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\begin{document}

\frame{\frametitle{\b Work and Energy III-1} \bigskip

{\bf Rock Climber (1):} Energy conservation.
$$ E = m\,g\,h= \frac{k}{2}\,x^2\,,~~~~h=???\ne L{\rm !}\,.$$
% \bigskip

{\bf Loop the Loop (2):} \bigskip

$E_A=$ Difference of the potential energies.
\bigskip

Downward acceleration:
$$ a = \frac{v^2}{R}\,,~~~E_A=\frac{m}{2}\,v^2\,.$$
Minimum height:
$$ h_{\min}\,g\,M = E_{\rm potential}(A) + E_{\rm kinetic}(A)
   \,~~{\rm with}~~g=\frac{v^2}{R}\,. $$
% \bigskip
\vskip 20pt }

\frame{\frametitle{\b Work and Energy III-2} \bigskip

{\bf Ball on a Swing (3):}  \bigskip

1.\ Find $v_x$ from kinetic energy, then $R=v\,t$ with $t$ from $g\,t^2/2=h$.

2.\ $v=\sqrt{v_x^2+v_y^2}$.
\bigskip

{\bf Block and Loop the Loop (4):} \bigskip

$W=E_{\rm kinetic}-E_{\rm potential}$ with $E_{\rm kinetic}>0$.
\bigskip

{\bf Swinging Ball (5):} \bigskip

1.\ $v$ from kinetic energy. 2.\ $v$ from new kinetic energy at $2r$.
\bigskip

{\bf Trunk up an Incline (6):} % \bigskip

$$ P_I=P\,\cos(\theta)\,,~~W=P_I\,d\,,~~F_N=P\,sin(\theta)+
   m\,g\,\cos(\theta)\,,$$
$$ W_{\mu}=-F_{\mu}\,d\,,~~W_g = \dots\ .$$

\vskip 20pt }

\end{document}
