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Math Help - Calculating Normal Acceleration

  1. #1
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    Calculating Normal Acceleration

    Hi guys,

    I have a 3D curve that is defined by a parameterization in the form of x(t), y(t), and z(t). In the context of the problem the curve is a waterslide and I need to calculate the normal acceleration at various points along the curve. I can calculate the velocity at any point by a simple conservation of energy from the top of the slide to the point of interest, but what do I need to do to calculate normal acceleration?
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  2. #2
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    You have x(t), y(t), z(t).

    Velocity vector v = (dx/dt,dy/dt,dz/dt).
    Acceleration vector a=dv/dt = (\frac{d^2x}{dt^2},\frac{d^2y}{dt^2},\frac{d^2z}{d  t^2}).
    v direction tangential to the curve.
    Find vector \tau = v/|v| |n|=1 tangential to the curve.
    Tangential acceleration
    a_t=a\tau
    Normal acceleration
    {a_n}^2=a^2-{a_t}^2.
    Last edited by zzzoak; April 13th 2010 at 01:13 PM.
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  3. #3
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    Thank you for helping, but I must admit that I do not totally get it yet. Perhaps someone could help me with a simple example and then I will be able to apply it to my more complicated problem.

    Imagine that my curve is defined by the simple parametrization:

    x(t)=sin(t)
    y(t)=cos(t)
    z(t)=.01t

    I am interested in the point where t=90 degrees, which represents one specific point on the curve. Based on the conservation of energy and the height at that point, I know that the rider will be traveling at 10m/s. How do I calculate his normal acceleration?
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  4. #4
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    r(t) = ( \sin t, \cos t, 0.01t )

    r'(t) = \frac{dr(t)}{dt} = ( \cos t, -\sin t, 0.01 )

    a(t) = r''(t) = \frac{d^2r(t)}{dt^2} = ( -\sin t, -\cos t, 0 )

    T = \frac{r'(t)}{|r'(t)|} - unit vector tangential to the curve

    N = \frac{T'}{|T'|} - unit vector normal to the curve

    a(t) = Ta_T + Na_N where

    a_T = T\cdot a(t) = \frac{|r'(t) \cdot a(t)|}{|r'(t)|}

    |r'(t)| = \sqrt{1.01} .

    r'(t) \cdot a(t) = ( \cos t, -\sin t, 0.01 )( -\sin t, -\cos t, 0 ) = 0 there is no tangetial component only normal, thus

    a(t) = Na_N = ( -\sin t, -\cos t, 0 )\: and\: |a(t)| = 1 .
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