16.8 + fix some formatting
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2025-04-23 16:20:39 -04:00
parent ac3c984f18
commit 3faf1ab200
4 changed files with 35 additions and 7 deletions

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@@ -31,7 +31,7 @@
This leaves the z direction. To find that, we first need to find the first moment in z: `int_{-2}^{2} int_{-sqrt(4 - x^2)}^{sqrt(4 - x^2)} int_{0}^{4-x^2-y^2} 2z dz dy dx`.
I categorically refuse to do this in Cartesian. For those who are interested, the integral in cylindrical coordinates is just `int_{0}^{2pi} int_{0}^{2} int_{0}^{4-r^2} 2zr dz dr d theta`,
which is a lot easier to evaluate. We'll see more discussion of this problem in the post on 15.7, but for now, just believe me when I say that it evaluates to 67.0206. Dividing
`frac {67.0206} {50.2655}` gives us a fairly nice number: `frac {4} {3}`. So our center-of-mass is `[0, 0, frac {4} {3}]`. Nice!
`frac {67.0206} {50.2655}` gives us a fairly nice number: `frac {4} {3}`. So our center-of-mass is `\[0, 0, frac {4} {3}]`. Nice!
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<p>
The next thing to talk about is moment of inertia. This is actually a physics concept, but it's an interesting illustration of another property of triple integrals.