Newton’s Universal Law of Gravitation

Physics 100Chapt 10

Newton’s Universal Law of Gravitation

Very fast horizontal toss

t = 0st = 1sx= 8km

5m

t = 2sx=16km

t = 3sx=24km

20m 45m

V=8km/s

Orbital motion is free fall

V = 4 km/sV = 6 km/sV = 8 km/sV = 10 km/s

Circular Orbit!Eliptical Orbit

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

Artificial satellite

av

a = v2/r = g

Moon-earthv

a=v2/r

Is the Moon in free-fall around the Earth?

r=3.84x1

05 km

v

a = v2/r

what is v?v = dist/time = 2r

28d =2x3.14x 3.84x108 m

28dx(24h/d)x3.6x103s= 24 x 108 m

2.4x106s = 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 g1

3600

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 force of gravity gets weaker as

distance squaredThe 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 t=1/3600

times weaker

Gravity gets weaker as 1/dist2

Universal law of gravity

m Mr

F mF M

F 1r2

combine: F mMr2 F = G mM

r2

Proportionality constant:

“Newton’s Constant”

Universal

Universal:applies to all objects!!!

What is G?

W= GmMERE

2W

W= mGMERE

2

W= m g

g=

GMERE

2

Force of gravity between “ordinary-sized” objects

80kg60kg

1mF = G mM

r2

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

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= GmMMRM

2

W= mGMMRM

2

W= m gM

gM=

GMMRM

2

WRM=1.7 x 106m

mMM

gM on the Moon

g =

GMMRM

2

6.7x10-11Nm2/kg2 x 7.4x1022Kg(1.7x106m)2

=

gM = 1.7 m/s2

1/6 x gEarth

summary

Obey the law!