Newton’s Universal Law of
Gravitation
Physics 100
Chapt 10
Very fast horizontal toss
t = 0s
V=8km/s
t = 1s
x= 8km
5m
t = 2s
x=16km
20m
t = 3s
x=24km
45m
Centripetal acceleration
a = v2/r for a circular orbit
(v = 8km/s = 8x103m/s)
(8 x103 m/s)2
64 x106 m2/s2
a=
=
6
6.4 x 10 m
6.4 x 106 m
= 10 m/s2
Toward Earth’s center
=g
Orbital motion is free fall
Circular Orbit!
Eliptical
Orbit
V = 10
4 km/s
6
8
km/s
Artificial satellite
v
a
a = v2/r
= g
Moon-earth
v
a=v2/r
Is the Moon in free-fall around the
Earth?
a = v2/r
what is v?
2pr
v = dist/time =
28d
24 x 108 m
=
6
x
2.4 10 s
2x3.14x 3.84x108 m
= 28dx(24h/d)x3.6x103s
= 1.0 x 103 m/s
Moon’s centripetal acceleration
amoon = v2/r;
v = 1.0 x103 m/s)
(1.0 x103 m/s)2 1.0x106 m2/s2
amoon=
=
8
3.84 x 10 m
3.84 x 108 m
= 2.7
x
10-3
m/s2
Toward Earth’s center

1
3600
g
Newton’s dreams
Hmmmmm…….
The Moon is in free-fall
around the Earth
It’s acceleration is only
1/3600 g (accel at the
Earth’s surface)
Distances
The moon is 60x further from
the Earth’s center than objects
on (near) the Earth’s surface
r=3.84x108m  60 x (6.4x106 m)
= 60 x RE
RE =
6.4x106m
(
1 2
1
60 ) = 3600
Newton’s
big idea
The moon is 60x
further from
the Earth’s center
than objects on
(near) the Earth’s
surface
The strength of
Earth’s gravity
near the Moon is
(1/60)2 =1/3600
times weaker
Gravity gets weaker as 1/dist2
Universal law of gravity
m
r
M
F  m
F  M
combine:
1
F 
r2
mM
F 
r2
Proportionality constant:
“Newton’s Constant”
mM
F = G r2
Universal
Universal:
applies to all objects!!!
mME
W= G R 2
E
What is G?
W
GME
W= m R 2
E
W= m g
GME
g= R 2
E
Determine G from g, RE & ME
G =
gRE2
ME
G =
10m/s2x(6.4x106m)2
6x1024kg
10m/s2x 41x1012m2
6x1024kg
G =
410x1012m3/s2
6x1024kg
G =
G = 6.7.x10-11 m3/kg s2
G = 6.7x10-11 Nm2/kg2
A very small number
Force of gravity between
“ordinary-sized” objects
80kg
mM
F = G r2
1m
60kg
F = 6.7x10-11Nm2/kg2
60 kg 80kg
(1m)2
F = 6.7x60x80x10-11N
F =
32160.x10-11 N
= 3.2x10-7 N
 30x109 times bigger!
Boy’s weight = mg = 80kg x 10m/s2 = 800 N
Measuring gravity force between
“ordinary-sized” objects is very hard
Cavendish’s measured the gravitational
Force between known masses & from
this deduced the value of Newton’s
Constant G. From this and the
relation for g he deduced ME, the mass
of the Earth, which turned out to be
about twice the value people had
guessed it to be at that time.
I weighed the
Earth
Moon’s gravity causes tides
So does the Sun’s
Half Moon
New Moon
Full Moon
Measuring Weight
N
mg
Weightlessness
N =0
N =mg
N >mg
N <mg
Weightlessness means N =0
compensating upward
Weightlessness in action
“Floating” is space is
really free-falling in space
What is g on
the moon?
mMM
W= G R 2
M
W
MM
GMM
W= m R 2
M
W= m gM
GMM
g M= R 2
M
gM on the Moon
GMM
g = R 2
M
=
6.7x10-11Nm2/kg2 x 7.4x1022Kg
(1.7x106m)2
gM = 1.7 m/s2
1/6 x
gEarth
Does Minertial = Mgravitational ?
..
..
Eotvos experiment
.. ..
Lorand Eotvos
1848-1919
Devised a sensitive test of the equality between inertial & gravitational mass
Miv2/r
q
M
q i
MG
If Mi = MG,
q is the same
for every object
If Mi = MG,
different materials
twist different amounts
same masses
different materials
.. ..
Eotvos saw no effect, all materials
felt the same twist to 1 part in 109
Minertial = Mgravitational
to very high precision
summary
Obey the law!