29:61 General Astronomy
Fall 2004
Lecture 9 ...September 27,2004
Vector Cross Products, Newton's Laws, the Gravitational Force

Just the facts, Ma'am

$\bullet$ The definition of a vector cross product: The magnitude of a cross product is defined as follows. If

$$\vec{C} = \vec{A} \times \vec{B}$$ (1)

then the direction of $\vec{A} \times \vec{B}$ is given by the right hand rule. The magnitude of $\vec{A} \times \vec{B}$ is given by

$$\vert C\vert = \vert A\vert \vert B\vert \sin \theta$$ (2)

where $\theta$ is the angle between the two vectors. Another way of expressing it as follows. Draw up an array

$$\left[ \begin{array}{ccc} \hat{e}_x & \hat{e}_y & \hat{e}_z \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{array}\right]$$ (3)

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

$$\vec{C} = (A_y B_z - B_y A_z) \hat{e}_x + (A_z B_x - B_z A_x) \hat{e}_y + (A_x B_y - B_x A_y) \hat{e}_z$$ (4)

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

Newton's Laws of Motion

1. 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.
2. If a force acts on an object, its momentum changes according to
   

   $$\vec{F} = \frac{\Delta \vec{p}}{\Delta t} = \frac{d \vec{p}}{dt}$$ (5)

   
   
   If the mass of the object acted upon stays constant, this simplifies to
   

   $$\vec{F} = m\frac{\Delta \vec{v}}{\Delta t} = m\frac{d \vec{v}}{dt} = m\vec{a}$$ (6)

   
   
   where $\vec{a}$ is the acceleration.

3. 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''.

$\bullet$ 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

$$a = \frac{v^2}{r}$$ (7)

$\bullet$ 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

$$\vert F\vert = \frac{GMm}{r^2}$$ (8)

where $G$ is the gravitational constant, $G= 6.6720 \times 10^{-11}$ N-m$^2$-kg$^{-2}$.

$\bullet$ 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

$$v = \sqrt \frac{GM}{r}$$ (9)