Chapter 8 Universal Gravitation

8.1 Motion in the Heavens and On Earth

Keplers Laws of Planetary Motion

The paths of the planets are ellipses with the center of the sun at one focus.
An imaginary line from the sun to a planet sweeps out equal areas in equal time intervals. Thus, planets move fastest when closest to the sun, slowest when farthest away.
The ratio of the squares of the periods of any two planets revolving about the sun is equal to the ratio of the cubes of their average distances from the sun

(T_a/T_b)² = (r_a/r_b)³

Universal Gravitation

Newtons Law of Universal Gravitation:

The gravitational force between any two bodies varies directly as the product of their masses and inversely as the square of the distances between them.

F_g = Gm₁m₂/d²

***Note: Newtons Law of Universal Gravitation is an inverse square law with respect to distance.

Newtons Use of His Law of Universal Gravitation

Calculating the period of a planet or satellite around the sun or planet:

GM_sM_p/r_sp² = 4M_pp²r_sp/T_p²

G (the Gravitational Constant) = 6.67 X 10^-11 Nm²/kg²

T_p² = 4p²r_sp³/GM_s ***Note: This is Keplers third law.

Weighing Earth

F_g = GM_em/d²

F_g = mg

mg = GM_em/r_e²

g = GM_e/r_e²

M_e = gr_e²/G

8.2 Using the Law of Universal Gravitation

Motion of Planets and Satellites

The Period of a Satellite or planet

F_g = F_c

GM_em/r²= mv²/r

V = (GM_e/r)^1/2

V = 2pr/T

2pr/T = (GM_e/r)^1/2

T = 2p(r³/GM_e)^1/2

Weight and Weightlessness

F = GM_em/d²= ma

a = GM_e/d²

on Earth

g = GM_e/R_e²

ratio

a/g = (GM_e/d²)/( GM_e/R_e²)

a = g(R_e/d)²

The Gravitational Field

g = F/m

Einsteins Theory of Gravity