The document discusses the laws of universal gravitation and how they relate to weight, mass, and satellite motion. It describes how the force of gravitational attraction decreases with the square of the distance between masses and explains that weight is a measure of the gravitational force on a mass. It then applies these concepts to discuss how gravity and weight differ on other planets and elevations from Earth's surface. The document concludes by summarizing Kepler's laws of planetary motion, including that the square of an orbiting body's period is directly proportional to the cube of the average orbital radius.

1. Copyright Sautter 2015

2. Gravitation
• The Law of Universal Gravitation is based on the
observed fact that all masses attract all other
masses. The force of attraction decreases as the
distance between the masses increases.
• This relationship is called an inverse square law
since the decrease in attraction between objects
is relative to the square of that distance.
• If the distance between the masses doubles, the
force of gravitational attraction becomes ¼ of the
original force. If the distance triples, the force
becomes 1/9 of the original, as so on.
• For example, 22 = 4, 32 = 9, etc.
2

3. Weight & Mass
• Weight and mass measure different things.
• Mass measures the quantity of matter which is present. It
represents the inertia property of matter meaning its ability of
resist changes in motion. Mass is measured in grams,
kilograms or slugs.
• Weight is a force resulting from the effect of gravity on a mass.
A mass without gravity is weightless. Weight is measured in
dynes, newtons or pounds.
• Gravity on the Earth’s surface is measured as – 980 cm/s2, - 9.8
m/s2 or – 32 ft/s2. (The negative sign means that gravity always
acts downward)
• As we move above the Earth’s surface or to other planets, the
strength of the gravitational field changes and so does the
weight. 3

4. Planet
Force of gravity
Force of gravity (weight) at the
Earth’s surface
Fearth = G m1 me
re
2
Force of gravity (weight) at
a point (P) above the
Earth’s surface
Fpoint P = G m1 me
rp
2
w = m1g = G m1 me
r2
g = G me
r2
g r2 = Gme
gearth r2
earth = Gmearth
gpoint p r2
point P = Gmearth
Therefore
gearth r2
earth =gpoint p r2
point P 4

5. Radius of Earth = 4000 miles
scale150 lbs
Two Radius of Earth = 8000 miles
scale37.5 lbs
Three Radius of Earth = 12000 miles
scale16.7 lbs
¼ wt
1/9 wt
Normal
wt
5

6. scale
150 lbs
scale
25.6 lbs
scale
406 lbs
g = 9.81 m/s2 g = 1.67 m/s2 g = 26.6 m/s2 6

7. Satellites
• When satellites orbit a planet the force which
supplies the centripetal force, and thereby the
circular motion, is the pull of gravity of the planet
which is orbited.
• Without the force of gravity, the satellite would
move in a straight line due to inertia.
• When the satellite orbits, the force of gravity must
equal the centripetal force. If the force of gravity
exceeded the centripetal force the satellite would
spiral into the planet. If the centripetal force
exceeded the force of gravity, the satellite would
seek a wider orbit or move off in a straight line. 7

8. Velocity
Vectors
Acceleration
Vectors
Force
Vectors
8

9. Earth
A satellite is a projectile shot from a very high elevation
and is in free fall about the Earth. 9

10. Inertial position
Centripetal force
Centripetal force
Centripetal force
Centripetal force
Gravity supplies
centripetal force
inward towards
the center of the
circular path
10

11. Fg = gravity force between
m1 and m2 separated by
a distance r
G is the Universal
Gravitational Constant
The weight of an object
is its mass times g’, the
gravity value at location r
11

12. Planet
Force of gravity
Fg
Centripetal force
Fc
Fg = Fc
Fg = G m1 m2
r2
Fc = m v2
r
G m1 m2 = m1 v2
r2 r
Canceling m1 & r on both sides
V2 = G m2
r
12

13. (1) V2 = G m2
r
(2) V2 r= G m2
(3) V = ωr
(4) ω= 2πf
(5) T = 1/f
(6) ω = 2π / T
(7) (ωr)2r = Gm2
(8) ω2 r3 = Gm2
(9) ( 2π / T)2 r3 = Gm2
(10) 4π2 r3 / T2 = Gm2
(11) T2 / r3 = 4π2/ Gm2 = a constant
T2 / r3 = a constant
Kepler’s
Third
Law
13

14. Kepler’s Laws of Satellite Motion
• (1) Satellites travel in elliptical paths. (The Earth and
the inner planets as well as the moon travel in nearly
circular orbits. The orbits of the outer planets are
more ellipsoid. Comets orbits are very elliptical.)
• (2) Areas swept out in equal times are equal even
though the speed of the satellite varies. Satellite
velocity is least when it is furthest from the central
body (apogee) and greatest when it is nearest
(perigee).
• (3) The period of motion squared divided by the
average orbital radius cubed gives a constant for all
satellites orbiting the same body. ( T2
1/ r3
1 = T2
2/ r3
2)
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