Question

What is the path that traces the circle in the plane

Trigonometric Functions
ANSWERED
asked 2021-01-10
What is the path that traces the circle in the plane =1 with radius =5 and center (−7,1,1) with constant speed 15?

Answers (1)

2021-01-11

Technically the path is independent of the speed, what you're talking about is a parametrization of the path. And even then, your description does not determine a single parametrization because you did not specify where on the path the parametrization should start, which way around the circle we want to go, and how many times we should loop around.
Nonetheless, we can find a parametrization that fits this bill. Firstly, let's look at the "standard" parametrization of the circle of radius 5 centered at the origin going around at a speed of \(\omega\) exactly once:
\(\displaystyle{r}{\left({t}\right)}={5}{\cos{{\left(\omega{t}\right)}}}{i}+{5}{\sin{{\left(\omega{t}\right)}}}{j},{t}\in{\left[{0},\omega{2}\times\pi\right)}\)
To tilt this on its side so that it's in the y=0 plane instead of the \(z=0\) plane, we can just replace j with k:
\(\displaystyle{r}{\left({t}\right)}={5}{\cos{{\left(\omega{t}\right)}}}{i}+{5}{\sin{{\left(\omega{t}\right)}}}{k},{t}\in{\left[{0},{2}\pi\omega\right)}\)
To have it go at a speed of 15 radians per unit of time, we just insert 15 for \(\omega \):
\(\displaystyle{r}{\left({t}\right)}={5}{\cos{{\left({15}{t}\right)}}}{i}+{5}{\sin{{\left({15}{t}\right)}}}{k},{t}\in{\left[{0},{2}\pi{15}\right)}\)
\(\displaystyle{r}{\left({t}\right)}={5}{\cos{{\left({15}{t}\right)}}}{i}+{5}{\sin{{\left({15}{t}\right)}}}{k},{t}\in{\left[{0},{152}\pi​\right)}\)
And finally to move it so that it's center is at (−7,1,1) (which incidentally will also move it from the \(y=0\) plane to the \(y=1\) plane), we just add the constant vector \(−7 i+j+k\) to it:
\(\displaystyle{r}{\left({t}\right)}={\left[−{7}+{5}{\cos{{\left({15}{t}\right)}}}\right]}{i}+{j}+{\left[{1}+{5}{\sin{{\left({15}{t}\right)}}}\right]}{k},{t}\in{\left[{0},{2}\pi{15}\right)}\)
\(\displaystyle{r}{\left({t}\right)}={\left[−{7}+{5}{\cos{{\left({15}{t}\right)}}}\right]}{i}+{j}+{\left[{1}+{5}{\sin{{\left({15}{t}\right)}}}\right]}{k},{t}\in{\left[{0},{152}\pi​\right)}\)
And we're done!

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