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\begin{document}

\frame{\frametitle{\b Gravity-1} \bigskip

{\bf Orbiting Planet (1):} 
$$ a=\frac{M\,G}{R^2}=\frac{v^2}{R}\,,~~~T=\frac{2\pi R}{v}\,.$$
\bigskip

{\bf Orbiting Satellite (2).} Kepler's 3rd law: 
$$ \frac{M\,G}{4\pi^2}\,T^2 = R^3\,,~~~v^2=\frac{M\,G}{R}\,.$$
\bigskip

{\bf Gravitational Acceleration (3):} 
$$ a=\frac{G\,M}{R^2}\,,~~~R_{\rm Earth}=6.37\times 10^6\,{\rm m}\,.$$
\bigskip
\vskip 20pt }


\frame{\frametitle{\b Gravity-2} \bigskip

{\bf Escape Velocity (4).}  \bigskip

Energy conservation:
$$ \frac{1}{2}\,m\,v_f^2 = \frac{1}{2}\,m\,v_i^2 - \frac{m\,M\,G}{R}\,.$$
% \bigskip

{\bf Two masses (5):} 
$$ a_1=\frac{M_2\,G}{d^2}\,,~~~\triangle t =\ ?\,,~~~v_f =\ ?\,.$$
% \bigskip

{\bf Tidal Force (6):}
$$ \triangle F = \frac{m\,Mm\,G}{(r-R)^2} - \frac{m\,Mm\,G}{(r+R)^2} 
   =\ ?~~{\rm (expand)}\,.$$
Taylor expansion:
$$ f(x)=f(0)+f'(0)\,x + \dots\,\,.~~~{\rm Let}~~
   f(x)=\frac{1}{(r\pm x)^2}\,,~~x=R\,.$$
\vskip 20pt }


\frame{\frametitle{\b Gravity-3} \bigskip

Denote the distance of one of the star from the center of the
triangle by $R$, 
$$ \frac{L}{2} = R\,\cos(30^0)\,, $$
and calculate the force towards the center. Then $a=F/M$, 
$a=v^2/R$, and $T=2\pi\,R/v$.
\vskip 120pt }

\end{document}
