Math 55 Differential Equations Project 1 20 Poe s Pendulum Emilia Brinckhaus and Liya Zhu Introduction This project is based on a story in Edgar Allan Poe s story The Pit and the Pendulum which tells of a prisoner tied on the floor facing a sharp edged pendulum descending toward him Poe describes that the sweep of the descending pendulum increases as the swinging velocity goes faster We ll try to model the pendulum s motion mathematically and discover whether Poe s description is accurate 2 20 Poe s Description of the Pendulum s Motion Looking upward I surveyed the ceiling of my prison I fancied that I saw the pendulum in motion In a instant afterward the fancy was confirmed Its sweep was brief and of course slow It might 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 increased in extent by nearly a yard As a natural consequence its velocity was also much greater I now observed with what horror it is needless to say that its nether extremity was formed of a crescent of glittering steel about a foot in length from horn to horn the horns upward and the under edge as keen as that of a razor and the whole hissed as it swung through the air long long hours of horror more than mortal during which I counted the rushing oscillations of the steel Inch by inch line by line which a descent only appreciable at intervals that seemed ages down and still down it came The vibration of the pendulum was at right angles to my length I saw that the crescent was designed to cross the region of my heart 3 20 4 20 The Model of a descending Pendulum r 5 20 R t L t r Length of the wire with respect to time L t The angle that the wire makes with the downward vertical t Unit vector in direction of the wire r Unit vector perpendicular to r The position vector for the pendulum bob R t L t r Position vector R Lr Doing some work with the derivatives of r and and using the chain rule we find and write the velocity and acceleration vectors with the inclusion of the terms and 6 20 R0 L0r L 0 R00 L00 L 0 0 r 2L0 0 L 00 From the last equation it is seen that the component of the acceleration vector in the direction is 2L0 0 L 00 Taking the mass of the pendulum bob to be m and g being the gravitational constant the component of the gravitational force in the direction is mg sin Equation of angular motion By using Newton s second law we have the equation of the pendulum s angular motion m 2L0 0 L 00 Setting our 2 equations equal to each other mg sin m 2L0 0 L 00 1 7 20 Since the angle is relatively small we use to replace sin in our equation After cancelling the mass factor m we have the equation of the pendulum s linear motion 2L0 0 L 00 g 0 2 L and L 0 represent the pendulum s sweep and curvilinear velocity respectively The decaying angle of a steadily descending pendulum 1 5 8 20 1 theta 0 5 0 0 5 1 0 20 40 60 80 100 time Here is a graph drawn with the help of Matlab s ode45 solver of versus time The Sweep of the Pendulum 4 9 20 3 2 1 0 1 2 3 4 0 20 40 60 80 100 This is the graph of L the pendulum s sweep The Curvilinear Velocity 6 10 20 4 2 0 2 4 6 0 20 40 60 80 100 Lastly this graph shows L 0 over time which is the velocity of the pendulum The Differential Equation of the Descending Pendulum 11 20 Assuming that the pendulum is descending at a constant rate the length of the wire is expressed as L t a bt 3 a and b are positive constants Substituting that equation for L in 2L0 0 L 00 g 0 we get the equation of the pendulum s angular motion a bt 00 2b 0 g 0 4 The Differential Equation Transformed We would like to be able to change our equation into the form of an equation with known properties Some intense work is required to figure out what new variables will transform our original equation But we will simply introduce the two new variables x and y 2p x a bt g y a bt b With the use of these new variables and the chain rule we get the following equation 2 2d y x dx2 x dy x2 1 y 0 dx 5 12 20 Our New Equation Bessel s Equation 13 20 2 2d y x dx2 dy x2 1 y 0 6 dx The new equation is a Bessel s equation of order one Bessel equations are special differential equations with solutions known as Bessel functions Any solution y x of this Bessel equation determines a solution of our original equation For all solutions y x the solution to our original equation is 2p 1 y t a bt g b a bt x Bessel Functions One solution of the first order Bessel s equation is J1 x which is the Bessel Function of order one of the First Kind For a second independent solution because we have an integer order Bessel s equation we must turn to the first order Bessel Function of the Second Kind depicted as Y1 x Therefore our general solution is given by 14 20 y x AJ1 x BY1 x where A and B are arbitrary constants No other solution can have any form but this The Solutions The only problem with the solutions to the Bessel equations is that they generally have the form of an infinite series a series of ascending powers of x We could solve these and get numerical approximations but we are going to take a whole different approach All Bessel s equations have several known properties Properties of solutions of our first order Bessel equation become properties of the angular motion of the steadily descending pendulum 15 20 Properties of Bessel Functions y x have an infinity of zeros 16 20 As x y x decays to 0 Therefore y x is oscillating with decreasing amplitude All these properties translate into properties of t the solution to our original differential equation J1 x The Bessel Equation of the First kind 0 6 17 20 0 4 y 0 2 0 0 2 0 4 0 20 40 60 x 80 100 Y1 x The Bessel Equation of the Second kind 1 18 20 y 0 5 0 0 5 1 0 20 40 60 x 80 100 Conclusion Has our mathematical model fit Poe s description Well the sweep of our steadily descending pendulum was seen to increase over time but this seemed to happen much more …