Poe's Description of the Pendulum's MotionThe Mathematical Model of a descending PendulumMatlab ODE45 resultsThe decaying angle of a steadily descending pendulumThe Sweep of the PendulumThe Curvilinear VelocityThe Differential Equation TransformedA Bessel's EquationThe Solutions:Bessel FunctionsCylinder FunctionsSide-noteNormal FormThe First PropertyThe Second PropertyWhat we now knowJ1(x): The Bessel Equation of the First kindY1(x): The Bessel Equation of the Second kindConclusionAcknowledgementsPoe’s Description of . . .The Mathematical . . .Matlab ODE45 resultsThe Differential . . .A Bessel’s EquationNormal FormConclusionAcknowledgementsHome PageTitle PageJJ IIJ IPage 1 of 22Go BackFull ScreenCloseQuitPoe’s PendulumEmilia Brinckhaus and Liya ZhuMay 7,2002AbstractThis project is based on a story in Edgar Allan Poe’s literary classic, The Pit and thePendulum, written in 1842. In this story Poe tells of a prisoner tied on the floor,facing a s harp-edged p e ndulum descending toward him. Poe describes that the sweep of the descending pendulumincreases as the velocity goes faster. In this activity, we will examine this model from a mathematicalstandpoint, and discover whether Poe’s description of the pendulum’s motion is accurate.1. Poe’s Description of the Pendulum’s MotionWe begin our project with a quotation directly from Poe’s work, The Pit and the Pendulum describingthe pendulum....Looking upward, I surveyed the ceiling of my prison. It was some thirty or forty feetoverhead, and constructed much as the side walls. In one of its panels a very singularfigure riveted my whole attention. It was the painted figure of Time as he is commonlyrepresented, save that, in lieu of a scythe, he held what, at a casual glance. I s upposed toto be the pictured image of a huge pendulum, such as we see on a antique clocks. Therewas something, however, in the appearance of this machine which caused me to regard itmore attentively. While I gazed directly upward at it...I fancied that I saw it in motion. Ina instant afterward and fancy was confirmed. Its sweep was brief, and of course slow...Itmight has been half an hour, perhaps even an hour...before I again cast my eyes upward.What I then saw confounded and amazed me. The sweep of the pendulum had increasedin extent by nearly a yard. As a natural consequence its velocity was also much greater.Poe’s Description of . . .The Mathematical . . .Matlab ODE45 resultsThe Differential . . .A Bessel’s EquationNormal FormConclusionAcknowledgementsHome PageTitle PageJJ IIJ IPage 2 of 22Go BackFull ScreenCloseQuitFigure 1: Edgar Allan PoeBut what mainly disturbed me was the idea that it had perceptibly descended. I nowobserved–with what horror it is needless to say–that its nether extremity was formed of acrescent of glittering steel, about a foot in length from horn to horm; the horns upward,and the under edge as keen as that of a razor...and the whole hissed as it swung throughthe air..long, long hours of horror more than mortal during which I counted the rushingoscillations of the steel! Inch by inch–line by line–which a descent only appreciable atintervals that seemed ages–down and still down it came!...The vibration of the pendulumwas at right angles to my length. I saw that the crescent was designed to cross the regionof my heart...its terrifically wide sweep (some thirty feet or more)... Down–steadily downit crept. I took a frenzied pleasure in contrasting its downward with its lateral velocity. Tothe right to the left–far and wide–with the shriek of a damned spirit!... Down–certainly,relentlessly down!...In his story, Poe claims that the pendulum’s sweep was brief and of course slow, but that as the pendulumdescended, the sweep...had increased by nearly a yard...its velocity also was much greater, until finally ithad a terrifically wide sweep some thirty feet or more. We will set up a model of a descending pendulumand s ee if it agrees with Poe’s pendulum.Poe’s Description of . . .The Mathematical . . .Matlab ODE45 resultsThe Differential . . .A Bessel’s EquationNormal FormConclusionAcknowledgementsHome PageTitle PageJJ IIJ IPage 3 of 22Go BackFull ScreenCloseQuitθˆθˆrR(t) = L(t)ˆrFigure 2: Poe’s Pendulum2. The Mathematical Model of a descending PendulumLet the pendulum be supported by a wire which is descending at a constant(steady) rate. The lengthof the wire is a function of time, L(t). The angle that the wire makes with the downward vertical fromthe point of support can also be expressed as a function of time, θ(t). Assuming that ˆr andˆθ are unitvectors starting at the point of support, with ˆr parallel to the wire andˆθ perpendicular, the positionvector for the pendulum bob is R = Lˆr.We obtain two more equations, ˆr0=θ0ˆθ andˆθ0=-θ0ˆr, by using the following procedures: From theunit circle, we know that any point on the circle has the coordinates (cos α, sin α), where α is the anglebetween the vector going to the point from the origin and the vector going from the origin in the positivedirection of the horizontal axis. Thus:ˆr =cos θsin θPoe’s Description of . . .The Mathematical . . .Matlab ODE45 resultsThe Differential . . .A Bessel’s EquationNormal FormConclusionAcknowledgementsHome PageTitle PageJJ IIJ IPage 4 of 22Go BackFull ScreenCloseQuitθFigure 3: Unit Circleandˆθ =cos(θ + π/2)sin(θ + π/2)=−sin θcos θFrom the above two equations, we can easily derive the following:dˆrdθ=−sin θcos θ=ˆθanddˆθdθ=−cos θ−sin θ= −ˆrThenˆr0=dˆrdt=dˆrdθdθdt=ˆθθ0ˆθ0=dˆθdt=dˆθdθdθdt= −ˆrθ0Poe’s Description of . . .The Mathematical . . .Matlab ODE45 resultsThe Differential . . .A Bessel’s EquationNormal FormConclusionAcknowledgementsHome PageTitle PageJJ IIJ IPage 5 of 22Go BackFull ScreenCloseQuitSince R = Lˆr,ˆr0=ˆθθ0andˆθ0= −ˆrθ0, the velocity and acceleration vectors are:R0= (Lˆr)0= L0ˆr + Lˆr0= L0ˆr + Lθ0ˆθ,R00= L00ˆr + L0ˆr0+ L0θ0ˆθ + L0θ00ˆθ + Lθ0ˆθ0= ˆr + L0ˆr0+ L0θ0ˆθ + L0θ00ˆθ − Lθ0θ0ˆr= (L00− Lθ0θ0)ˆr + (2L0θ0+ Lθ00)ˆθ(1)Three forces are affe cting the pendulum: the force of gravity, the tension of the wire, and thefrictional force due to air. Since the magnitude of the friction force is small, we can ignore it. We shall alsoignore the tension force because we only need to consider the forces in theˆθ direction perpendicular tothe