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Homework Statement
>Problem:<br>Find the Moment of Inertia of a circular disk of uniform density about an axis which passes through the center and makes an angle of $\dfrac{\pi}{6}$ with the plane of the disc.
Homework Equations
Moment of Inertia ($I$) is $$\int r^2dm$$ where $r$ is the perpendicular distance from the chosen axis (and can vary) and $dm$ is an elemental mass.
The Attempt at a Solution
I interpreted the disk as a 2 Dimensional structure, hence it has mass per unit area ($\sigma$)
$$dm=\sigma dA$$
Thus, $$I=\int r^2dm=\int \int_A \sigma dA$$
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Using a change of variables to Cylindrical Coordinates,
$$dA=rdrd\theta$$
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$$\Longrightarrow I=\int\int \sigma r^3 dr d\theta$$
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However, this is the Integral for finding the Moment of Inertia for an axis perpendicular to the plane of the body. I cannot understand how to modify it for the given question.
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I would be very grateful if somebody could please solve this question without resorting to differential equations.
Many thanks in anticipation!
PS. This is not a homework question. I came across it while browsing the Web for problems on Moment of Inertia.
PPS. I don't know why the Tex code isn't working on top of the page. I'm really sorry for that.
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