- #1
fluidistic
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I've been trying to calculate the moment of inertia (with respect to the center of mass) of several rigid bodies (including 2 dimensional ones) but I never reached any good answer.
For example a cube whose edges are equal to [tex]a[/tex].
My work : From wikipedia and at least 2 physics books, [tex]I_{CM}=\int r^2 dm[/tex]. From my notes it's equal to [tex]\int _{\Omega} \rho \zeta ^2 dV[/tex].
My alone work : [tex]dV=dxdydz[/tex] and as [tex]\rho[/tex] (the density) is constant I can write [tex]I_{CM}=\rho \int_{-\frac{a}{2}}^{\frac{a}{2}} x^2 dx \cdot \int_{-\frac{a}{2}}^{\frac{a}{2}} y^2 dy \cdot \int_{-\frac{a}{2}}^{\frac{a}{2}}z^2dz[/tex]. Calculating this I get that [tex]I_{CM}=\rho \left( \frac{a^3}{12^3} \right)=\frac{M}{V}\cdot \frac{V}{12^3}=\frac{M}{12^3}[/tex].
I never took calculus III yet so I never dealt with triple integrals and even double ones. I'm guessing that I'm multiplying the 3 integrals and that instead I should be adding them or something like that, but I don't know why at all. I'd be glad if you could help me.
P.S. : I considered the origin of the system as being the center of mass of the cube.
For example a cube whose edges are equal to [tex]a[/tex].
My work : From wikipedia and at least 2 physics books, [tex]I_{CM}=\int r^2 dm[/tex]. From my notes it's equal to [tex]\int _{\Omega} \rho \zeta ^2 dV[/tex].
My alone work : [tex]dV=dxdydz[/tex] and as [tex]\rho[/tex] (the density) is constant I can write [tex]I_{CM}=\rho \int_{-\frac{a}{2}}^{\frac{a}{2}} x^2 dx \cdot \int_{-\frac{a}{2}}^{\frac{a}{2}} y^2 dy \cdot \int_{-\frac{a}{2}}^{\frac{a}{2}}z^2dz[/tex]. Calculating this I get that [tex]I_{CM}=\rho \left( \frac{a^3}{12^3} \right)=\frac{M}{V}\cdot \frac{V}{12^3}=\frac{M}{12^3}[/tex].
I never took calculus III yet so I never dealt with triple integrals and even double ones. I'm guessing that I'm multiplying the 3 integrals and that instead I should be adding them or something like that, but I don't know why at all. I'd be glad if you could help me.
P.S. : I considered the origin of the system as being the center of mass of the cube.