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\begin{document}

\frame{\frametitle{\b Angular Momentum - 1} \bigskip

{\bf Angular Momentum and Torque (1):} 
$$ \vec{L}=\vec{r}\times M\,\vec{v}\,,~~~
   \vec{\tau}=\vec{r}\times\vec{F}\,.$$
% \smallskip

{\bf Rotating Chair (2).}  Angular momentum conservation:
$$ L = I_i\,\omega_i = I_f\,\omega_f\,. $$
Kinetic energy is not conserved:
$$ \triangle K = \frac{1}{2}\,I_f\,\omega_f^2 -
                 \frac{1}{2}\,I_i\,\omega_i^2\,.$$
% \smallskip % \bigskip

{\bf Rotating Nucleus (3).} Angular velocity:
$$ L = I\,\omega~~{\rm with}~~I=\frac{2}{5}\,M\,r^2\,. $$
The other questions:
$$ v=\omega\,r\,,~~K_{\rm rot}=\frac{1}{2}\,I\,\omega^2\,,
   ~~{\rm Ratio} = \frac{K_{\rm rot}}{M\,c^2}\,. $$
\smallskip % \bigskip

\vskip 20pt }


\frame{\frametitle{\b Angular Momentum - 2} \bigskip

{\bf Children on a Merry-Go-Round (4).} Rotational inertia:
$$ I = 2\,M_c\,R^2+\frac{M_b\,L^2}{12}$$
with $M_c$ the mass of one child, $M_b$ the mass of the board
and $R=L/2$. Angular momentum: $L=I\,\omega$. Angular momentum
conservation when the children are moving:
$$ L = L' = I'\,\omega'~~{\rm with}~~I'= 
   2\,M_c\,\left(\frac{R}{2}\right)^2+\frac{M_b\,L^2}{12}\,.$$
As in (2) change in kinetic energy: 
$$ \triangle K = \frac{1}{2}\,I'\,\omega'^2 -
                 \frac{1}{2}\,I\,\omega^2\,.$$
{\bf Putty on Turntable (5):} 1.~Angular momentum conservation:
$$ I_0\omega_0 = I_f\omega_f~~{\rm with}~~I_f=I_0+m\,R^2\,.$$
2.~Angular momentum carried away.
\vskip 20pt }


\frame{\frametitle{\b Angular Momentum - 3} \bigskip

{\bf Mass on Turntable (6).} Energy and angular momentum
are conserved:
$$ \frac{1}{2}\,k\,x^2 = \frac{1}{2}\,m\,v^2 + 
   \frac{1}{2}\,I\,\omega^2\,,~~~b\,m\,v=I\,\omega\,. $$
1.~Insert $\omega=b\,m\,v/I$ in the first equation and solve 
for $v$. 2.~Insert $v=I\,\omega/(b\,m)$ in the first equation
and solve for $\omega$.
\medskip

{\bf Time-Dependent Torque (7):} 
$$ L(t) = \int_0^t dt'\,\tau(t') \,.$$
\smallskip
\vskip 20pt }
\end{document}

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