Newton’s Universal Law of Gravitation

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!