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{\bf 29:61 General Astronomy\\
     Fall 2004\\
     Lecture 9 \ldots September 27,2004\\
     Vector Cross Products, Newton's Laws, the Gravitational Force}\\

{\em Just the facts, Ma'am}
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\noindent
$\bullet$ The definition of a vector cross product: 
The {\em magnitude} of a cross product is defined as follows.  If 
\begin{equation}
\vec{C} = \vec{A} \times \vec{B}
\end{equation}
then the {\em direction} of $\vec{A} \times \vec{B}$ is given by the right hand rule.  
The {\em magnitude} of $\vec{A} \times \vec{B}$ is given by 
\begin{equation}
|C| = |A| |B| \sin \theta
\end{equation}
where $\theta$ is the angle between the two vectors.  Another way of expressing it as follows.  Draw up an array 
\begin{equation}
\left[ 
\begin{array}{ccc}
\hat{e}_x & \hat{e}_y  & \hat{e}_z  \\
A_x  &  A_y  &  A_z  \\
B_x  &  B_y  &  B_z  
\end{array}
\right]
\end{equation}
For each component, knock out the column corresponding to that coordinate, and form the product of the remaining terms.  The result for the vector is
\begin{equation}
\vec{C} = (A_y B_z - B_y A_z) \hat{e}_x + (A_z B_x - B_z A_x) \hat{e}_y + (A_x B_y - B_x A_y) \hat{e}_z
\end{equation}

\noindent
$\bullet$ Remember that in a vector equation,  the equation must be satisfied {\em component by component}, in other words, you must satisfy the equation for the $x$ component of the vector, then the $y$ component, etc.  

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{\bf Newton's Laws of Motion}
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\begin{enumerate}
\item An object in motion remains in motion with constant vector momentum $\vec{p}$, unless acted upon by an external force. An object at rest has zero momentum, and therefore remains at rest.  
\item If a force acts on an object, its momentum changes according to 
\begin{equation}
\vec{F} = \frac{\Delta \vec{p}}{\Delta t} = \frac{d \vec{p}}{dt}
\end{equation}
If the mass of the object acted upon stays constant, this simplifies to 
\begin{equation}
\vec{F} = m\frac{\Delta \vec{v}}{\Delta t} = m\frac{d \vec{v}}{dt} = m\vec{a}
\end{equation}
where $\vec{a}$ is the {\em acceleration}. 

\item If an object A exerts a force on B, B exerts a force on A which is equal in magnitude and opposite in direction to that exerted by A on B.  Rather lyrically said, ``to every action there is an opposite and equal reaction''.  
\end{enumerate}

\noindent
$\bullet$  Centripetal acceleration:  If an object moves in a circle of radius $r$ with speed $v$, it undergoes a centripetal acceleration which points toward the center of the circle and has a magnitude 
\begin{equation}
a = \frac{v^2}{r}
\end{equation}

\noindent
$\bullet$ The gravitational force: If two objects possessing masses $M$ and $m$ are a distance $r$ apart, there is an attractive force between them, the magnitude of which is 
\begin{equation}
|F| = \frac{GMm}{r^2}
\end{equation}  
where $G$ is the gravitational constant, $G= 6.6720 \times 10^{-11}$ N-m$^2$-kg$^{-2}$.

\noindent
$\bullet$ The circular orbit equation.  If $M \gg m$, and the orbit of $m$ about $M$ is circular, there is a relation between the radius of the orbit $r$, the orbital speed $v$, and the mass $M$ which is called the circular orbit equation.  It says 
\begin{equation}
v = \sqrt \frac{GM}{r}
\end{equation}

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