Lab 13: The Undamped Pendulum

In this lab we will study the motion of a pendulum, following exercises 18.3.7, 18.3.8, and 18.3.9 in the text.  You may work in groups of 2-3.  One written lab report (hard copy) should be submitted per group, by next Thursday’s lab.

Your report should have a title and an introductory paragraph, stating the purpose of the report and your principal findings.  Then your report should include several paragraphs explaining what you did (procedure) and what you discovered (results).  (Make sure that your paragraphs answer all the questions that are asked.)  Incorporate well-labeled supporting graphs or tables into your report as appropriate.  Finally, your report should include a paragraph with your conclusions and insights.

Introduction

We model the motion of a pendulum using a differential equation that can be obtained from Newtonian mechanics, assuming that the pendulum is a point mass attached to a massless rod that does not stretch or bend and that there is no friction or air resistance.

Despite these simplifying assumptions, the differential equation we obtain does not have an elementary solution.   We will approximate the solution by hand first, then use Mathematica to obtain approximate solutions.

We describe the motion in terms of the angular displacement θ (in radians) as it changes over time.  The derivative θ' is the angular velocity (in radians per unit time).  Mathematica always assumes that arguments for trigonometric functions are in radians, so there is no need to convert to degrees, except when it will help you interpret your results.  If you want the tick marks on your axes to be multiples of π, you can set the tick marks manually using the Ticks option with the Plot command.  For example,

Plot[Sin[x],{x,0,10},Ticks->{{0,Pi,2 Pi,3 Pi},{-1,1}}]

will produce a graph of sin(x) on the interval [0,10] with tick marks on the x-axis occuring at integer multiples of π.

Please read the introductory material on page 267, which precedes the statement of the exercises.

Approximation by hand (18.3.7a)

First we find an approximate solution using Taylor polynomials at t=0.  Our initial conditions give us θ(0)=0 and θ'(0)=2, but in order to obtain values for higher order derivatives at t=0, we will need to do some work.  In particular, we need to take the derivative (with respect to t, not θ) of both sides of the given differential equation.  Make sure to use the chain rule!

First approximation with Mathematica (13.3.7b,d)

Next we use Mathematica to obtain a graph of an approximate solution.  The following commands find and plot an approximate solution:

Approx1 =NDSolve[{theta''[t]+ 4*Sin[theta[t]]==0,theta[0]==0,theta'[0]==2},theta,{t,0,15}];
Plot[Evaluate[theta[t]/.Approx1],{t,0,15},PlotRange->{-2, 2}]

Notice the Evaluate command inside of the Plot command.  The Evaluate command is converting the output of NDSolve into a form that can be plotted.  When you want to plot your Taylor polynomial with the Mathematica approximation, on the same set of axes, make sure to use curly braces: Plot[{...,...}, {t, 0, 15}, ...].

Shahriari discusses the stepsize parameter in Maple (in 13.3.7c), but you may omit this.

Period and amplitude (13.3.8)

By varying the initial conditions, we explore the relationship between period and amplitude.  Use Mathematica to find and plot approximate solutions corresponding to ten different initial angular velocities.  Record the period and amplitude of each, and make a plot of your data.  You can make a plot of data points in Mathematica using the ListPlot command.  For example, the following command plots the points (1,2), (2,4), and (3, 8).

ListPlot[{{1,2},{2,4},{3,8}}]

Linearizing the differential equation (18.3.9)

By replacing sin(θ) with a local linear approximation, we obtain a simpler differential equation.  In fact, we are able to obtain an exact solution when we assume θ'(0)=1.

In part (c) of this exercise, you are invited to make a conjecture about the relationship of the period of the solution to the non-linear equation and the period of the solution to the linear differential equation.  It might help to consider the following: for what θ is the linear differential equation a good approximation for the non-linear differential equation?

Varying the initial displacement

(★★ Extra Credit ★★)  What happens if you have a nonzero initial angular displacement instead of a nonzero initial angular velocity?  (First let θ(0) be some small but nonzero angle, in radians, and let θ'(0) be zero.)  What happens if you vary the initial angular displacement?