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\begin{document}

\frame{\frametitle{\b Oscillations II - 1} \bigskip

{\bf Oscillations (1):} Consider
$$ x(t) = A\,\cos(\omega\,t+\pi) $$
and its derivatives with respect to time $t$.
\bigskip

{\bf Energy and Amplitude in simple Harmonic Motion (2):} 
$$ E_{\rm tot} = \frac{1}{2}\,M\,v^2_{\rm max}\,,~~~
   x_{\max} = \frac{v_{\max}}{\omega}\,. $$
\bigskip

{\bf Spring and Rolling Sphere (or Cylinder) (3).} Energy:
$$ U = \frac{1}{2}\,k\,x^2\,,~~~K=\frac{7}{10}\,M\,v^2
   ~~~\left({\rm or}~~K=\frac{3}{4}\,M\,v^2\right)\,.$$
Angular velocity:
$$ 0 = k\,x\,\dot{x}+\frac{7}{5}\,M\,\dot{x}\,\ddot{x}~~~
   \left({\rm or}~~\frac{3}{2}\,M\,\dot{x}\,\ddot{x}\right)\,.$$
\vskip 20pt }


\frame{\frametitle{\b Oscillations II - 2} \bigskip

{\bf Pendulum in an Accelerating Frame (4):} 
$$ T \sim \sqrt{\frac{1}{g\pm a}}\,. $$ 
\medskip

{\bf Pendulum in Rocket Ship (5):} \vskip -20pt
$$ L\,\ddot\theta = -m \,g\,\sin(\theta) \approx -
   m\,g\,\theta~\Rightarrow~\ddot\theta = -\omega^2\,\theta
   ~~{\rm with}~~\omega = \sqrt{\frac{m\,g}{L}}\,,
   ~T=\frac{2\pi}{\omega}\,.$$

{\bf Pendulum (6).} Equate the maximum kinetic with the maximum 
potential energy:
$$ K_{\max} = \frac{1}{2}\,M\,\dot{\phi}^2\,L^2 = U_{\max}
   = M\,g\,L\,[1-\cos(\phi_{\max})]\,.$$
CAPA seems to use the approximation $1-\cos(\phi_{\max})=
\phi_{\max}^2/2$.
\bigskip

{\bf Damped SHM (7).} With probability $p$, reduction
factor $r$:
$$ p = \exp[x\,\ln(r^2)]\,,~~~t=x\,T\,. $$
\vskip 20pt }
\end{document}
