I have a an assignment and I'm stuck on this question:
First of all I can't figure out the equation for a $2$-dimensional donut as shown in the diagram. For the calculation of the center of mass, I have tried polar coordinates and took the limits of $r$ to be from $1$ to $2$ and $\theta$ from $0$ to $2\pi$. Is it right? What should I do? plz help!!
1 Answer
As far as I can understand, the question implies that:
For uniform areal density the center of gravity is at center of inner circle.
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Suggest a function for areal density such that the unbalanced mass center of gravity is forced out of the inner circle.
Try out some odd functions like:
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$$ \rho (y) = a + b y , a > b , \rho(y) = c + d y^3, c>0, d>0. $$
EDIT1
$$ y \_{CofM} = \frac{\int y \, dx dy }{\int dx dy } = \frac{\int r \sin \theta \, r dr d\theta}{\int r dr d\theta }. $$
Can take forward?