Newton’s Universal Law of Gravitation Physics 100 Chapt 10

Very fast horizontal toss t = 0s t = 1s x= 8km t = 2s x=16km t = 3s x=24km V=8km/s 5m 20m 45m

Centripetal acceleration a = v2/r for a circular orbit (v = 8km/s = 8x103m/s) (8 x103 m/s)2 6.4 x 106 m 64 x106 m2/s2 6.4 x 106 m a = = = g = 10 m/s2 Toward Earth’s center

Orbital motion is free fall Circular Orbit! Eliptical Orbit V = 10 km/s V = 8 km/s V = 4 km/s V = 6 km/s

Artificial satellite a = v2/r = g v a

Moon-earth v a=v2/r

Is the Moon in free-fall around the Earth? v a = v2/r r=3.84x105km what is v? 2pr 28d 2x3.14x 3.84x108 m 28dx(24h/d)x3.6x103s v = dist/time = = 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) (1.0 x103 m/s)2 3.84 x 108 m 1.0x106 m2/s2 3.84 x 108 m amoon= =  g 1 3600 = 2.7 x 10-3 m/s2 Toward Earth’s center

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 ( )2= 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 1 60 1 3600 ( )2= RE = 6.4x106m

Newton’s big idea The force of gravity gets weaker as distance squared 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 m r F  m F  M Proportionality constant: 1 r2 F  “Newton’s Constant” mM r2 mM r2 combine: F  F = G

Universal Universal: applies to all objects!!!

What is G? mME RE2 W= G W GME RE2 W= m W= m g GME RE2 g=

Determine G from g, RE & ME 10m/s2x(6.4x106m)2 6x1024kg G = 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 mM r2 F = G 80kg 60kg 1m 60 kg 80kg (1m)2 F = 6.7x10-11Nm2/kg2 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

I weighed the Earth 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 =0 compensating upward

Weightlessness in action

“Floating” is space is really free-falling in space

What is g on the moon? MM mMM RM2 m W= G GMM RM2 W W= m W= m gM GMM RM=1.7 x 106m GMM RM2 W W= m W= m gM MM GMM RM2 gM=

gM on the Moon gM = 1.7 m/s2 GMM RM2 g = 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

q Miv2/r Mi MG MGg q If Mi = MG, q is the same for every object

twist different amounts If Mi = MG, different materials twist different amounts inertia same masses different materials

Minertial = Mgravitational .. .. 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!