content(warnings): Add W12-W16, N5-N6 — rolling, pendulum, and inertia enhancements

concepts/mechanics/u3/m3-3-potential-energy.tex

@@ -17,6 +17,8 @@ Thus the work done by the conservative force is
 W_c=-\Delta U.
 \]}
 
+\nt{The formula $U=mgh$ is a special case valid only near Earth's surface where $g$ is approximately constant. The general gravitational potential energy between two masses $M$ and $m$ separated by distance $r$ is $U=-GMm/r$. When $h\ll R_{\oplus}$, expanding $-GMm/(R_{\oplus}+h)$ to first order in $h/R_{\oplus}$ gives $U\approx- GMm/R_{\oplus} + (GMm/R_{\oplus}^2)h = \text{const} + mgh$, since $g=GM_{\oplus}/R_{\oplus}^2$. Setting $U=0$ at ground level drops the constant term.}
+
 \wc{Potential energy belongs to a system}{Gravitational $U=mgh$ requires both the object \emph{and} the Earth. Spring $U=\tfrac12 kx^2$ requires both the block \emph{and} the spring. An isolated single object cannot have potential energy --- it is stored in the \emph{interaction}.}
 
 \thm{Equivalent conservative-force relations}{Let $\vec{F}_c$ denote a conservative force and let $d\vec{r}$ denote an infinitesimal displacement. Then the following relations hold: