Physics 100 Chapt 10 Newton’s Universal Law of Gravitation
Jan 06, 2016
Physics 100
Chapt 10
Newton’s Universal Law of Gravitation
Very fast horizontal toss
t = 0s
t = 1sx= 8km
5m
t = 2sx=16km
t = 3sx=24km
20m 45m
V=8km/s
Centripetal acceleration
a = v2/r for a circular orbit (v = 8km/s = 8x103m/s)
a =(8 x103 m/s)2
6.4 x 106 m=
64 x106 m2/s2
6.4 x 106 m
= 10 m/s2
Toward Earth’s center = g
Orbital motion is free fall
V = 4 km/sV = 6 km/sV = 8 km/sV = 10 km/s
Circular Orbit!Eliptical Orbit
Artificial satellite
a
v a = v2/r = g
Moon-earthv
a=v2/r
Is the Moon in free-fall around the Earth?
r=3.8
4x105 km
v
a = v2/r
what is v?
v = dist/time = 2r28d =
2x3.14x 3.84x108 m28dx(24h/d)x3.6x103s
= 24 x 108 m2.4x106s = 1.0 x 103 m/s
Moon’s centripetal acceleration
amoon = v2/r; v = 1.0 x103 m/s)
amoon=(1.0 x103 m/s)2
3.84 x 108 m=
1.0x106 m2/s2
3.84 x 108 m
= 2.7 x 10-3 m/s2
Toward Earth’s center
g13600
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
r=3.84x108m 60 x (6.4x106 m)
The moon is 60x further fromthe Earth’s center than objectson (near) the Earth’s surface
= 60 x RE
160
13600( )2=
RE = 6.4x106m
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
The force of gravity
gets weaker as distance squared
Gravity gets weaker as 1/dist2
Universal law of gravity
m Mr
F m
F M
F 1r2
combine: F mMr2 F = G
mMr2
Proportionality constant:
“Newton’s Constant”
Universal
Universal:applies to all objects!!!
What is G? W= GmME
RE2
WW= m
GME
RE2
W= m g
g=
GME
RE2
Determine G from g, RE & ME
G = gRE
2
ME
G = 10m/s2x(6.4x106m)2
6x1024kg
G = 6.7.x10-11 m3/kg s2
G = 6.7x10-11 Nm2/kg2
G = 10m/s2x 41x1012m2
6x1024kg
G = 410x1012m3/s2
6x1024kg
A very small number
Force of gravity between “ordinary-sized” objects
80kg60kg
1mF = G
mMr2
F = 6.7x10-11Nm2/kg260 kg 80kg(1m)2
F = 6.7x60x80x10-11N
Boy’s weight = mg = 80kg x 10m/s2 = 800 N
30x109 times bigger!
F = 32160.x10-11 N = 3.2x10-7
N
Measuring gravity force between “ordinary-sized” objects is very hard
Cavendish’s measured the gravitationalForce between known masses & fromthis deduced the value of Newton’sConstant G. From this and therelation for g he deduced ME, the massof 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
Full Moon
New Moon
Half Moon
Measuring Weight
mg
N
Weightlessness
N =mg
N >mg N <mg
N =0
Weightlessness means =0N
compensating upward
Weightlessness in action
“Floating” is space isreally free-falling in space
What is g on the moon?
W= GmMM
RM2
W= mGMM
RM2
W= m gM
gM
=
GMM
RM2
WRM=1.7 x 106m
mMM
gM on the Moon
g =
GMM
RM2
6.7x10-11Nm2/kg2 x 7.4x1022Kg(1.7x106m)2
=
gM = 1.7 m/s2
1/6 x gEarth
Eotvos experiment
Lorand Eotvos1848-1919
.. ..
.. ..
Devised a sensitive test of the equality between inertial & gravitational mass
Does Minertial = Mgravitational ?
M Gg
Miv2/r
Mi
MG
If Mi = MG,is the same
for every object
inertia
same masses different materials
If Mi = MG,different materials
twist different amounts
Eotvos saw no effect, all materialsfelt the same twist to 1 part in 109
Minertial = Mgravitational
to very high precision
.. ..
summary
Obey the law!