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\begin{document}
\header{8.012}{7}
\problem{1}
\[ \vec{L} = \vec{r} \times \vec{p} \]
\[ {\vec{L}}_{z} = m(xV_y - yV_x)  \]

\part{a}

\[ x = y = 0 \hspace{1 in}  L = 0 \]
\[ V_x = V_y = \frac{V_0\sqrt{2}}{2} \]

\part{b}
\[ t = \frac{v_0\sqrt{2}}{2g} \hspace{.5 in} y = V_{y_{0}}\frac{t}{2} = \frac{V_{0}^{2}}{4g} \]
\hspace{2 in}$ x = v_{x_{0}}t $
\[x = \frac{V^{2}_{0}}{2g} \hspace{.25 in} y = \frac{V^{2}_{0}}{4g}  \]
\[V_x = \frac{V_0\sqrt{2}}{2} \hspace{.25 in} V_y = 0 \]
\[ L = -\frac{mV^{3}_{0}\sqrt{2}}{8g} \]
\vspace{-.25 in}
\part{c}
\[ x = \frac{V^{2}_{0}}{g} \hspace{.25 in} y = 0 \]
\[ V_x = \frac{V_0\sqrt{2}}{2} \hspace{.25 in} V_y = \frac{V_0\sqrt{2}}{2} \]
\[ L = \frac{mV^{3}_{0}\sqrt{2}}{2g} \]


\problem{2}
\part{a}
\[ r = R\cos\phi  \hspace{.25 in} dS = r dr d \phi \hspace{.25 in} I = \int r^2 \sigma dS \]
\[ = \int^{2\pi}_{0}\int^{2\pi}_{0} \frac{R^2(\cos^2\phi)Mr}{{\pi}R^{2}} dr d\phi \hspace{.25 in} = \left.\int^{2\pi}_{0}\frac{(\cos^{2}\phi)Mr^{2}}{2{\pi}R^2}\right|^{R}_{0} d\phi \]
\[ = \frac{M}{2\pi} \int^{2\pi}_{0}(\cos^{2}\phi) d\phi \hspace{.25 in} = \frac{MR^{2}}{4\pi}\left(\left.\frac{\sin2\phi}{2} + \phi\right)\right|^{2\pi}_{0} \hspace{.25 in} = \frac{MR^2}{2} \]

\part{b}

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