content(warnings): Add W3-W5, N2-N3 — drag, work, and rolling warnings + notes

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2026-05-04 23:31:32 -05:00
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@@ -39,6 +39,8 @@ W=F\,\Delta s.
\nt{Work is positive when the force has a component in the same direction as the displacement, negative when that component is opposite the displacement, and zero when the force is perpendicular to the displacement. For a general force, the value of $W=\int_C \vec{F}\cdot d\vec{r}$ can depend on the path $C$, not just on the endpoints. In AP problems, this often appears when the force changes with position or when different paths make the parallel component $F_{\parallel}$ different. Typical zero-work cases include a normal force on motion along a surface or a centripetal force in uniform circular motion, because those forces are perpendicular to the instantaneous displacement.}
\wc{Holding something still does no work}{If you hold a heavy object stationary, you expend biological energy but do \emph{zero mechanical work} because $\Delta\vec{r}=\vec{0}$. Work requires both a force \emph{and} a displacement along that force: $W=\int\vec{F}\cdot d\vec{r}$.}
\pf{Why the line integral gives total work}{Break the path into many small displacement vectors $\Delta \vec{r}_1,\Delta \vec{r}_2,\dots,\Delta \vec{r}_n$. Over each small segment, the work is approximately
\[
\Delta W_k\approx \vec{F}_k\cdot \Delta \vec{r}_k.