How to calculate curvature of a vector in Mathematica.

[november1992]

Homework Statement

r(t)={(4+cos20t) cost,+(4+cos20t) sint,+0.4sin20t}
Calculate the curvature of r[t] for 0≤t≤4pi

Homework Equations

k = | r' x r'' | / | r' |^3

The Attempt at a Solution

r[t_]:={4+Cos[20t]*Cos[t],4+Cos[20t]*Sin[t],0.4Sin[20t]}
k[t_]:=Norm[Cross[r',r'']]/Norm[r']^3
Plot[k[t],{t,0,4Pi}]
I don't get any error messages, but the graph is blank.

http://i.imgur.com/woWlabm.png
http://i.imgur.com/ptUZcVG.png

 

[SteamKing]

t or t_ in the Plot function?

 

[november1992]

It doesn't seem to make a difference.

http://i.imgur.com/woWlabm.png
http://i.imgur.com/ptUZcVG.png

 

[MednataMiza]

If you have defined a function using the "standard" form, id est, f[x_] := you do not want the x_ on the right hand side of the definition.
Now to your real problem:
You forgot to specify that the derivatives are also functions of t :)

r[t_] := {4 + Cos[20 t]*Cos[t], 4 + Cos[20 t]*Sin[t], 0.4 Sin[20 t]}
k[t_] := Norm[Cross[r'[t], r''[t]]]/Norm[r'[t]]^3
Plot[k[t], {t, 0, Pi/4}]

 

[november1992]

Haha, I actually managed to plot it, but I thought it was wrong because I wasn't expecting a huge wave.

I was also told to calculate the length of the curvature. Do you think this would be an acceptable answer?

http://i.imgur.com/HHYzFy7.png

Is there a way to simplify this expression?

 

[MednataMiza]

If I am not mistaken
$$\kappa = \left | \frac{f''(x)}{(1 + f'(x)^{2})^{\frac{3}{2}}} \right |$$

is the formula one would use to calculate the curvature.
Following this we would end up right here:

Well, in theory you can apply //Simplify, but Mathematica has done this already, so in my opinion - no :)

 

[november1992]

Okay, thanks. I have one last question. How can I use the inte\frac{}{}rval <0,4pi> in this equation?
I'm just a little confused about why this formula, k = |$\frac{r(t)' X r(t)''}{r(t)'^3}$| gives a different answer to the formula you posted

 

[MednataMiza]

Something like this should do the trick:

Sum[Limit[b, t -> \[Omega]], {\[Omega], 0, 4 Pi}]

where 'b' is

Where did you get this formula ?

 

[november1992]

From my textbook.

I scanned the page I found it from. http://i.imgur.com/AivojRL.jpg

 

[MednataMiza]

Well, I am probably using wrong/incorrect formula.
Sticking to your notebook we get 26.8967 as an answer.
Can you verify that ?

 

[november1992]

I'm not getting a number.
http://i.imgur.com/mQ4Pfac.png

I'm guessing I have incorrect syntax.

Edit:

I realized i didn't capitalize the 'p' in Pi. [STRIKE]now I"m getting 13k though[/STRIKE] I got 20.39

 

[MednataMiza]

Well, using the simple

Sum[k[t],{t,0,4Pi}]

I, again, receive 26.8967.

 

[november1992]

I tried it 3 different ways and I still get 20.39

http://i.imgur.com/mQW6TaO.png
http://i.imgur.com/DmTseJc.png
http://i.imgur.com/QKfTfoW.png

 

[MednataMiza]

You should check the initial settings in r, because I see errors there.
For example a missing 't' in the first part :)
Check again the problem and repost what is actually given.

 

[november1992]

I can't believe I overlooked that. I also removed the parentheses. Now I'm getting 26.89. Thanks for the help

 

[MednataMiza]

You are welcome :)