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I've derived equations for a 2D polygon's moment of inertia using Green's Theorem (constant density $\rho$)

$I_y = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + x_i x_{i+1} + x_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$

$I_x = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( y_i^2 + y_i y_{i+1} + y_{i+1}^2 ) ( x_{i+1} y_i - x_i y_{i+1} )$

And I'm trying to add them up for calculating $I_0 = I_x + I_y$.

$I_0 = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 - y_i^2 + x_i x_{i+1} - y_i y_{i+1} + x_{i+1}^2 - y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$

But I found a different(?) equation for $I_0$ on the internet. and many people say the equation given below is correct.

$I_0 = \frac{\rho}{6} \frac{ \sum_{i=0}^{i=N-1} ( x_i^2 + y_i^2 + x_i x_{i+1} + y_i y_{i+1} + x_{i+1}^2 + y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i ) }{ \sum_{i=0}^{i=N-1} ( x_i y_{i+1} - x_{i+1} y_i ) }$

So I'm confused now. I think my equations for $I_x$ and $I_y$ are correct. But how am I gonna calculate $I_0$ (moment of inertia with respect to origin axis)? I couldn't prove both equations are equal.

Could you help me out please ?

(This post has been cross-posted at MathOverflow)

  • 1
    It's good style to tell people that you're cross-posting (http://mathoverflow.net/questions/73556/calculating-moment-of-inertia-in-2d-planar-polygon); else efforts will be unnecessarily duplicated.2011-08-24

2 Answers 2

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Sorry for my mistake. both equations was slightly incorrect. Let me write correct equations

$I_y = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + x_i x_{i+1} + x_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$

$I_x = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( y_i^2 + y_i y_{i+1} + y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$

$I_0 = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + y_i^2 + x_i x_{i+1} + y_i y_{i+1} + x_{i+1}^2 + y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$

and

$I_0 = \frac{m}{6} \frac{ \sum_{i=0}^{i=N-1} ( x_i^2 + y_i^2 + x_i x_{i+1} + y_i y_{i+1} + x_{i+1}^2 + y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i ) }{ \sum_{i=0}^{i=N-1} ( x_i y_{i+1} - x_{i+1} y_i ) }$

Note that latter equation changed mass density term($\rho$) to mass(m).

Both equations are equal.

2

Your moments don't pass two straightforward tests: They should be invariant under reversal of the vertex order (instead they change sign); and they should be quadratic under scaling (instead they scale with the fourth power). The expression you quote from the net passes both tests, so there's a good chance it's correct.

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    I know this is an old post but second moments of area should scale with the fourth power (and not quadratic). For example, by definition, $I_x=\iint y^2 dx dy$.2017-01-23