- #1
minimario
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Homework Statement
Find the moment of inertia of a spherical shell (hollow) with mass M and radius R.
Homework Equations
## I = \int r^2 dm ##
The Attempt at a Solution
This is method I use to find Moment of Inertia of solid sphere:
We use circular cross sections.
At some radius r, ##\frac{dm}{M} = \frac{\pi * r^2 dz}{4/3 \pi R^3} \Rightarrow dm = \frac{3Mr^2}{4R^3} dz##.
Therefore, the total moment of inertia is ## \int \frac{1}{2} (dm) r^2 = \frac{3M}{8R^3} \int r^4 dz ##. We use ## r = R^2 - z^2 ##, and integral is ##\int (R^2-z^2)^2 dz ##, and evaluated from -R to R, it is ##\frac{16}{15} R^5 ##. So the total integral is ##\frac{2}{5} MR^2##.
But with spherical shell, something is different. ##\frac{dm}{M} = \frac{2 \pi * r dz}{4 \pi R^2} \Rightarrow dm = \frac{Mr}{2R^2} dz ##. (Again, we use circles).
But when taking this integral, the answer is incorrect. So is there something wrong with my expression for dm?