- #1

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- Homework Statement
- Find the moment of inertia of a uniform square plate of mass M and side L rotating around an axis through its diagonal.

- Relevant Equations
- ##I=\int r^2 dm##

I placed my Oxy coordinate system at the center of the square, the ##x##-axis pointing rightwards and the ##y##-axis pointing upwards.

I divided the square into thin vertical strips, each of height ##h=2(\frac{L}{\sqrt{2}}-x)##, base ##dx## and mass ##dm=\sigma h dx=\frac{M}{L^2}2(\frac{L}{\sqrt{2}}-x)dx##, so ##I=2\int_{x=0}^{x=\frac{L}{\sqrt{2}}}x^2 \frac{M}{L^2}2(\frac{L}{\sqrt{2}}-x)dx=\frac{5}{12}ML^2##.

However, the solution, according to the exercise, should be ##\frac{1}{12}ML^2## but I haven't been able to see what I am doing wrong so I would appreciate if someone would help me see where my mistake is. Thanks.

I divided the square into thin vertical strips, each of height ##h=2(\frac{L}{\sqrt{2}}-x)##, base ##dx## and mass ##dm=\sigma h dx=\frac{M}{L^2}2(\frac{L}{\sqrt{2}}-x)dx##, so ##I=2\int_{x=0}^{x=\frac{L}{\sqrt{2}}}x^2 \frac{M}{L^2}2(\frac{L}{\sqrt{2}}-x)dx=\frac{5}{12}ML^2##.

However, the solution, according to the exercise, should be ##\frac{1}{12}ML^2## but I haven't been able to see what I am doing wrong so I would appreciate if someone would help me see where my mistake is. Thanks.