Experimental Physics - Mechanics - Gravity 1

Experimental Physics EP1 MECHANICS

- Gravity –

Rustem Valiullin

https://bloch.physgeo.uni-leipzig.de/amr/

Experimental Physics - Mechanics - Gravity 2

Our “Cavendish experiment”

a0s

a

s L

0=wX

M

wX20

2

2

smMG

dtsdm » 0,0

0,0

0 ====vtst 2

202

1 tsMGs =als »

aLx 2= xLl

s2

=

wXLls20.12

29.1=

2202

229.10.12 t

sMG

lLXw =

( )252

2 1076.2808.022.10

1.04.192

29.10.12 GtGtXw ´=

´´

=

0 10000 20000 30000 40000 50000-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

5,5

6,0

X w

t2

Equationy = Intercept + B1*x^1 + B2*x^2 + B3*x^3 + B4*x^4Weight No Weighting

Residual Sum of Squares

0,02345

Adj. R-Square 0,99937Value Standard Erro

C

Intercept -0,04922 0,02868B1 1,89473E-4 1,29127E-5B2 -3,0878E-9 1,389E-9B3 6,47675E-14 5,06395E-14B4 -5,7302E-19 5,84816E-19

45 1089.11076.28 -´=´ G

21311 skgm1057.6 ---´=G

l

Experimental Physics - Mechanics - Gravity 3

Kepler’s law

BornDecember 27, 1571Weil der Stadt near Stuttgart, Germany

Died November 15, 1630) (aged 58)Regensburg, Bavaria, Germany

Residence Württemberg; Styria; Bohemia; Upper Austria

Fields Astronomy, astrology, mathematics and natural philosophy

Institutions University of Linz

Alma mater University of Tübingen

Known for Kepler's laws of planetary motionKepler conjecture

Law 1: All planets are moving in elliptical orbits with the Sun being in at one of the focuses.

Law 2: A line joining the Sun and a planet sweeps out equal areas in equal times.

Law 3: The square of the period of a planet is proportional to the cube of the planet’s distance from the Sun.

tD

1A

tD

2A

12

2

2

2

=+ay

bx

222 cba +=

Focal point

Experimental Physics - Mechanics - Gravity 4

Second and third Kepler’s law

0=÷øö

çèæ´=´=rrFrFr!

!!!!t

( ) 0=´==

dtvmrd

dtLd !!!

!t

tD

1A

tD

2A

r!

v!

dtmLdtvrsdrdA22

121

=´=´=!!!!

22

2 wmrrmv

rGmM

==

const4 2

3

2 ==TrGM

p

TVenus = 224.7 d TMars = 686.98 d

Experimental Physics - Mechanics - Gravity 5

Newton’s law of gravity

1687: Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

1m

2m21F!

12F!

r

12r̂

12212

2112 r̂

rmmGF -=

!

rrr!

=12ˆ

2211 kgNm1067.6 --´=GUniversal gravitational constant:

22 2÷øö

çèæ==T

RRvaplanet

p

2

24RKmFcf

p=

!"#$ = &'()* ≡ ,

planetR

- . 2

24RKmFg

p-=

/0 = −2.-!$ = −44$,-

.-!$

Experimental Physics - Mechanics - Gravity 6

Measuring the gravitational constant

qt k=2

2 LF=t

2rmMGF =

kIT p2=

LskmT p2=

222

222 mLLmmlI =÷

øö

çèæ==å

Lr

GmMFLk 2==qGMrL

GmMLrmLT

22

22

222 qpqp ==

2

222MT

rLG qp=

Cavendish's value for the Earth's density, 5.448 g cm−3

Þ G = 6.74 × 10−11 m3 kg−1 s−2

The Cavendish experiment, performed in 1797–98 by British scientist Henry Cavendish(not exactly shown below)

Experimental Physics - Mechanics - Gravity 7

Gravitational force: extended objects

ïî

ïí

ì

<=

>-=

Rr0

Rrr̂2

g

g

Fr

GmMF!

!

ïïî

ïïí

ì

<-=

>-=

Rrr̂

Rrr̂

3

2

RGmMrF

rGmMF

g

g

!

!

r! r!

r!j

Experimental Physics - Mechanics - Gravity 8

R

An object within a spherical shell

j

rRj

Rseg Rs j=

segsa

a

b

c

a

b

g

cbagba sinsinsin

==

rRaj sinsin

=anglessmallrR

aj»Þ

°=-°++ 180)180( Rjaj

( )RrR /1+=+= jjaj

( )rRsseg +=j

( ) 22 distance~~ =+Þ rRAm segseg

'2 Krm

KF seg == - the same from both sides

Experimental Physics - Mechanics - Gravity 9

r!

An object within a planet

2)(

rrGmMF =

ïî

ïíì

=

=3

34

334)(

RM

rrM

tot rp

rp

r̂3RrGmM

F tot-=!

Does not contribute

?

makxF =-=

xmk

dtxd

-=2

2

Experimental Physics - Mechanics - Gravity 10

Interaction with ring

dmsGMdF 2= acos2 dms

GMdFx -=

aa coscos2ring

2 sGMm

dms

GMFx =-= ò

integration over the ring

222 xas +=sx

=acos

( ) 2/322

ring2ring

xa

xGMmsx

sGMm

Fx+

==

R

r

dm

s

axa

M

Fd!

xdF

ydF

ydF

Experimental Physics - Mechanics - Gravity 11

R

r

sdm

s

axa

M

An object outside a hollow sphere

qd

qdF

densitylengthwidth ´´=dm

( )( )rpq aRddms 2= rp 2shell 4 Rm =

aR =qsin qqdmdms sin2shell=

qqa dmsGM

dFx sin21

cos shell2-=Now we have to integrate over all angles q, but keep in mind that s and a are functions of q.

qcos2222 rRRrs -+= qqdrRsds sin=Þ

acos2222 rsrsR -+=rsRrs

2cos

222 -+=Þ a

dssRr

RrGMm

dFx ÷÷ø

öççè

æ -+-= 2

22

2shell 1

4

îíì

+==-==RrsRrs

pqq 0 integration

limits

Experimental Physics - Mechanics - Gravity 12

2shell

22

2shell

2

22

2shell

41

4 rGMm

sRr

sRr

GMmds

sRr

RrGMm

FRr

Rr

Rr

Rr

Rr

Rrx -=

÷÷

ø

ö

çç

è

æ ---=÷÷

ø

öççè

æ -+-=

+

-

+

-

+

-ò

Extended objects: results

A What happens if we move the mass M into the spherical shell? (integration limits)

04

14

22

2shell

2

22

2shell =

÷÷

ø

ö

çç

è

æ ---=÷÷

ø

öççè

æ -+-=

+

-

+

-

+

-ò

rR

rR

rR

rR

rR

rRx s

Rrs

RrGMm

dssRr

RrGMm

F

R

rs

M

q

B What if object is not empty? (integrate over all shells)

Hollow sphere:

ò-= shell2 dmrGM

Fx

ò-= shell2 dmrGM

Fx

Experimental Physics - Mechanics - Gravity 13

What causes tides

2S

SS rGmMF = 2

M

MM r

GmMF =

2

2

SM

MS

M

S

rMrM

FF

=

kg1098.1 30´=SM

km1049.1 8´=Srkg1035.7 22´=MMkm1084.3 5´=Mr

200»MS FF

dr Fdr

drrdFdF )(

= drrmGmdF 3212

=

rR

rdr

FdF Earth42

=-=

4.0»=DD

S

M

M

S

M

S

RR

FF

FF

Experimental Physics - Mechanics - Gravity 14

Gravitational potential energy

sdFdW !!×=

drrFrdFdW )(=×=!!

0=arcdW

÷÷ø

öççè

æ--=÷

øö

çèæ--=-=D ò

122

112

1rr

GmMdrr

GmMWUr

r

Let us choose r1

such that U(r1)=0 rGmMU -=

rGmMmvE -= 2

21

rmv

rGmM 2

2 =

2mvr

GmM=2

21mvE -=

Bound systems; m << M

Experimental Physics - Mechanics - Gravity 15

Escape velocity

rGmM

RGmMmv -=-2

21

Sun - 617.7 km/s

Mercury - 4.25 km/s

Venus - 10.46 km/s

Earth - 11.186 km/s

Moon - 2.38 km/s

Mars - 5.027 km/s

Jupiter - 59.5 km/s

Saturn - 35.5 km/s

Uranus - 21.3 km/s

Neptune - 23.5 km/s

Pluto - 1.27 km/s

M

rGM

RGMv 22

-=

RGMvesc2

=

1 a.m.u. = 1.66 10-27 kgkB = 1.38 10-23 J/K

Hydrogen ~ 1

Oxygen ~ 16

Experimental Physics - Mechanics - Gravity 16

Ø Three empirical Kepler’ laws:

- Planets are moving in elliptical orbits.

- Line joining the Sun and a planet sweeps out equal

areas in equal times.

- The square of the period of a planet is proportional to

the cube of the planet’s distance from the Sun.

Ø The Newton’s law of gravitation:

Every particle attracts every other particle

with a force directly proportional to their

masses and inversely proportional to the

square of the distance between them .

To remember!