proof of newton's laws

8/9/2019 Proof of Newton's Laws

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Analytical Proof of Newtons Force Laws Page 1

Analytical Proof of Newton's Force Laws

1 Introduction

Many students intuitively assume that Newton's inertial

and gravitational force laws, F ma= and

( )F G

Mm

distance M m=

, are true since they are clear and

sim!le" #owever, there is an analysis that ties the two e$uationstogether and demonstrates that they must %e true" &he analysis!rovides answers to $uestions such as, (s the inertial masse)actly the same as the gravitational mass* +hy is thee)!onent of distance, , and not 1" or "-1 or 1* +hy is aconstant re$uired in one law and not in the other*

&he ideal way to !rove new theoretical laws is to forecastthe outcome of an e)!eriment using the laws, !erform thee)!eriment, and find that the result is as forecast" .ut nature hadalready !erformed the e)!eriment with !lanets in the solarsystem, and /e!ler had determined the results" 0o, Newton, inhis 1 !a!er, Mathematical Princi!les of Philoso!hy, 2now!art of the Great .oo3s 0eries in local li%raries4, a!!lied his forcelaws to the solar system and o%tained the same results that/e!ler had stated" &his confirmed Newton's ideas, !ut !hysicson a firm mathematical %asis and answered the a%ove $uestions"

2 Summary of Analytical Proof of Newton's Force Laws

(n the 5 ste! !rocedure that follows, Newton's force laws

are a!!lied to the !lanetsun system, and the !lanet 2earth4 !atharound the sun is shown to %e an elli!se" &his !rocedure %elowuses the mathematics found in first year college te)ts ande)!lains the mathematics within the derivations as they are%eing evolved"

1" 6%servations show that the !lanets follow a smooth curvearound the sun" 03etch the !lanet at !osition P, using

!olar coordinates,r and , within an orthogonalcoordinate system"


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Page Analytical Proof of Newtons Force Laws

" 7ifferentiate !lanet !osition functions to o%tain !lanet

velocities, v vx yand "

8" 7ifferentiate !lanet velocities to o%tain !lanet

accelerations, a ax yand "

9" :$uate Newton's inertial and gravitational force laws asa!!lied to the !lanet" (n this ste! we assume that inertialmass is identical to gravitational mass and that the forceof gravity decreases as the s$uare of distance" &heacceleration re$uired in the inertial law is also assumed to%e the !lanet radial acceleration" G must also %eassumed to ma3e the e$uations consistent" &he endresult of this analytical !rocedure must show that all theseassum!tions are correct, or Newtons e$uations would not%e true"

;"


3/37


P

r

)

y

-

y

!osition of P

) !osition of P

Figure 1

0un

:arth

=adius vector, r, is attached to !lanet, P, and varies in

length asPmoves" Also, angle and its rate of change vary asPmoves" &herefore, the velocity and acceleration ofPvarycontinuously as the !lanet moves along its !ath" =ecall thatacceleration, velocity and force have magnitude and direction"

Newton had !reviously !roved that, as far as the force ofgravity was concerned, the entire mass of the !lanet and suncan %e considered to %e at the center of their s!heres" &heradius vector starts at the center of the sun and ends at the

center of the !lanet" 7etermine the ) and y !ositions of P, as a function of r

and , %y using the trigonometric functions that are indicated %yFigure 1"

&he ) distance ofPfrom the origin? Px@rcos "

&he y distance ofPfrom the origin? Py@rsin "

As time !asses,Pmoves along its curve, ma3ingrand de!endent u!on time, t" &he !ositions of Pas functions of timeare indicated as?

P t r t t ) @ cos ,

P t r t y @ sin t .

&his com!letes ste! 1"


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Page 9 Analytical Proof of Newtons Force Laws

2.2 Planet Velocity in and y !irections

Figure indicates that the change in ) and y distances is a

function of %othrand asPmoves in time along its !ath"

P

r

)

y

-

) velocity of P y!osition of P

y velocity of P

) !osition of P

Figure

&he velocity of the !lanet, P, is the change of distancealong the curve !er the change in time" 6r,

change in distance

change in time"

s

t=

&he calculus e)!ression for velocity in the ) direction, as

the change in time is made very small is?d

dtPx t "

elocity in the y direction is? ddt

P ty .


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Analytical Proof of Newtons Force Laws Page ;

&herefore, the velocity ofPin the ) direction is?

v ddt

r t tx @ cos .

And the velocity of Pin the y direction is?

v d

dtr ty @ sin t .

&he calculus rule for o%taining the derivative of the !roductof two varia%les is to multi!ly the first term times the derivative ofthe second term !lus the second times the derivative of the first"

Also, the derivative of the sine of an angle is the cosine ofthe angle times its derivative, and the derivative of the cosine ofan angle is minus the sine of the angle times its derivative"

Bsing these differentiation rules?

&he e)!ression for the velocity of Pin the ) direction is,

( ) "dt

dsinr

dt

drcoscosr

dt

dv )

==

&he e)!ression for the velocity ofPin the y direction,following the same rules, isC

( ) "dt

dcosr

dt

drsinsinr

dt

dv y

+==

&his com!letes ste! "


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2." Planet Acceleration in and y !irections

&he ne)t ste! is to o%tain e)!ressions for the !lanetaccelerations in the ) and y directions indicated in Figure 8"

Acceleration in y direction

PAcceleration in ) direction

-

Figure 8

)

y

r

=ecall that acceleration is the rate of change of velocity"

Let the acceleration of the !lanet in the ) direction %e a)"

Let the acceleration of the !lanet in the y direction %e ay"

&hen? a d

dtvx x@ ,

and a d

dtvy y@ .

Finding acceleration causes us to ta3e the derivative, withres!ect to time, of velocity" elocity, in turn, is the derivative of

distance with res!ect to time" &herefore, acceleration is thesecond derivative of distance with res!ect to time" &he derivativeof a derivative is called the second derivative" &he sym%ol for the

second derivative, in this case is?d r

dt

2

2

varia%le or"


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Analytical Proof of Newtons Force Laws Page >

=e!lace vx and vywith their derived e)!ressions listed in0ection "" Follow the same differentiation rules as given a%oveand o%tainC

a d rdt

r ddt

rddt

drdt

ddt

x =

+

cos sin "

a d r

dtr

d

dtr

d

dt

dr

dt

d

dty =

+ +

sin sin "


Accelerations a)and aymust %e used to o%tain e)!ressions

for the radial and transverse accelerations of the !lanet in 0te!;"

2.# $%uate &raitational Force to Planet Inertial Force

Newton's force of gravity law as a!!lied to earth mass, m,and sun mass, M, is?

F GMm

Gravitational=r

"

+hereris the radius vector, the varying distance fromearth to sun, and G is the gravitational constant"

FGravitationalis the amount of force that acts in a straight line%etween the !lanet and sun" &his force would !lace the !lanet infree fall toward the sun if it were not for the counteracting !lanetinertial force"

+hat is the inertial force on the !lanet*

Newtons inertial force law states that the inertial force ise$ual to the acceleration of the !lanet times the mass of the!lanet"

F m(nertial =adial= a "


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&he inertial and gravitational forces must %e e$ual to eachother in magnitude %ut o!!osite in direction, or else the !lanetwould leave its or%it" +ith une$ual forces, the !lanet would fallinto the sun, or attain a different or%it in a new e$uili%rium !ath,or go s!inning off into s!ace" 0ince the !lanet does maintain itsor%it, the sum of the two forces must %e Dero"

F FGravitational (nertial+ = -"

GMm

m =adialr

a

-+ = "

7ivide through %y m and o%tain?GM

=adialr

a

-+ = "

&hen? GM

raRadial = "

&his is an im!ortant !lace in the !roof where the inertialmass is assumed to %e identical to gravity mass and the radialacceleration is shown o!!osite to the attraction of gravity" +emust continue to %e s3e!tical of these assum!tions, includingdistance to the second !ower, until we derive the elli!tical !ath ofthe !lanet around the sun" &his e$uation of the radialacceleration shows that a=adialis !ro!ortional to the inverse ofdistance s$uared" .y a!!lying some mathematics we will modify

the e$uation to o%tain a=adialas a function ofrand " &his radialacceleration e$uation is the %asic e$uation that will evolve intothe e$uation showing that the earth or%it is an elli!se"


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Analytical Proof of Newtons Force Laws Page

Notice also that Newton's inertial force law can %econsidered sim!ly as the definition of the unit of force" 6nce thestandards of 3ilogram, meter and second are agreed u!on, theunit of inertial force is esta%lished" +e need a constant, 2G4, toma3e the gravitational units of force have the same dimensionsand the same magnitude as inertial force units" .ut we have noreason 2as yet4, to %elieve that the inertial force, %ased onrandom %ut agreed u!on standards, is directly !ro!ortional to thegravitational force" +e Eust assumed the e$uivalence when wecanceled m in the a%ove derivation" (f the !ath of the eartharound the sun is analytically determined to %e an elli!se, thenthe assum!tion is correct"


&he ne)t !art of this !roof to find e)!ressions for radial andtransverse !lanet accelerations in ste! ; of the !rocedure"


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Page 1- Analytical Proof of Newtons Force Laws

2.( Planet )adial and *ranserse Accelerations

Figure 9 shows the geometric construction to determinethe radial and transverse !lanet accelerations"

a)sin

aycos

aysin

a)cos

P

r

a)

ay

aa&

a=

)

y

-

Figure 9

A vector re!resenting an assumed total !lanetacceleration, a, is drawn at some angle to the radius vector, r"(n Figure 9, it is convenient to draw a u!ward and away fromthe direction of the radial acceleration, a=, 2which is o!!osite tothe line of attraction %etween earth and sun4"


11/37


6ne com!onent of assumed !lanet acceleration,a, must%e in line with 2%ut in the o!!osite direction to4 the force ofgravity %etween the sun and !lanet" &his accelerationcom!onent, a=, is the radial acceleration " &he other com!onentof assumed !lanet acceleration, a, !laced !er!endicular to theradial acceleration, is the transverse acceleration, a&" 2&histransverse acceleration, is not to %e confused with the tangential,i"e" tangent to the !ath, acceleration" &angential acceleration, notrelevant in this !roof, is discussed in 0ections >"1 and >"4

6f course, we 3now that if the !lanet actually has atransverse acceleration, a transverse force must %e a!!lied" .utif a transverse force is a!!lied, the !lanet will %e !ushed out ofits or%it" 0o the transverse force must %e Dero and the transverse

acceleration must also %e Dero" &his conce!t of an assumedtransverse acceleration, will !rovide one e$uation needed for this!roof"

(f the assumed acceleration, a, had %een !laced in linewith the radius vector, it would have %een identical to a=and nonew information could %e gained from the geometry" Placed as itis though, acceleration a is com!osed of two vectors, a=and a&"&he radial acceleration, a=is drawn in line with the radius vector,r" &he transverse acceleration, a&, is drawn !er!endicular to the

radial acceleration"

(t is seen in Figure 9 that acceleration a is e$ual to twodifferent sets of com!onent vectors that !rovide the informationneeded to continue the !roof" 6ne set of these com!onents is a)and ay"

&he second set of acom!onents is a=and a&" &he

geometric construction in Figure 9 shows how aH, ayand areused to determine a=and a&"


12/37


&he construction shows that?

a a aR x y= +cos sin "

a a aT y x= cos sin "

&he alge%raic e)!ressions for aHand aIwere derived in0ection "8"

&herefore, re!lace aHin a=and a&with

cos sin "

d r

dtr

d

dtr

d

dt

dr

dt

d

dt

+

Also, re!lace aIin a=and a&with

sin cos " d r

dtr

d

dtr

d r

dt

dr

dt

d

dt

+ +

Ma3e the su%stitutions and o%tain?

a d r

dt

r d

dtR =

"

a rd r

dt

dr

dt

d

dtT = +

"

&his com!letes ste! ;"


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2.+ )elace *ime !eendent *erm, ,d

dt

in a)

From 0ection "9, GM

=adialr

a

= "

=e!lacing a=adialwith its e$uivalent from a%ove?

GM

r

d r

dtr

d

dt

=

"

A term that is not a function of time must %e develo!ed to

re!laced

dt

, %ecause

d

dt

is a factor in a="

=ecall that we have already determined that a&is Dero"

From a%ove? a rd r

dt

dr

dt

d

dt& = +

"

Lets ta3e the derivative of r d

dt

and see what results"

d

dt r d

dt r d

dtr

dr

dt

d

dt

= + "

Now multi!ly %oth sides %y1

rand o%tain?

1

r

d

dtr

d

dtr

d

dt

dr

d

d

dt

= + ,which is a&"


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.ut a&is Dero" 0o,1

r

d

dtr

d

dt

is also Dero"

0ince1

r

cannot %e Dero, the derivative of r d

dt

must %e

Dero" &he rate of change of a constant is Dero" &herefore, r d

dt

must %e a constant" &hat constant is designated /"

Let /= r d

dt

, and

/

r

d

dt = "

0u%stitute/

for

r

d

dt

in the e$uation of radial acceleration,

=

GM

r

d r

dtr

d

dt

"

&hen, =

GM /

r

d r

dtr

r

" -r, =GM /

r

d r

dt r

8 .

&his com!letes ste! "

Nowd r

dt

must %e changed to a term containingrand ,

and %e time inde!endent"


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Analytical Proof of Newtons Force Laws Page 1;

2.- )elace *ime !eendent *erm,d r

dt

, in a)

&he factor to re!laced r

dt

is found through the following

!rocedure"

Let rn

=1

" 7ifferentiate1

nwith res!ect to time %y

following the calculus rule for differentiating a varia%le to a!ower" &he rule isC Place the e)!onent in front of thevaria%le, su%tract one from the e)!onent and differentiatethe varia%le"

&he result is, ddt n n

dndt

1 1

= "

&hen,dr

dt n

dn

dt=

1

"Multi!ly 1n

dn

dt%y 1 in the form of

d

d

"

dr

dt n

dn

d

d

dt=

1

.

(n 0ection ", we found that ddt

is e$ual to /r

"

0u%stitute and o%tain?dr

dt n

dn

d r

dn

d= =

1

// "

+e want to re!lace the second derivative of rwith res!ect

to t, therefore we must differentiate the first derivative,dr

dt, once

more to o%tain the second derivative"

&his second derivative is,d r

dt

d n

d

d

dt

@ /

.


16/37


+e again su%stitute/

rfor

d

dt

, and o%tain?

d r

dt

n d n

d

= /

"

&he result from 0te! a%ove is = GM /

r

d r

dt r

8"

0u%stituting ford r

dt

we o%tain?

=

GM/

/

rn

d n

d r

8

"

0ince nr

=1

,

GM / /

r r

d n

d r

8= +

"

&his is the !oint in the analytical !roof where distance

s$uared must %e in the denominator of the gravitational force" (norder for the analysis to conclude with a closed curve we musthave distance, r, the radius vector, e)actly to the first !ower"+hen we multi!ly through with r,, we will have r to the first!ower within the e$uation" (t will then %e !ossi%le for the radiusvector to trace a smooth curve as we assumed in Figures 1 to9"&he second derivative term will determine the sha!e of thecurve"

GM /

/

= +

d n

d r "

&his com!letes 0te! >"


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Analytical Proof of Newtons Force Laws Page 1>

2. /0tainr as a Function of and Confirm eler's First Law

0im!lify the a%ove e$uation %y re!lacing1

rnwith "

d n

d n

- + =

GM

/ "

&his is the differential e$uation to %e solved in order to getras

a function of " +e 3now, from 0ection " 0te! , that the derivativeof a cosine function is a negative sine function and the derivative ofthe sine function is a cosine function" (t a!!ears that the cosine

function of will fit into the differential e$uation and solve it" &he!rocedure to solve the e$uation is to let,

n= +AGM

/cos ,

where A is another constant"

+hat is the first derivative of n with res!ect to *

dn

d

d

d =

= Acos J

GM

/A

sin "

+hat is the second derivative of nwith res!ect to *

d n

d

= A cos "

&o test this solution, !ut n and the second derivative of n

with res!ect to %ac3 into the original e$uation and chec3 theresult"

d n

dn

-

GM

/+ = "

+ + =A AGM

/

GM

/cos cos "

-

&he left side of the test e$uation is also Dero"&he result shown in 0te! > isC

GM / /= +

d nd r

"

0u%stitute Acos for the second derivative of n withres!ect to .

( )GM / A/

= +

cos "r


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0olve forr and o%tain?

r=

+

/

GM

/

GMA

1 cos

"

&he radius vector determining the earths !ath around thesun is a function of the mass of the sun, the cosine of thegenerated angle and constants G, /, and A" &his is Newton'sderived e$uation for the !lanet's !ath around the sun and it hasthe form of the !olar e$uation of a conic" 20ee 0ection 8"4 &hisanalytically derived !ath turns out to %e an elli!se and agreeswith /e!ler's first law" Notice again that the mass of the earth

!lays no !art in its e$uation of motion" An o%Eect of a far differentsiDe and mass would occu!y the same !ath if it had the sameradial acceleration"

&his same !henomenon occurs in


19/37


" Polar $%uation of Conics

=ecall that one !olar e$uation of a conic is? r=

d

cos

1.

+here r is the radius vector, and d is the distance fromfocus to directri)" &he radius vector generates the angle andtraces out the conic" &he !lanet or%it starts in Figure 1 andcom!letes the elli!se in Figure ;"

%

a a

a

r@a

r

a21K 4

0un FocusFocus

7irectri)

d

Figure ;

(f the eccentricity, ,is less than one and greater thanDero, the !lotted e$uation is an elli!se with the sun at a focus"

2+hen e$uals one the e$uation is a !ara%ola" +hen isgreater than one the e$uation is an hy!er%ola"4

(f the !ath of the !lanet is an elli!se, the !lanet will returnto some starting !oint once every or%it" &he earth returns to a

randomly selected starting !oint, as do all the !lanets" 0o, usingonly his e$uations, Newton !roved that the !ath of the earth isan elli!se as /e!ler had o%served" &herefore Newton'se$uations are !roved to %e correct" 0ince all the !lanets andtheir moons follow elli!tical !aths, they all demonstrate Newtonslaws"


20/37

Page - Analytical Proof of Newtons Force Laws

&o follow the !roofs of /e!lers laws, we need moreinformation on the mathematical characteristics of an elli!se"

(n Figure ;C

r@ radius vector"

@ angle generated %y radius vector, -oK 8-

o" +hen

goes %eyond 8-othe curve re!eats Eust as the !lanet re!eats

its or%it"

a @ semiKmaEor a)is" &his is also the mean !lanetKsundistance"

% @ semiKminor a)is"

@ eccentricity"

of earth or%it @"-1>" of moon or%it @ "-"

Area of elli!se @ a%"

Polar e$uations of elli!se?

r r=+

=

d and

a

1J

1

1

cos cos"

&he second e$uation shows that when the eccentricity, ,

is Dero, the conic is a circle of radius a" 0ince the !lanet'so%served distance to the sun varies, the !lanet !ath cannot %e a

circle and cannot %e Dero"


21/37


# Proof of eler's Second Law

/e!ler's second law states that the radius vector, from!lanet to sun, swee!s e$ual areas in e$ual times as the !lanetor%its the sun" &his law can %e shown as followsC

&he first ste! is to determine the area of a small triangularsegment, dA, of the elli!tical sha!ed area shown in Figure "

Figure

d

r

h

-

d rA 1

h"

For the small angle, d, the sine of the angle in radians is

e$ual to the angle? sin " = =h

rd &hen? d r dA @

1

"

(ndicate the derivative with res!ect to time on %oth sides of

the e$uation"d

dt

d

dt

A"

1

r

=ecall in 0ection ",

d

dt r

=/

"

&herefore?d

dt

A /=

,and d dtA

/=

"&hen,

( )A=/

times a s!ecified time !eriod "

&he e$uation shows that area swe!t is a constant times anela!sed time" &his is /e!ler's second law" &he swee! of area %ythe radius vector in any ela!sed time !eriod is inde!endent of

where the !lanet is in its or%it" &his means that the !lanet'svelocity is faster when it is closer to the sun"


22/37


( Proof of eler's *ird Law

/e!ler's third law states that, the s$uare of the !lanet'stime for one or%it, divided %y the cu%e of the mean distance of!lanet to sun, is e$ual to the same constant for all !lanets"

From the !roof of /e!ler's second law 20ection 94 we note

that? d dtA/

=

"

&he !lanet swee!s through the whole area of the elli!se,

a %, in the time & that it ta3es to or%it the sun"

&he total area swe!t %y vectorrin time &is,

a% / &"

&hen, &/a%= .

/e!ler's law re$uires &" 0o, s$uaring each side,

&/

a %

9

=

"

From 0ection 8 discussion of elli!se characteristics andFigure ;C

a % a = + "

% a a a = =

1 "

&hen&

/

9

a=

9

1 "


23/37


Ne)t e$uate the numerators of the analytic e$uations of

the elli!se? 1 =

d

a"

&hen, &a /

8 d= 9

"

&he e$uation of an elli!se in !olar coordinates is,

r=d

1J

cos"

And Newton's !lanetary or%it e$uation is,

r=/GM

1J/

GMAcos

"

0o for !lanetary or%iting motion,d/

GM

= "

&hen,&

a GM

8 =9

"

&herefore, for every !lanet&

a

8is e$ual to the same

constant" 2Note that each !lanet has a different constant in/e!ler's second law"4

Newton e$uations again !roved an astronomicallyo%served /e!ler law, gave the mathematical !rinci!les involved

and determined e)actly the value of the constant "


24/37


+ Newton's Analytical $stimate of &

&he following !rocedure is one of Newtons estimate of G%ased on his own force laws"

1" (nertial force, F(@ m a"

" At the :arths surface? inertial force F(@ mass of anyo%Eect times the acceleration due to earth's gravity" &heacceleration due to earth's gravity, g, was found %y

Galileo to %e "5 meters sec"(nertial force, F(, at earth's

surface @ mass of any o%Eect H g"

8" Gravitational force, F GM ) m

=adiusG

:arth 6%Eect

:arth

= "

9" &he unit of inertial force is 3gKmeter !er second" &o ma3ethe unit of gravitational force consistent, G has thedimension of 3gKmeter !er secondtimes meterover3g"&he unit of force, 3gKmeter !er second, is now namedthe newton"

;" F(@ FG@ m6%EectH "5 msec@G

M ) m

=adius

:arth 6%Eect

:arth

"


25/37

Analytical Proof of Newtons Force Laws Page ;

&he mass of the o%Eect is on %oth sides of the e$ual signand cancels"

&hen, 5" "=GM

=adius:arth

Newton estimated that the average density of the earthwas %etween ; and times the density of water" Bsing ;"; timesthe density of water, and an estimated radius and volume ofearth, the value of G is determined to %e? > H 1-K11m8 3g sec"

&he !resent day value is "> H 1-K11m8 3g sec"

&his value of G ena%led astronomers to estimate the mass

of many %odies in the solar system and correlate the estimateswith the measured distances of moons, !lanets and sun, andvelocities of moons and !lanets" &his also indicated that G is auniversal constant"

- *wo 3etods of Calculatin4 3oon )adial Acceleration

&here are two methods of calculating the radialacceleration of the moon using Newtons lawsC

1" &he first method re$uires calculating g times the ratio of

the earth radiusto moon's distance" +hen we 3now theradial acceleration of the moon at its mean distance fromearth we can calculate G M:arth, and modify the estimatesof earth mass or G %y using Newton's force laws"

" &he second method re$uires that astronomers !rovidethe tangential velocity of the moon at its mean distancefrom the earth" &he tangential velocity s$uared divided %ythe mean distance moonKearth also results in radialacceleration"


26/37


-.1 First 3etod of Calculatin4 3oon )adial Acceleration.

=ecall that the weight of an o%Eect on the earth surface isthe mass of the o%Eect times the acceleration due to earth'sgravity, g" .y using g and Newtons e$uations we can calculate

the radial acceleration of the moon"

Astronomical dataC

Mean distance 2semiKmaEor a)is of elli!se? moon center toearth center4 is 8"59 H 1-5meters"

Moon's tangential velocity is 1"- meters !er second at itsmean distance from earth"

:arth radius is "85H1-meters"

Acceleration due to earth's gravity, g, is "5 msec"

First method !rocedureC

Gravitational force @GM ) m

=adius@ m )g"

:arth 6%Eect

:arth

6%Eect

g is the acceleration e)!erienced %y the o%Eect on the earth'ssurface !ointing to the earth center"

gGM

=adius

:arth

:arth

.


27/37

Analytical Proof of Newtons Force Laws Page >

&he moon is staying in its or%it around the earth for the same reasonthat the earth stays in its or%it around the sun" &he radial accelerationof the moon is e)actly e$ual and o!!osite the acceleration caused %yearthKmoon gravity attraction on the moon" :$uate moon inertial forceto gravitationalforce"

m a @ Gm M

7istancemoon

:arth=adial moon

moon

:arthKmoon

.. . .

(t is only at this !oint in s!ace, when the moon is at its meandistance from earth, that the moon has e)actly this radialacceleration" +hen the moon is closer to earth in its elli!tical or%it,the magnitude of the radial acceleration is greater" +hen the moon isfurther from earth, the magnitude of radial acceleration is less"


28/37


-.2 Second 3etod of Calculatin4 3oon )adial Acceleration

&he second method used for calculating the radialacceleration of the moon, re$uires dividing the s$uare of themoon tangential velocity, at its mean distance from earth, %y that

distance" 2&he semiKmaEor a)is of an elli!se is its mean distance,designated %y the letter, a"4

As an e$uation?

Mean 7istancem L sec

&angential

:arthKmoon

= " "-->

&his result confirmed Newton's ratio method of calculatingthe radial acceleration of the moon"

#ow do we !rove analytically that,

Mean 7istancea

&angential

:arthKmoon=radial moon= *

An a!!ro)imation of radial acceleration can %e made %yassuming that the or%it of the moon around the earth is a circle".ut Newton has shown that the !ath is an elli!se" &heassum!tion is then wrong for four reasonsC

1" &he moon's or%it is an elli!se"

" &he tangential velocity is not constant"

8" &he radial acceleration is not constant"

9" &he radial acceleration of the moon is in line with thefocus of an elli!se 2the center of the earth4, and not at thecenter of a circular !ath" Bsing moon's velocity s$uareddivided %y an assumed radius gives an assumedcentri!etal acceleration"

&herefore we must find the radial acceleration of the moon,as a function of tangential velocity and the elli!se radius vector,to show that %oth methods of calculating the moon's radialacceleration are correct"


29/37

Analytical Proof of Newtons Force Laws Page

-.2.1 Conseration of /r0ital $ner4y

(n order to o%tain an e$uation of radial acceleration as afunction of tangential velocity, we have to consider theconservation of or%ital, i"e" mechanical, energy of the moon as it

or%its the earth" +e will com%ine the or%ital energy e$uation withNewton's !lanet elli!se e$uations in order to o%tain an e$uationfor the moon's tangential velocity" &hen we can e$uate thistangential velocity to Newton's e$uation and o%tain the moon'sradial acceleration"

&he conservation of energy conce!t, as a!!lied to themoon, means that the total mechanical energy of the moon must%e constant for the moon to maintain its elli!tical or%it"

&he total or%ital energy of the moon is the alge%raic sum ofits 3inetic energy, / :, and its !otential energy, P :" &here is ane)change of small !ercentages of P : and / : as the moonor%its the earth"


30/37

Page 8- Analytical Proof of Newtons Force Laws

-.2.1a !eeloment of First /r0ital $ner4y $%uation

&he P : of the moon is considered to %e Dero at an infinitedistance from the earth" +hen the infinite distance from the earthis selected as the reference level, the P : of the mass of the

moon, m at a distance r, from the earth, M, isC

P :G M m

=

r

.

&his e$uation shows that the increase in the moon's P :,when its mass was %rought to the or%ital !osition, is e$ual to thenegative of the wor3 done %y the earth's gravity field"

&he / : of the moon is e$ual to1

mv , where v is the

tangential or or%ital velocity that varies during the elli!tical or%it"

: 2total mechanical or or%ital energy4 @ / : J P :"

:1

mv

G M m6r%ital

=

r

.

&he e$uation shows that when the moon a!!roaches its!erigee and gets closer to earth, the P : changes to a larger

negative value %ut the / : grows larger as the moon's velocity isincreasing, and the or%ital energy remains constant"

&he a%ove or%ital energy e$uation is the first e$uation, ofthe two that are needed, for this calculation method" (t showsor%ital energy as a function of the elli!se radius vector andtangential velocity" G, M, and m are constant" &he seconde$uation will give the or%ital energy at a certain !oint in the or%it,2at !erigee4" .ut since or%ital energy is constant, the designatedor%ital energies can %e e$uated"


31/37


-.2.10 !eeloment of Second /r0ital $ner4y $%uation

+e will now o%tain the second e$uation of :6r%ital" &hesetwo e$uations will ena%le us to finally show the connection%etween tangential velocity and radial acceleration"

As derived %elow, the differential calculus e$uation for thevaria%le tangential velocity of an o%Eect moving some distancealong any smooth curve 2such as an elli!se4 isC

&angential =

+

dr

dtr

d

dt

"

&his e$uation is develo!ed %y considering a small increase

in angle and distance r, with a small increase in time, indicated

in Figure >"

r

r

r

h s

s distance

Figure >

rd

-

For small angles, where sin e$uals , h @rd"

( ) ( ) ( ) s = +r r "

7esignate a change of, s, rand due to a increase intimeC

s

t

=

+

r

tr

t

"


32/37


As the change in time is made smaller and a!!roachesDero , we o%tain the calculus e)!ression for the tangentialvelocity along the curve"

&angential = + drdt r ddt

"

&he change in swe!t angle with res!ect to time, is thesame function ofrfound in section " ?

d

dt r

=

3

"

&his e$uation a!!lies to the !lanetKmoon systems 2with the

different constant %ecause the elli!tical !ath is different4 as wellas to the sunK!lanet system"

0u%stituting? 3

&angential

=

+

dr

dt r

"

As seen in Figure , there are several things to noteconcerning the moons elli!tical !ath at !erigee"

r@a

!erigee

rmin

aKa

a

:arthcenter

Focus

&angential

velocity, A

Figure

a

Moon


33/37


&he moon's radius vector goes through a minimum length

at !erigee" &he radius vector length at !erigee is, a21K4"

At the !erigee the rate of change of the radius vectorlength is Dero"

For the elli!se, with %eing generated %y the radius vectorstarting at the !erigee, the derivative ofr with res!ect to time is

-, when @--"

&herefore,dr

dtis Dero and can %e removed from the

standard e$uation for the o%Eect tangential velocity on an elli!seat !erigee"

&he tangential velocity e$uation reduces to?

3

&angential

Perigee

=r

"

&he !olar e$uation for the moon's elli!tical or%it is the sameform as the one that Newton determined for earth"

r=+

3

GM

3

GM.cos

:arth

:arth

1

"

From 0ection 8, a general e$uation for an elli!se isC

r=

a 1K

1J cos

.a is the semiKmaEor a)is of the elli!se and

is called the mean distance of an elli!se"


34/37


:$uating the numerators of the e$uivalent elli!see$uationsC

3

GM

a 1K

:arth

=

"

From Figure ,rat !erigee @ a21K4"

&he ne)t ste! is to calculate the moon's energy at !erigee"

:nergy at !erigee @ / : K P :"

0ince @3

&angential

Perigee

r,

: m3 GMm

Perigee

Perigee

Perigee

= 1

r r"

From a%ove?

3 GM a 1K =

"

0u%stituting for 3in the energy at !erigee e$uation andsim!lifying?

:GMm

a:Perigee 6r%ital= = "

&he or%ital energy of the moon at !erigee is the same asthe or%ital energy at any other !lace in its or%it"

&his is the second e$uation of :6r%ital that is needed to show

the relationshi! %etween &angentialand radial acceleration"


35/37

Analytical Proof of Newtons Force Laws Page 8;

-." )elation of 3oon *an4ential Velocity and )adial Acceleration

+e will first o%tain the moon's tangential velocity as afunction of its radius vector %y e$uating the two energye$uations"

First e$uation? : mGMm

6r%ital &angential= 1

r"

0econd e$uation? :GMm

a6r%ital= "

:$uate the or%ital energy e$uations and solve for

&angential "

GM

a&angential =

r1 "

+e will now find another e)!ression for radialacceleration"

(n 0ection "9, we found that the general e)!ression for theradial acceleration of earth or%iting the sun 2or moon or%iting theearth4 is?

ar

=adialGM=

"+herein, M is the mass of the sun"

(n the earthKmoon system, M is the mass of the earth"

ar

=adial:arthGM=

"

+hen r @mean distance, a,

a=adial:arth

GM

a= "


36/37


+e Eust found that, GM

a&angential :arth= "

7ividing %oth sides %y mean distance a results in?

a

GM

a

&angential

earth

= "

&his is the radial acceleration of the moon at the meandistance, a, from the earth as !roven in 0ection >"1"

+e have Eust shown whyA

a

can %e used for the

radial acceleration of the moon and the e$uation loo3s

li3e the e$uation for the centri!etal acceleration of ano%Eect revolving in a circle" .ut we must always use&angentialat the mean 2semiKmaEor a)is4 distance of moonto earth 2or earth to sun4 and, of course, use the semiKmaEor a)is for the distance" And this radial acceleration istrue at only two !laces 2or times4 on each or%it"

As shown a%ove , we have evolved two more e$uations of!lanetary 2or moon 4 motionC

6neC &he radial acceleration of a !lanet 2or moon4 is

a

&angential

at the mean distance, a, from the sun 2or !lanet4"

&woC GM

a&angential =

r1

is a!!lica%le to o%servations

of all the !lanets 2or moons4 and ena%les accurate crossKchec3ing of o%served distances and tangential velocities"


37/37

Analytical Proof of Newtons Force Laws Page 8>

Conclusions Sown 0y tis Analysis of Newton5s Laws

&he conclusion to %e drawn here is that each one ofNewton's laws is !roven %y his analysis of !lanetary motion" #econfirmed e)actly the em!irical data of /e!ler, and in addition,

he has shown mathematically why the data is true" (f any !art ofNewton's wor3 was incorrect, he could not have arrived at thee$uation of !lanet elli!tical motion a%out the sun"

+hen e$uating force of gravity and inertial force, we foundthat the mass of the earth canceled out" &his means that inertialmass and gravity mass are e)actly the same" Also, any siDeo%Eect in the earth's or%it will or%it the sun e)actly as earth if ithas the same radial acceleration as earth"

Newton !roved conclusively that the or%ital !ath of a !lanet2or moon4 is an elli!se with the sun 2or !lanet4 at a focus" &his isa remar3a%le fact in itself" .ut more remar3a%le is that Newton!roved his own laws at the same time, and introduced calculusinto his wor3" #e !roved that all the !lanets, and the moons orsatellites or%iting the !lanets 2and comets4 follow his laws"

Newton !roved F @ ma,

the force of gravity Gm m

7istance m to m

1

1

=

,and action

e$uals reaction, in that the !lanet attracts the sun with the sameforce that the sun attracts the !lanet" Newton !roved that theforce of gravity decreases e)actly as 1r"

Newton estimated G %y using his own laws, and showedthat G is a universal constant"

Newton's laws are always in use in !ro%lem solving" &heyform the %asis of mechanical !hysics and engineering,

momentum, wor3, satellite !ositioning, moon landings andcalculations involving %inary stars"

proof of newton's laws

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