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Physics IClass 17

Newton’s Theory ofGravitation

Rev. 03-Nov-04 GB

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Forces Known to Physics

There are four fundamental forces known to physics: Gravitational Force (today) Electromagnetic Force (later in Physics 1 and 2) Weak Nuclear Force Strong Nuclear Force

(All forces we observe are comprised of these fundamentalforces. Most forces observable in everyday experience areelectromagnetic on a microscopic level.)

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Forces in Physics 1(so far)

We have encountered the following forces in Physics 1: Gravity Ideal Springs (Hooke’s Law) Pushes and Pulls Friction

What makes gravity different from the other three?(Hint: The ideal spring force is also conservative,so that isn’t the answer.)

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Newton’s Theory of Gravitation

Isaac Newton, 1642-1727

In 1666, our old friend, Isaac Newton, was musingon the motions of heavenly bodies while sitting in agarden in Lincolnshire England, where he had goneto escape the plague then ravaging London.

What if the force of gravity, the same force that causes an apple tofall to the ground in this garden, extends much further than usuallythought? What if the force of gravity extends all the way to themoon? Newton began to calculate the consequences of hisassumption…

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Newton’s Law ofUniversal Gravitation

r̂rmmGF 221

The meaning of each term:

F: Gravitational force on object 1 from object 2.G: Universal gravitational constant = 6.673 x 10–11 N m2/kg2.1m: Mass of object 1.

2m:Mass of object 2.2r: Center distance from object 1 to object 2, squared.r̂: Unit vector from object 1 to object 2.

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Properties of Gravity

Object 1

Object 2

Gravitational Force on 1 from 2

Every object with mass is attracted by every other object with mass. Gravity is a force at a distance (through occupied or empty space). Gravity is a “central” force (center-to-center for spherical bodies). Gravity varies as the inverse square of the center distance. Gravity varies as the product of the masses.

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If Gravity Varies As 1/r2,Where Does g = 9.8 m/s2 Fit In?

C o n s i d e r t h e f o r c e o n a n o b j e c t n e a r t h e s u r f a c e o f t h e e a r t h .( A s s u m e t h e e a r t h i s a s p h e r e a n d i g n o r e r o t a t i o n e f f e c t s . )R = r a d i u s o f t h e e a r t h .M = m a s s o f t h e e a r t h .m = m a s s o f t h e o b j e c t .

gmr̂R

MGmr̂R

MmGF 22

( W h a t i s t h e d i r e c t i o n ? )

g = 9 . 8 m / s 2 o n l y s e e m s c o n s t a n t b e c a u s e w e d o n ’ t g o v e r y f a rf r o m t h e s u r f a c e o f t h e e a r t h .

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Gravity is aConservative Force

B o t h t h e m a t h e m a t i c a l f o r m o f N e w t o n ’ s L a w o f U n i v e r s a lG r a v i t a t i o n a n d e x p e r i m e n t a l e v i d e n c e s h o w t h a t g r a v i t y i s ac o n s e r v a t i v e f o r c e . T h e r e f o r e , w e c a n f i n d a g r a v i t a t i o n a lp o t e n t i a l e n e r g y f o r a n o b j e c t w i t h m a s s m b e i n g a t t r a c t e d b ya n o t h e r o b j e c t w i t h m a s s M .

T h e g r a v i t a t i o n a l p o t e n t i a l e n e r g y i s d e f i n e d ( f o r c o n v e n i e n c e )t o b e z e r o a t i n f i n i t y . W e c a n c a l c u l a t e i t b y f i n d i n g t h ep o s i t i v e w o r k f r o m a n y p o i n t t o i n f i n i t y – y o u c a n f i n d t h ed e t a i l s i n t h e b o o k i n s e c t i o n 1 3 - 6 .

rMmGrd

)r(MmGrdF)r(U

r2

rg

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We Have Two Formulas for Gravitational Potential Energy!

O l d : )yy(gm)y(U 0g

N e w :r

MmG)r(U g

H o w c o u l d t h e s e b e t h e s a m e ?C o n s i d e r a l o c a t i o n n e a r t h e s u r f a c e o f t h e e a r t h , y

0 = R , y = R + h .

T h e o n l y t h i n g t h a t m a t t e r s i s U , n o t U i t s e l f .

O l d : hgm)RhR(gmU g

N e w :

hR1

R1MmG

RMmG

hRMmGU g

hRRhMGm

hR1

R1MGm 2

( h < < R )

hgmhR

MGmR

hMGm 22

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Class #17Take-Away Concepts

1 . F o u r f u n d a m e n t a l f o r c e s k n o w n t o p h y s i c s : G r a v i t a t i o n a l F o r c e E l e c t r o m a g n e t i c F o r c e W e a k N u c l e a r F o r c e S t r o n g N u c l e a r F o r c e

2 . N e w t o n ’ s L a w o f U n i v e r s a l G r a v i t a t i o n

r̂r

mmGF 221

3 . G r a v i t a t i o n a l P o t e n t i a l E n e r g y ( l o n g - r a n g e f o r m )

rMmG)r(U g

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Class #17Problems of the Day

___1. To measure the mass of a planet, with the same radius asEarth, an astronaut drops an object from rest (relative to thesurface of the planet) from a height h above the surface of theplanet. (h is small compared to the radius.) The objectarrives at the surface with a speed that is four (4) times whatit would be if dropped from the same distance above Earth’ssurface. If M is the mass of Earth, the mass of the planet is:

A. 2 M D. 16 MB. 4 M E. 32 MC. 8 M

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Class #17Problems of the Day

2 . C a lc u la te th e a c c e le ra tio n d u e to g ra v i ty a t th e s u r fa c e o f th ep la n e t M a rs . A s s u m e M a rs is a p e r fe c t s p h e re a n d n e g le c t e f fe c t sd u e to ro ta t io n . U s e M = 23104.6 k g a n d R = 6104.3 m .

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Activity #17Gravitation

(Pencil and Paper Activity)Objective of the Activity:

1. Think about Newton’s Law of Universal Gravitation.2. Consider the implications of Newton’s formula.3. Practice calculating gravitational force vectors.

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Class #17 Optional MaterialPart A - Kepler’s Laws of Orbits

Material on Kepler’s Lawsthanks to

Professor Dan Sperber

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Kepler’s Three Lawsof Planetary Motion

1. The Law of Orbits: All planets move in elliptical orbits having the Sun at one focus.2. The Law of Areas: A line joining any planet to the Sun sweeps out equal areas in equal times.3. The Law of Periods: The square of the period of any planet about the Sun is proportional to the cube of the semi-major axis of its orbit.

Newton showed through geometrical reasoning (without calculus)that his Law of Universal Gravitation explained Kepler’s Laws.

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Kepler’s Three Lawsof Planetary Motion

Try this link to see an animation:http://home.cvc.org/science/kepler.htm

The Law of Areas

A r rdAdt

r ddt

r

LL rmv rm r

L mr

12

12

2 12

2

2

( )

constant

The Law of Periods

F maGMmr

m r

GMr T

TGM

r

22

32

2

22

3

2

2

( )

ENERGY IN CIRCULARORBITS

K mv mGMr

K GMmr

U GMmr

E U K GMmr

12

2 12

2

2

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Class #17 Optional MaterialPart B - General Relativity

Material on General Relativitythanks to

Albert Einstein

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Where Did Newton Go Wrong?(Again!)

Albert Einstein (1879–1955)

(Check back to the optional material for classes 3 and 6 first…)

Einstein realized that something must be wrong with Newton’stheory of gravity, because it implied that the force of gravity istransmitted instantaneously to all points in the universe. Thiscontradicts the fundamental limitation in the Theory of SpecialRelativity that the fastest speed information or energy of any typecan travel is the speed of light.

To overcome this problem Einstein postulated a third principle, thePrinciple of Equivalence, to go with his two principles of SpecialRelativity. (1907)

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The Principle of Equivalence

In broad terms, the Principle of Equivalence states that there is noexperiment that one can perform to distinguish a frame of referencein a gravitational force field from one that is accelerating with acorresponding magnitude and direction.

This is sometimes called the “Elevator Postulate” because we canimagine a physicist in a closed elevator cab trying to determinewhether he is at rest on earth, or accelerating at 9.8 m/s2 far fromany planet, or perhaps on a planet where gravity is half that of earthand the elevator is accelerating upward at 4.9 m/s2. According toEinstein, there is no experiment that could detect a difference.

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The Principle of Equivalence

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General Theory of Relativity

By 1915, Einstein had worked through all the math (with some help)to show that his postulates led to a new theory of gravity based onthe effect of mass and energy to curve the structure of space andtime. His theory has some startling implications, one being theexistence of “black holes” – regions of space where the gravity fieldis so high that even light cannot escape. The predictions of GeneralRelativity, including the existence of black holes, have beenconfirmed by all experiments to date.

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Black Holes

Black holes are detected by the characteristicx-rays given off by matter falling into them.

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If Newton’s Gravity isn’t true, why do we still use it?

It’s a good approximation for most engineering purposes.

Massive Black Holes In Galaxies NGC 3377, NGC 3379 And NGC 4486B