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

concepts/mechanics/u6/m6-4-rolling.tex

@@ -18,6 +18,8 @@ These relations are called the \emph{rolling constraint}. They apply only when t
 
 Also, the friction in rolling without slipping is \emph{static} friction. Let $\vec{N}$ denote the normal force, let $N=|\vec{N}|$ denote its magnitude, and let $f_s$ denote the magnitude of the static friction force. Static friction is not automatically equal to $\mu_s N$. Instead, its magnitude is whatever value is required to prevent slipping, provided that value satisfies $f_s\le \mu_s N$. On level ground at constant speed, the needed static friction can even be zero.}
 
+\wc{Static friction does not always oppose motion}{Static friction opposes \emph{impending relative motion at the contact point}, not the motion of the object as a whole. In rolling without slipping down an incline, the object moves downward but static friction points \emph{up} the incline, because without friction the contact point would slide downward. The friction prevents that slip by pulling the contact backward.}
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 \ex{Illustrative example}{A wheel of radius $R=0.30\,\mathrm{m}$ rolls without slipping on level ground with angular speed $\omega=8.0\,\mathrm{rad/s}$. Find the speed of its center of mass and the speed of the top point of the wheel relative to the ground.
 
 From the rolling constraint,
@@ -43,6 +45,8 @@ For an object accelerating down the incline, the static friction force on the ob
 
 \nt{When several objects roll without slipping down the same incline from the same height, the one with the smallest $I_{\mathrm{cm}}/(MR^2)$ ratio reaches the bottom first. Race order: solid sphere ($\tfrac25=0.40$) $>$ solid cylinder ($\tfrac12=0.50$) $>$ hollow cylinder ($\simeq1$) $>$ thin hoop ($1.0$). Mass and radius do not appear in the comparison --- only the shape-dependent coefficient matters.}
 
+\wc{Rolling race order depends on moment of inertia}{Objects with a \emph{smaller} $I/(MR^2)$ ratio accelerate faster and reach the bottom first. For example: solid sphere ($\tfrac25$) beats solid cylinder ($\tfrac12$) beats thin hoop ($1$). The mass and radius cancel out --- only the \emph{shape distribution} matters. More mass near the rim means a larger fraction of energy goes into rotation, leaving less for translation.}
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 \qs{Worked example}{A solid cylinder of mass $M=2.0\,\mathrm{kg}$ and radius $R=0.20\,\mathrm{m}$ is released from rest on an incline that makes an angle $\beta=30^\circ$ with the horizontal. The cylinder rolls without slipping through a vertical drop $h=0.75\,\mathrm{m}$. Let $g=9.8\,\mathrm{m/s^2}$.
 
 Find: