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Finding the curvature of a space curve

  1. Oct 18, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the curvature of the car's path, K(t)
    Car's Path: [itex] r(t) = \Big< 40cos( \frac {2 \pi}{16}t ) , 40sin( \frac {2 \pi}{16}t ), \frac{20}{16}t \Big> [/itex]

    2. Relevant equations
    [itex] K(t) = \frac { |r'(t)\:X \:r''(t)|}{|r'(t)|^3 } [/itex]

    3. The attempt at a solution
    This is part of a massive 6 part question, a,b,c,d,e,f. This is part e. I already have r'(t) and r''(t) as well as r'(t) x r''(t). I'm just getting totally lost in the algebra, and I don't know if I am on the right track:
    [itex] r'(t) = \Big< -5 \pi sin( \frac{2 \pi}{16}t) , 5 \pi cos( \frac{2 \pi}{16}t) , \frac{20}{16} \Big> [/itex]
    [itex] r''(t) = \Big< \frac{ -5 \pi ^2 cos( \frac{2 \pi}{16}t )}{8} , \frac{ -5 \pi ^2 sin( \frac{2 \pi}{16}t )}{8} , 0 \Big>[/itex]
    [itex] r'(t) \: X \: r''(t) = \big< \frac{ -125 \pi sin( \frac{2 \pi}{16}t )}{32} , \frac{ -25 \pi ^2 cos( \frac{2 \pi}{16}t )}{32} , \Big[ \frac{25 \pi ^3}{8} \Big]\Big[cos(\frac{2 \pi}{16}t) \Big]^2 + \frac{125 \pi ^2}{8} \Big> [/itex]

    So I know the next step will be to get the magnitude of this. Does this look right so far? I have to admit, this is probably the hardest problem (algebra-wise) I have ever done in college so far.
     
  2. jcsd
  3. Oct 18, 2016 #2

    LCKurtz

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    I didn't check your work, but I can tell you that I certainly would have started with$$
    \vec r(t) = \langle a\cos(bt),a\sin(bt),ct\rangle$$and put the constants in at the end.
     
  4. Oct 18, 2016 #3
    Well sure, I'm positive I have done the derivatives right. I was told by someone that the 't' will cancel somewhere, and the answer will simply be a number. I just can't see how 't' will cancel. I was also told it was related to the identity sin^2(t) + cos^2(t) = 1, but when I did take the magnitude of this it was horrendous.
     
  5. Oct 18, 2016 #4

    LCKurtz

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    I will just repeat my advice: do it again with ##a,~b,~c##. It will cut down mistakes and make it apparent where to use the sine-cosine identity. It isn't "horrendous".
     
  6. Oct 19, 2016 #5
    Okay I took your advice, which helped a ton. I'm just hoping I didn't make any stupid arithmetic errors. When I crossed r'(t) X r''(t) I got:
    [itex] \big< ab^2c\:sin(bt), -ab^2c\:cos(bt),a^2b^3 \big> [/itex]
    I took the magnitude and got:
    [itex] ab^2\sqrt{a^b+b^2+1} [/itex]
    I then took the magnitude of r'(t) cubed and got:
    [itex] (a^2 + b^2 + c^2)^{3/2}[/itex]
    I plugged everything back in for a,b and c and got 0.01. Seems kind of low, I'm not sure if i have made some kind of arithmetic error
     
  7. Oct 19, 2016 #6

    LCKurtz

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    I agree with the cross product, but check that last line. I get a different value.
     
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