N

mg

Surface of spandex

Gary D. WhiteNational Science Foundation and

American Institute of Physicsgwhite@nsf.gov

04/22/23 1Gulf Coast Gravity Meeting, Oxford

The Spandex, for demonstrating celestial phenomena:

• The Solar System– Orbits, precession– Escape velocity– Planetary Rings– Roche Limit– Density differentiation– Early solar system

agglomeration models• Earth and moon

– Binary Systems– Tidal effects

• See ‘Modelling Tidal Effects’, AJP, April 1993, GDW and students

NOTE: “Gravity wells” rather than “curved space-time” or “embedding diagrams”

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Video fun

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From XKCD (A webcomic of romance,sarcasm, math, and language, http://xkcd.com/681/)

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…but are spandex

gravity wells really like

3-D space?

Wrong things that I thought I knew about the shape of the Spandex

– “It is like a soap bubble between rings.”

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rotate

Better known as the curve of a hanging chain

…or perhaps in St. Louis, the curve of

the Arch

…except data can’t

be fit to the appropriate hyperbolic cosine…

Pull middle ring down---this has long been known to

produce a catenary curve

– “Oh, right, it is like a weighted drum head.”

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Wrong things that I thought I knew about the shape of the Spandex

So, it solves Laplace’s

equation with cylindrical symmetry,

h=A + B*ln(R)

…except our original

data couldn’t be fit to any

logarithmic form…

M

This would make it like 2-D gravity, like orbits around

a long stick of mass M

…until we learned to stretch it as we attached it

thanks to Don Lemons and TJ Lipscombe, AJP 70, 2002

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Connection to general relativity

• Wilson (1920!- Phil. Mag. 40, 703) gives the metric for an infinite wire of mass to be (to leading order in m)

where ; …incredibly small for any reasonable linear density…in the slow speed, small mass density limit this means that the the Newtonian effective potential predicted by Einstein’s equations of a wire (or long stick or bar galaxy or other prolate distribution, perhaps) is given by

…In other words, logarithmic, as perhaps expected.04/22/23 Gulf Coast Gravity Meeting, Oxford 8

22 4 2 8 2 4 2 2 2m m mds dt d dz d

2 4 2 4 ln 2 200(1 ) / 2 (1 ) / 2 (1 ) / 2 (2 4 ln ...) / 2m m

Newton g c c e c m c

2 27/ [3 10 / ]m G c x m kg

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…so “pre-stretched” Spandex potential well is like the well around a skinny stick of mass m…but what about rolling marbles on Spandex? Is that really like planets moving in a logarithmic potential? To relay that

story, let’s recall some of the coolest early science

planets period, T radius from sun, R T-squared R-squared T-cubed R-cubed(in years) (in earth-sun distances)

Mercury 0.241 0.387 0.0580 0.150 0.0140 0.058Venus 0.616 0.723 0.379 0.523 0.2338 0.378Earth 1 1 1 1 1.0000 1.000Mars 1.88 1.52 3.54 2.321 6.65 3.54Jupiter 11.9 5.20 141.6 27.1 1685.16 140.8Saturn 29.5 9.54 870.3 91.0 25672.38 867.9

So, in natural units, T2 = R3 for planets.

(In unnatural units, T2 is proportional to R3)

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We determined Kepler’s law analog for unstretched Spandex for circular orbits by doing some experiments…

• For fixed M, unstretched Spandex has ln(T)=(1/3)ln(R2) +b– So, Spandex is T3/R2 = k…– Kepler Law for real

planets about sun is T2/R3

= c.• Curiously close, but no cigar; • What is pre-stretched spandex

Kepler’s law analog for circular orbits?

Let’s come back to that…for now notice how noisy the data is…

Kepler's Law analog

-1

-0.5

0

0.5

1

1.5

-6 -4 -2 0

ln(R^2/sqrt(M))

ln(T

) line has slope 1/3y-intercept ~ 1.35

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x

mgh(x)

About rolling on the Spandex…let’s first consider the lower dimensional case---modelling one

dimensional oscillations with motion in a vertical plane

• One-D motion

Diff. wrt time to get

Assume , then

Rolling in a vertical plane in a valley given by h(x):

butand no-slip rolling means

so

21

1

2D xE mV U x

0

2 20

( )

/ 1 ( )roll

rollroll

k mgh x

m m I a h x

2 2 21 1 1

2 2 2roll x yE mV mV I mgh x

tan( ) ( )y x xV V V h x

2 2 2 2( )x yV V V a 0 00 ( ) ( ) ...m U x U x

0x x

0 { ( )} xmx U x V

So for small we get SHM with 2 2

2

1 1 1 ( )

2roll x

IE m h x V mgh x

ma

00

( )U xk

m m

Conclusion:

You can model motion of a mass at the end of a

spring (1D motion) with a ball rolling in a vertical

plane if

1)the shape of the hill matches the potential, and

2) if you “adjust” the mass and

3) if the derivatives of the hill are “small”

Likewise for 2D(that is, when we want to model near circular 2D motion in a plane we can use near-circular motion on a Spandex)…WHY?

12

22

2 21 / ( )

22D LE mV mU

Diff. wrt time, assumeR

0 0

2 3 2 40 0

0 ( ) ( )

/ ( ) 3 / ( ) ...

m U R U R

L mR L mR

Again, SHM, constant terms give orbital frequency,

coefficient of gives frequency of small oscillations about orbit,

2 40 0

2

( ) 3 / ( )D oscillations

U R L mRk

m m

2 3 2 20 0 0 0 0 0( ) ( ) / / ( ) , .orbitalL mR U R R U R m T R U R Kepler etc

04/22/23 Gulf Coast Gravity Meeting, Oxford

First, let’s look at the planar case

Rolling adds more complications, but when rolling in a horizontal circle we have something

similar, but with a few new terms due to the rolling constraint

leading to,

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2 2 2 2 2

2

2

2 2

2

1 ( / ) (1 ) ( / (2 )

(

) 1 (1 ) / ( ) (2

/(1 )( )

1/ )

rolling

z z

LE m I a V h mgh I h ma

Lh mh a

m

I a ah

20 0 2

cos( )/ 1

cos( )cos( )orbital

IR gh

ma

20 0( ) /orbitalR U R m

instead of Kepler’s Law

• Forhave

or• So if h(R) is power law,

yielding Kepler’s law analog

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20 0 2

cos( )/ 1

cos( )cos( )orbital

IR gh

ma

2

0 0orbitalR gh

22

2

21

( )

T I

R ma gh R

1( )h R A R

2

2 2

21 =constant

R I

T ma gA

Effect of rolling on orbits

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• Forhave

or• So if h(R) is logarithmic,

yielding

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20 0 2

cos( )/ 1

cos( )cos( )orbital

IR gh

ma

20 0orbitalR gh

22

2

21

( )

T I

R ma gh R

( ) /h R A R

2constant

RV

T

Conclusion:

A wire of mass M (or any cigar-shaped matter

distribution, from far away, but not too far) has a

constant velocity profile…hmm…

Returning to the question of what is the pre-stretched Spandex vesion of Kepler’s Law

What if not going in a circle?

• For cones, the oscillations about near circular motion satisfy

• Can also derive neat analytical expression for “scattering angle” for cones…

• Spandex is more complicated…

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22 2 2 3/2

02 2

(1 )3 1 2 ( / ) / (1 )rolling oscillations z

I h Ia R h h

ma ma

21 sin ( ) / B

Another video, ball in cone

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Moving in a cone---exp. vs theory for near circular orbits

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exp 0.97orbitT s

exp 0.72devT s

exp 6.48 .2 /rad s

exp 8.73 .4 /rad s

8.96 .4 /theory rad s

Two comments1) Should imperfect models, like Spandex and cones be used to convey ideas about gravity, general relativity?•Yup,(Imperfect models are better than “perfect” ones (consider “full-scale” maps!))

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2) Recall that to get logarithmic potential

that is like real gravity for wire-shaped mass distributions, we have to stretch the Spandex taut and then add a heavy mass. Why is

that? …Why do you have to stretch the Spandex

for it to model the real gravity?

Was real gravity pre-stretched?

Thanks to

• My students, especially Michael Walker, Tony Mondragon, Dorothy Coates, Darren Slaughter, Brad Boyd, Kristen Russell, Matt Creighton, Michael Williams, Chris Gresham, Randall Gauthier.

• Society of Physics Students (SPS) interns Melissa Hoffmann and Meredith Woy

• Aaron Schuetz, Susan White,• SPS staff, AIP, APS, AAPT, NSF, NASA, and • You!

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