MATH1020 Calculus II	Maple Lab 9
Section 1-4	Due: Tuesday, November 23, 1999.
http://www.math.rpi.edu/~mitchj/math1020

This lab consists of one question, examining the behavior of a space curve. Be sure to follow the lab guidelines as described on the class handouts.

1.

Consider the motion of a particle whose position is given as function of time t by

$${\bf r}(t)=\cos(5t)(2+\cos(t))\,{\bf i}+\sin(5t)(2+\sin(t))\,{\bf j}+t\,{\bf k}.$$

We will consider the motion for values of t between 0 and 10.

(a)

Let's begin by plotting the particle path. Type

> x:=cos(5*t)*(2+cos(t)); y:=sin(5*t)*(2+sin(t)); z:=t;
> plot3d([x,y,z],t=0..10,s=0..1,grid=[500,2]);

In the plot3d command, s is a dummy parameter and the grid option tells Maple to plot using 500 subintervals in t and 2 in s. Choose an appropriate view (or views) for your plot so that you can get a good feel for the motion of the particle. Describe the motion qualitatively.

(b)

In a typical application you might be interested in the speed of the particle as a function of time. Recall that the speed is given by $v= \Vert{\bf r}'\Vert=\sqrt{{\bf r}'\cdot{\bf r}'}$. The dot product is part of Maple's linear algebra package, so to compute the speed using Maple type

> with(linalg):
> r:=vector([x,y,z]);
> rp:=map(diff,r,t);
> speed:=sqrt(dotprod(rp,rp)) ;

Plot the speed versus t for $0\le t\le10$. Discuss the behavior of this curve. Does it agree with what you would expect?

(c)

The curvature $\kappa$ of the particle path plays a role in the calculation of the acceleration of the particle. The formula for $\kappa$ is

$$\kappa={\vert{\bf r}\sp\prime\times{\bf r}\sp{\prime\prime}\vert\over \vert{\bf r}\sp\prime\vert\sp3}$$

To compute the curvature using Maple type

> rpp:=map(diff,rp,t);
> u:=crossprod(rp,rpp);
> kappa:=sqrt(dotprod(u,u)/dotprod(rp,rp)**3);

Plot the curvature versus t for $0\le t\le10$ and discuss its behavior. Notice that there are rapid fluctuations on top of a gentle oscillation. In view of the form of ${\bf r}(t)$ above and its curve in part (a), can you explain this behavior?