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Thread: Find the Curvature

  1. #1
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    Find the Curvature

    Let k = curvature.

    Find k given r (t) = t i + t^2 j + (1/2)t^2 k

    I found ||T'(t)|| to be (5t + 3)/(5t^2 + 1)^(3/2).
    Is this correct?

    I found ||r'(t)|| to be sqrt {5t^2 + 1}. Is this correct?

    I must find k = ||T'(t)||/||r'(t)||.

    The answer is (sqrt {5})/(1 + 5t^2)^(3/2).

    Can someone set up the correct fraction to find k?
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  2. #2
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    Re: Find the Curvature

    Quote Originally Posted by USNAVY View Post
    Let k = curvature.
    Find k given r (t) = t i + t^2 j + (1/2)t^2 k
    Can someone set up the correct fraction to find k?
    \kappa (t) = \frac{{\left\| {R'(t) \times R''(t)} \right\|}}{{{{\left\| {R'(t)} \right\|}^3}}}.
    I think that I gave you this before. I have always advised students that if looking for say $\kappa(t_0)$
    first evaluate $R'(t_0)~\&~R''(t_0)$ then use the kappa formula. That makes the calculations easier.
    Thanks from topsquark and USNAVY
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  3. #3
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    Re: Find the Curvature

    Quote Originally Posted by Plato View Post
    \kappa (t) = \frac{{\left\| {R'(t) \times R''(t)} \right\|}}{{{{\left\| {R'(t)} \right\|}^3}}}.
    I think that I gave you this before. I have always advised students that if looking for say $\kappa(t_0)$
    first evaluate $R'(t_0)~\&~R''(t_0)$ then use the kappa formula. That makes the calculations easier.
    The kappa formula you suggested works best for polynomials or so I've read. But, what about for trigonometric functions mixed with e^t?
    Last edited by topsquark; Jan 9th 2017 at 09:31 AM.
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  4. #4
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    Re: Find the Curvature

    Quote Originally Posted by Plato View Post
    \kappa (t) = \frac{{\left\| {R'(t) \times R''(t)} \right\|}}{{{{\left\| {R'(t)} \right\|}^3}}}.
    I think that I gave you this before. I have always advised students that if looking for say $\kappa(t_0)$
    first evaluate $R'(t_0)~\&~R''(t_0)$ then use the kappa formula. That makes the calculations easier.
    I did apply the formula you suggested and found the answer easily
    Last edited by topsquark; Jan 9th 2017 at 09:32 AM.
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