created at April 25, 2021

Rectangle

Area:

A=a\cdot b

L=2\cdot (a + b)

I_{x}=\frac{b\cdot a^{3}}{12};\quad I_{y}=\frac{a\cdot b^{3}}{12};\quad I_{xy}=0;\quad I_{\rho}=I_{x} + I_{y}

W_{x}=\frac{b\cdot a^{2}}{6};\quad W_{y}=\frac{a\cdot b^{2}}{6};

i_{x}=\frac{a}{2\cdot \sqrt{3}};\quad i_{y}=\frac{b}{2\cdot \sqrt{3}};

Consider axis Ox1y1, rotated relative to Oxy by angle α (see figure above). Then moments of inertia relative to these rotated axis can be found as:

I_{x1}=I_{x}\cdot cos^{2}(\alpha) + I_{y}\cdot sin^{2}(\alpha) - I_{xy}\cdot sin(2\cdot \alpha)

I_{y1}=I_{x}\cdot sin^{2}(\alpha) + I_{y}\cdot cos^{2}(\alpha) - I_{xy}\cdot cos(2\cdot \alpha)

I_{x1y1}=\frac{I_{x} - I_{y}}{2}\cdot sin(2\cdot \alpha) + I_{xy}\cdot cos(2\cdot \alpha)