the time it reaches its victim, possess a “terrifically wide sweep (some thirtyfeet or more) and the hissing vigour of its descent, sufficient to sunder thesevery walls of iron” and which swings through an arc so great that “the vibrationof the pendulum was at right angles to [the victims] length” implying that thependulum increases not only in sweep, but in its angle of motion, and in itsspeed. Such a device would truly be terrifying. We will attempt to model thislengthening pendulum and to discover if its real-life physical motion truly reflectsPoe’s poetic description.IntroductionModeling Poe’s . . .Modeling Poe’s . . .A System of EquationsA Change of VariablesFinding Initial . . .Computing and . . .ConclusionAcknowledgementsAppendixHome PageTitle PageJJ IIJ IPage 4 of 36Go BackFull ScreenCloseQuitL(t)ˆθˆrˆiˆjmg sin(θ)mgθθFigure 1: Poe’s Pendulum Viewed in Terms of Newtonian Physics2. Modeling Poe’s Pendulum with NewtonianMechanicsThe main constraint of the problem is the changing length of the pendulum.Suppose that the length of the wire at time t is given as L(t). We will let θ(t)represent the angle that the wire of the pendulum makes with its resting verticalposition. Suppose that ˆr andˆθ are unit vectors (see Figure 1), ˆr lying along thewire andˆθ always pointing int he direction of motion and orthogonal to the wire.Neglecting the damping force because it is minimal, and the tensile force of theIntroductionModeling Poe’s . . .Modeling Poe’s . . .A System of EquationsA Change of VariablesFinding Initial . . .Computing and . .