- #1
dlacombe13
- 100
- 3
Homework Statement
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]
Homework Equations
[itex] K(t) = \frac { |r'(t)\:X \:r''(t)|}{|r'(t)|^3 } [/itex]
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.