Section 5.6 Calculating Centers of Mass and Moments of Inertia (Omitted)
References.
- Section 5.6 of OpenStax Calculus Volume 3.
1
openstax.org/books/calculus-volume-3/pages/5-6-calculating-centers-of-mass-and-moments-of-inertia - Section 15.6. of Calculus, Early Transcendentals by Stewart.
Physics: Mass and Center of Mass.
Two related physical quantities are the mass and center of mass of an object occupying the region \(E\) with density \(\rho(x,y,z)\text{.}\) The mass is
\begin{equation*}
m = \iiint\limits_E \rho(x,y,z) \, dV
\end{equation*}
The center of mass is the point \((\bar{x},\bar{y},\bar{z})\) which is the "density weighted average" location of the mass of a body, with components given by
\begin{equation*}
\bar{x} = \frac{\iiint_E x \rho(x,y,z) \, dV}{m} = \frac{\iiint_E x \rho(x,y,z) \, dV}{\iiint_E \rho(x,y,z) \, dV},
\quad
\bar{y} = \frac{\iiint_E y \rho(x,y,z) \ dV}{m},
\quad
\bar{z} = \frac{\iiint_E z \rho(x,y,z) \, dV}{m}.
\end{equation*}
Physics: Moments of Inertia.
Coming later.
