Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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)

share|improve this question
1  
It's good style to tell people that you're cross-posting (mathoverflow.net/questions/73556/…); else efforts will be unnecessarily duplicated. –  joriki Aug 24 '11 at 15:22
    
Thanks for pointing out that. I've fix my post. –  juhl Aug 24 '11 at 15:58

2 Answers 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.

share|improve this answer
up vote 1 down vote accepted

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.