Basic Gravitation

Important quantities and constants:

• R_earth = 6400 km (approx.)
• G = 6.67 * 10^-11 N m^2 / kg^2
• M_earth = 6 * 10^24 kg (approx.)

Central Relationships

• a_cent = v^2 / r
• F_grav = G M m / r^2
• a_grav = G M / r^2
• a_grav = 9.8 m/s^2 ( R_earth / r)^2 (approx. by proportionality)
• PE_grav = - G M m / r (gravitational PE with respect to infinite separation)
• phi_grav = - G M / r:  Gravitational potential phi_grav is gravitational potential energy per unit of mass.
• Condition for circular orbit:  F_cent = F_grav so m v^2 / r = G M m / r^2 so v = sqrt(G M / r) so KE = 1/2 G M m / r.
• Escape velocity from surface of planet:  v = sqrt(G M / r)

Graphical Relationships:

• A thin trapezoid under and |F_grav| vs. r curve, understood as force vs. distance from a planet's center for a given object, has average 'graph altitude' which represents the average gravitational force and a width which represents a displacement `dr, so its area represents the approximate magnitude of the work done by the gravitational force in moving between the two corresponding positions, which is in turn equal to the gravitational PE change between these positions.

The corresponding slope represents the average rate at which the gravitational force on the specific object changes with respect to distance from the planet.

• The slope of a thin trapezoid on a graph of |PE_grav| vs. r represents the change in gravitational PE divided by the change in position, i.e., represents `dPE / `dr.  Since `dPE = F_grav_ave * `dr, this slope represents just F_grav_ave.  The area of the trapezoid doesn't represent a physically meaningful quantity.

• If F_grav is dividied by the mass m of the object, we get F_grav / m = a_grav.  Since an area under F_grav vs. r represents change in gravitational PE, an area under the graph of a_grav vs. r represents the change in gravitational PE / m, or change in the gravitational potential phi_grav.