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\begin{document}

\frame{\frametitle{\b Rotational Energy and Torques - 1} \bigskip

{\bf Three masses on equilateral triangle (1):} 
$$R=\frac{L}{2\cos(\pi/6)}\,,~~\cos(\pi/6)=?\,;~~~R=\frac{L}{2}\,.$$
\smallskip

{\bf Disk with Hole (2).}  Solid disk:
$$ I=\frac{1}{2}\,M\,R^2\,,~~~M=\pi\,R^2\,\rho\,. $$
Small disk:
$$I=\frac{1}{2}\,m\,r^2+m\,r^2\,,~~~m=\pi\,r^2\,\rho\,,~~~r=R/3\,$$
\smallskip % \bigskip

{\bf Torque on a Merry Go Round (3):} 
$$ I = I_{\rm disk}+I_{\rm kids}\,,~~~\tau = I\,\alpha\,,
   ~~~\alpha=\frac{\omega}{t}\,,~~~F=\frac{\tau}{R}\,.  $$
\smallskip % \bigskip

\vskip 20pt }


\frame{\frametitle{\b Rotational Energy and Torques - 2} \bigskip

{\bf Hoop Rolling up Hill (4):} 
$$ K=m\,g\,h\,,~~~K =(1+f_{\rm hoop})\,\frac{1}{2}\,m\,v^2\,.$$
\smallskip

{\bf Solid Sphere Rolling down the Ramp (5):} 
$$ K = \frac{1+f}{2}\,M\,v^2 = M\,g\,h\,,~~~h=d\,?\,,~~?=\,.$$
\smallskip

{\bf Mass pullling a Pulley (6):} 
$$ \triangle U = U-K~~{\rm with}~~U=M_b\,g\,D~~{\rm and}~~
   K=I_p\,\frac{\omega^2}{2}+M_b\,\frac{v^2}{2}\,. $$
\smallskip

{\bf Rolling around a Loop the Loop (7):} 
$$ v^2_{\rm min}=g\,R\,,~~~ M\,g\,h_{\rm min} = (1+f)\,M\,
   \frac{v^2_{\rm min}}{2} + M\,g\,2R\ \Rightarrow\ h_{\rm min}\,.$$
\smallskip

\vskip 20pt }

\end{document}
