Worksheet #6 - PHY102 (Spr. 2005)

Generating and plotting lists of numbers,
``Do'' loops and animation - due Feb. 24th

We often ask a computer to do an operation many times. There are a large number of ways of doing these ``iterative''tasks in Mathematica. Here are two that you will need this week(look them up in the online help):

Table, Do

You will also need to learn how to plot lists of numbers using:

ListPlot, ListPlot3D

Finally, animation is very simple in mathematica. Simply generate a series of frames (e.g. using a ``Do'' loop) and then double click on one of the frames. This automatically animates the set of frames.

Problem 1. (i) Listplot plots a list of numbers on the y axis of a graph. To see how this works, enter the following code

sintable=Table[Sin[x],{x,0,20,.1}]

ListPlot[sintable]

(ii) Three dimensional plots are just as easy. Enter and run the following code

sintable3D=Table[Sin[x*y],{x,0,4,.1},{y,0,4,0.1}] ListPlot3D[sintable3D]

(iii) Using the Table function, generate points to represent a circle for y>0. Plot this data using ListPlot.

Problem 2.

Here is a code to sum the first n integers, with n running from 1 to 100. The first command sets up an array which is used to store the sums.

sumintegers=Range[100];

sumintegers[[1]]=1;

Do[

{sumintegers[[n]]=sumintegers[[n-1]]+n},

{n,2,100,1}

]

ListPlot[sumintegers]

The Riemann zeta function is defined by $\zeta(p) =\sum_{n=1,\infty} 1/n^p$. This sum is convergent for p>1(why?). Write a program to find $\zeta(p)$ as a function of the number of terms, N, included in the sum. Plot the value of this sum for p=3 as a function of N. How many terms do you need to take until your answer appears to be correct to 4 digit accuracy(how big does N need to be)?

Problem 3. Enter and run the following code which animates circular motion.

timestep = 0.1 Pi

tstart = 0

tend = timestep

x[a_]:= Cos[a]

y[a_]:= Sin[a]

Do[

{ParametricPlot[{x[t],y[t]},{t,tstart,tend}, PlotRange->{{-1,1},{-1,1}}];

tstart=tend;

tend=tend+timestep}, {i,1,20}

]
Modify this code to animate the following projectile motion problem: A mass of 20kg is fired from a height of 2000 meters, with initial angle to the horizontal of 60 degrees and initial speed of 500m/s (ignore drag). Your animation should begin at firing and end when the mass hits ground level. DO NOT print out the frames used to produce your animation. Instead show your animation to your TA. However DO hand in your Mathematica code.