Worksheet #6 – PHY102 (Spring 2012)Generating and plotting lists of numbers,“Do” loops and animationWe often need to ask the computer to do an operation many times. There are several ways of doingthese “iterative” tasks in Mathematica. The two that you will use this week are Table and Do.Look these up in the online help. Especially if you are familiar with C++, you may also want tolearn For at this time.)This week you will also learn how to make plots based on lists of numbers, using ListPlot andListPlot3D.Finally, animation is very simple to do in Mathematica. You will learn the commands Animateand Manipulate to do this.Problem 1. (This problem really belongs to the last week’s worksheet, but that worksheet was indanger of becoming too long; and anyway, reviewing is a good thing!)You lift a box of mass 30 kg vertically to a height of h meters by sliding it up an inclined planethat makes an angle θ with respect to the horizontal.(i) Assuming there is no friction, use Mathematica to evaluate the integralR~F · d~r to calculate thework done, and check that the result agrees with the change in gravitational potential energy.(ii) Now include a frictional force f = c x N, which increases as the box moves up the plane. Herex is the distance measured along the inclined plane, N is the force normal to the inclined plane,and c is a constant. How much additional work does it now take to slide the box to the top of theramp? Solve this problem analytically using Mathematica. (In most physics courses, the coefficientof friction is assumed to be a constant. But in this problem it is not: here the coefficient of frictionis µ = c x, i.e., the inclined plane is very smooth at the bottom, but it gets gets progressivelystickier, the farther up you go.)Problem 2.(i) ListPlot plots a list of numbers on the y axis of a graph. To see how this works, enter thefollowing codesintable=Table[Sin[x], {x, 0, 20, .1 } ]ListPlot[sintable](ii) Three dimensional plots are just as easy. Enter and run the following code1sintable3D=Table[Sin[x*y],{x,0,4,.1},{y,0,4,0.1}]ListPlot3D[sintable3D]Notice that if you left-click on the pretty picture, you can rotate the point of view!(iii) Using the Table function, generate points to represent a circle, and plot these data usingListPlot. You can use the qualifier AspectRatio → 1 in ListPlot to force it to choose equal axes,so that your circle looks round. (AspectRatio → Automatic seems to work also.)Problem 3.Here is some code that computes the sum of the first n integers, with n running from 1 to 100.(The Range command is used here to set up an array to store the sums. It also sets the elementsof that array to the integers, but that asp ect of Range is not relevant here, because each elementof array is overwritten inside the loop.)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 ζ(p) =P∞n=11/np. This sum converges for all p > 1(Why? – Hint: the behavior at large n is similar to the behavior of the integralRdx/xpat largex).Write a program to find the approximate value of ζ(p) as a function of the number of terms, N, thatare included in the sum. Plot the value of this sum for p = 3 as a function of N. How many termsdo you need to take (i.e., how big does N need to be) to get the approximation correct to 4 digitaccuracy? Note, you will have to make up an alternative name for this N, because Mathematicaalready uses N for its numerical value function.Problem 4. First type in and run the following code, which animates circular motion:tstep = 0.05*Pi;x = Cos[t];y = Sin[t];Animate[ParametricPlot[{x, y}, {t, tstart, tstart + tstep}, PlotRange − > {{-1, 1},{-1, 1}}] , {tstart, 0, 2*Pi}]2(Another useful command is illustrated by replacing “Animate” by “Manipu late” in the abovecode. This generates a graph with a mouse-controlled slider to vary the parameter; but it appearsalso to be capable of doing the same function as Animate as well.)Modify this code to animate t he following projectile motion problem: A mass of 20 kg is fired froma height of 2000 meters, with initial angle above the horizontal of 60 degrees and initial speed of500m/s (ignore drag). Your animation should begin at firing and end when the mass hits
View Full Document
Unlocking...