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Basic integration, geometry

  1. Jun 12, 2010 #1
    Integration is not one of my strong points, so I just wanted to make sure I did it correctly.

    I am working with bullets and need to calculate volume, mass, angular mass, and centre of mass for use in my ballistics model. I have area, volume, and mass...but also need to figure out the solutions for angular mass and centre of mass.

    To keep it simple, let's use a straight cone.

    Definitions

    png.latex?y(x)&space;=&space;s_{y}&space;-&space;\frac{xs_{y}}{s_{x}}.png

    png.latex?\delta&space;y(x)&space;=&space;s_{y}x&space;-&space;\frac{x^{2}s_{y}}{2&space;s_{x}}.png

    [URL]http://latex.codecogs.com/png.latex?r&space;=&space;y(x)[/URL]

    [URL]http://latex.codecogs.com/png.latex?s&space;\leq&space;s_{x}[/URL]

    Integrals

    [URL]http://latex.codecogs.com/png.latex?\delta&space;A&space;=&space;\int{\delta&space;r}[/URL]

    [URL]http://latex.codecogs.com/png.latex?\delta&space;V&space;=&space;2&space;\pi&space;\int{\delta&space;A}[/URL]

    [URL]http://latex.codecogs.com/png.latex?\delta&space;m&space;=&space;\rho&space;\int{\delta&space;V}[/URL]

    Solutions

    [URL]http://latex.codecogs.com/png.latex?A&space;=&space;\int_{0}^{s}{\delta&space;r}&space;=&space;\left(&space;{s_{y}s&space;-&space;\frac{s^{2}s_{y}}{2&space;s_{x}}}&space;\right)[/URL]

    [URL]http://latex.codecogs.com/png.latex?V&space;=&space;2&space;\pi&space;\int_{0}^{s}{\delta&space;A}&space;=&space;2&space;\pi&space;\left(&space;s_{y}s&space;-&space;\frac{s^{2}s_{y}}{2&space;s_{x}}&space;\right)[/URL]

    [URL]http://latex.codecogs.com/png.latex?m&space;=&space;\rho&space;\int_{0}^{s}{\delta&space;V}&space;=&space;2&space;\pi\rho&space;\left(&space;s_{y}s&space;-&space;\frac{s^{2}s_{y}}{2&space;s_{x}}&space;\right)[/URL]

    I am a little uncertain about angular mass and centre of mass though...

    [URL]http://latex.codecogs.com/png.latex?\delta&space;m_{(\phi)}&space;=&space;\int{\delta&space;r^{2}\delta&space;m}&space;=&space;2&space;\pi\rho&space;\int{\delta&space;r^{3}}[/URL]

    [URL]http://latex.codecogs.com/png.latex?m_{(\phi)}&space;=&space;2&space;\pi\rho&space;\int_{0}^{s}&space;\delta&space;r^{3}&space;=&space;2&space;\pi\rho&space;\left(&space;s_{y}s&space;-&space;\frac{s^{2}s_{y}}{2&space;s_{x}}&space;\right)^{3}[/URL]

    [URL]http://latex.codecogs.com/png.latex?\delta&space;s_{(m)}&space;=&space;\frac{1}{m_{1}}&space;\int{\delta&space;r\delta&space;m}&space;=&space;\frac{2&space;\pi\rho}{m_{1}}&space;\int{\delta&space;r^{2}}[/URL]

    [URL]http://latex.codecogs.com/png.latex?s_{(m)}&space;=&space;\frac{2&space;\pi\rho}{m_{1}}&space;\int_{0}^{s}&space;\delta&space;r^{2}&space;=&space;\frac{2&space;\pi\rho}{m_{1}}&space;\left(&space;s_{y}s&space;-&space;\frac{s^{2}s_{y}}{2&space;s_{x}}&space;\right)^{2}[/URL]

    Is this correct, or am I completely out of my mind?
     
    Last edited by a moderator: Apr 25, 2017
  2. jcsd
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