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Homework Statement
A thin square (4 ft side) metal sheet of homogeneous density ([tex]\sigma = M/A[/tex]is rotating around one of its diagonals at 10 rev/s. Develop a definite integral to express its kinetic energy.
Homework Equations
[tex]dK = \frac{1}{2}(r\omega)^{2}\sigma dA[/tex]
The Attempt at a Solution
I am using one side of the sheet, and plotting it as the area enclosed between:
[tex]y_{1}=x[/tex]
[tex]y_{2}=-x + 4\sqrt{2}[/tex]
[tex]0\leq x \leq 2\sqrt{2}[/tex]
Then:
[tex]v^{2}=(20\pi x)^{2}[/tex]
and my integral will be:
[tex]200 \pi^{2}\sigma\int_{0}^{2\sqrt{2}} x^{2}(-x + 4\sqrt{2}-x) \text{d}x[/tex]
This is half the total kinetic energy, by symmetry, so double the above should be the total.
Is this correct?
Thanks!