find curvature r = (4/5 Cost, 1-sint, -3/2 Cost)
"r = [\\frac{4}{5} cos(t), (1-sin(t)), -\\frac{3}{2} cos(t)]"
Since f(t), g(t), and h(t) are the components of the vector r(t), then f, g, and h are real-valued functions called the component functions of r and we can write
"r(t)=\\frac{4}{5} cos(t) i + (1-sin(t)) j - \\frac{3}{2} cos(t) k"
"r'(t)=-\\frac{4}{5} sin(t) i - cos(t) j + \\frac{3}{2} sin(t) k"
"r''(t)=-\\frac{4}{5} cos(t) i + sin(t) j + \\frac{3}{2} cos(t) k"
So the curvature is
"\u03ba(t)=\\frac{\\vert r'(t)\u00d7 r''(t)\\vert}{{\\vert r'(t)\\vert}^{3}}"
"r'(t)\u00d7 r''(t)=\\begin{vmatrix}\n i & j & k \\\\\n-\\frac{4}{5} sint & -cost & \\frac{3}{2} sint\\\\\n-\\frac{4}{5} cost & sint & \\frac{3}{2} cost\n\\end{vmatrix}"
"r'(t)\u00d7 r''(t)=(-\\frac{3}{2}cos^{2}t -\\frac{3}{2}sin^{2}t)i -(-\\frac{12}{10}cost.sint +\\frac{12}{10}cost.sint)j +(-\\frac{4}{5}sin^{2}t -\\frac{4}{5}cos^{2}t)"
"r'(t)\u00d7 r''(t)=-\\frac{3}{2}(cos^{2}t+sin^{2}t)i -0j -\\frac{4}{5}(sin^{2}t +cos^{2}t)k"
"r'(t)\u00d7 r''(t)=-\\frac{3}{2}i +0j -\\frac{4}{5}k"
So Now
"\\vert r'(t)\u00d7 r''(t)\\vert ={[(-\\frac{3}{2})^2+(-\\frac{4}{5})^2]}^{1\/2}"
"={[\\tfrac{9}{4}+\\tfrac{16}{25}]}^{1\/2}" "={[\\frac{289}{100}]}^{1\/2}" "=\\frac{17}{10}"
"{\\vert r'(t)\\vert}^3={[(-\\frac{4}{5}sint)^2+(cost)^{2}+(-\\frac{3}{2}sint)^2]}^{3\/2}"
"{\\vert r'(t)\\vert}^3={[\\frac{16}{25}sin^2t+cos^2t+\\frac{9}{4}sin^2t]}^{3\/2}"
"{\\vert r'(t)\\vert}^3={[\\frac{189}{100}sin^2t+cos^2t]}^{3\/2}"
"{\\vert r'(t)\\vert}^3={[\\frac{189}{100}sin^2t+1-sin^2t]}^{3\/2}={[1+\\frac{189-100}{100}sin^2t]}^{3\/2}"
"{\\vert r'(t)\\vert}^3={[1+\\frac{89}{100}sin^2t]}^{3\/2}"
"\u03ba(t)=\\frac{\\vert r'(t)\u00d7 r''(t)\\vert}{{\\vert r'(t)\\vert}^{3}} =\\frac{\\frac{17}{10}}{{[\\frac{189}{100}sin^2t+cos^2t]}^{3\/2}}"
So the curvature is ...
Comments
Leave a comment