One of the things I detested about being a little kid was that every
time I thought of something really cool to do, I was invariably
thwarted by my little brother shouting, “Mom! Kelvin's
mixing rocket fuel in the bathtub again!” or
“Mom! Kelvin's making a submarine out of the old
refrigerator!”. Well, middle age has its drawbacks, but at
least you can undertake a project like this without fear of getting
nipped in the bud at the cry, “Mom! Kelvin's down in the
basement bending spacetime!”. It's important to recall the
distinction between “grownup” and “grown up”.
Let's us grownups head for the basement to bend some serious
spacetime.
Matters of Gravity
Apart from rare and generally regrettable moments of free-fall, we
spend our entire lives under the influence of the Earth's
gravity, yet rarely, if ever, do we experience the
universal nature of gravitation. It's a tremendous
philosophical leap from “stuff falls” to “everything in the universe
attracts everything else”. That leap, made by Isaac Newton in the
17th century, not only allowed understanding the motion of the Moon
and the planets, but inoculated in Western culture the idea that
the universe as a whole was governed by laws humans could discover.
This realisation fueled the Enlightenment and the subsequent
development of science and technology.
This page presents a “basement science” experiment which
reveals the universality of gravitation by demonstrating the
gravitational attraction between palpable objects on the human scale.
The experiment deliberately uses only
the crudest and most commonplace materials, permitting
anybody who's so inclined to perform it.
Einstein's 1915 theory of General Relativity explains
gravitation as spacetime curvature created by
matter and energy. So, by demonstrating how every object
in the universe attracts everything else, we're
bending spacetime in the basement.
But, if gravitation is ubiquitous, why was it not discovered millennia
before Newton's 1687 Philosophiæ
naturalis principia mathematica? The reason lies in the
extraordinary weakness of the gravitational force.
Feeble Attraction
Now you might say, “What do you mean, weak? I fell down a
flight of stairs a couple of years ago, and gravity sure didn't feel
weak to me!“. And yet, of the four forces of nature known to
physics, gravitation is the weakest, by the mindboggling factor of
4.17×1042 (4 followed by 42 zeroes) times weaker than
the electromagnetic force.
The stark difference in the strength of the electromagnetic and
gravitational forces is evident in the picture to the left. The
bright square in the jaws of the pliers is a 4 mm cubical magnet. It
is lifting a spherical steel pétanque (a lawn bowling game popular in
southern France and Switzerland) ball which weighs 550 grams.
Consider this picture in the following way: we're pitting our valiant
little magnet, with a volume of 0.064 cubic centimetres, weighing
less than one gram, pulling up with the electromagnetic force,
against the entire Earth, pulling down with gravity. And the
winner is…the magnet. A one gram magnet (I'm being generous: I
don't have a scale on which its weight reads other than zero)
out-pulls the Earth, which weighs 5.9736×1027 grams and has
a volume of more than 1027 cubic centimetres.
(The apparent discrepancy between the ratio of masses of the Earth
and the magnet and the 4.17×1042 strength ratio
of electromagnetism and gravitation is due to the
fact that only an infinitesimal fraction of the mass of the magnet
contributes to the [electro]magnetic attraction on the ball,
while every gram of the Earth's mass exerts gravitational force.
To obtain the correct ratio of force strengths, one must compare
the gravitational attraction between two electrons at a given
distance with the electromagnetic repulsion
resulting from their charge. This calculation arrives at the
correct strength ratio for the two forces.)
If gravity were not so weak compared to the electromagnetic force, you
wouldn't be reading this page; it's only because the electromagnetic
force that bonds the atoms in your body together so easily defeats the
Earth's gravity that you, along with all other solid objects,
don't slump into a puddle and eventually merge into a perfectly
spherical (actually, slightly ellipsoidal thanks to rotation) planet.
If your browser supports JavaScript, you can use the following calculator
to determine the gravitational force between any two objects. The
gravitational force between two masses m and m'
whose centres of gravity are separated by a distance r is
given by:
where G, the gravitational constant, is:
6.67408×10−8 cm³/g-sec²
Do the Twist!
Even though the force of gravity between objects of modest mass is
palpable compared to the weight of objects one can see,
detecting such a tiny force seems a daunting, if not hopeless,
endeavour for the basement tinkerer. Certainly, painstakingly
designed and constructed laboratory apparatus has allowed measuring
the gravitational constant to great precision and verifying the
equivalence of gravitational and inertial mass, and precision
gravitometers are routinely used in oil and gas exploration and
mineral prospecting, but we're trying to see if we can experience
the universal attraction of gravity without any high tech, high
budget gear.
Measuring tiny gravitational forces would be easy if we were in deep
space, far from any massive bodies. The only forces on objects in our
space laboratory, then, would be those entirely under our control. As
long as we made sure none of the objects we were experimenting with
were magnetic or electrically charged (easily arranged, assuming they
are conductive, simply by bringing them into contact so all excess
charges equilibrate), the only force remaining between objects would
be gravity, so however weak it be, we need only be sufficiently
patient to observe its effects. (The other two forces, the strong and
weak nuclear interactions, are limited in range to distances on the
order of the size of an atomic nucleus and can be neglected on the
human scale.)
What we'd like to do, then, is cancel the Earth's gravity so
that the much smaller gravitational forces between objects that fit in
the basement become evident. Fortunately, we don't need a 25th
century WarpMan to accomplish this, only a modest helping of 18th
century technology.
Differential Cleverness
One of the great all-purpose sledgehammers in the toolbox of
physicists and engineers is differential measurement; in
other words, don't worry about the absolute value of something, but
only the difference between things you can measure. For
example, it is common practice for linemen repairing high-voltage
power transmission lines to work on them, without cutting power, from
insulated baskets raised by a crane. As long as the lineman is
insulated from the ground, only the voltage difference between his
hands and the line he's working on matters; after attaching the basket
to the line, this is zero, so he might as well be repairing a grounded
conductor. Now if, while working on a conductor at, say, 200,000
volts above Earth potential, he should happen to touch the tower,
grounded to Earth, that would make for a really bad day. The trick is
keeping the difference small; you can live your entire life
at 1 million volts, and as long as everything around you is near that
value, there is no way, even in principle, you could discover the
absolute potential. This is the consequence of all the
forces of physics being gauge invariant:
absolute values don't exist—only differences matter.
The Torsion Balance
What we're looking for, then, is a device which responds only to
differences in gravitational attraction, canceling out the
much stronger constant gravitational attraction of the
Earth. We need look no further than a slightly modified version of
the same device Henry Cavendish used in 1798 to first measure
the gravitational constant, G in the equations above.
Ever since, the torsion balance has been the primary tool
used both for measuring the gravitational constant
and testing the equivalence principle, which states
that all bodies experience the same gravitational force regardless
of composition; Einstein's General Relativity showed this to be
a fundamental consequence of the structure of space and time.
State of the art torsion balances have measured the gravitational
constant to better than one part per million and confirmed
the equivalence principle to more than 11 decimal places. This
requires extraordinarily refined and delicate laboratory apparatus and
experimental design, in which a multitude of subtle effects must be
compensated for or canceled out. We, however, aren't going to
measure anything—we're only interested in observing
universal gravitation. This allows simplifying the torsion balance to
something we can set up in the basement.
The principle of the torsion balance is extremely simple. Suspend a
horizontal balance arm from a vertical elastic fibre. At
each end of the balance arm are masses, much denser than material of
the arm, which respond to the gravitational force. Once the
suspending fibre, balance arm, and weights are set up and brought into
balance, the downward force of gravitation acts equally on every
component. The balance arm is then free to rotate without any
hindrance from the Earth's gravity. It is constrained only by air
friction and the torsional strength of the support fibre—its
resistance to being twisted. We can then place
test masses near the ends of the balance arm and observe
whether the gravitational attraction between them and the masses on
the arm causes the balance arm to move. When measuring the
gravitational constant one must precisely calibrate the torsional
strength of the fibre, but to simply observe gravitation we need only
make sure the fibre is sufficiently limp to allow the gravitational
force to overcome its resistance to twisting.
In practice, the balance arm is so free to move that once any
force sets it into motion, it oscillates for a long period, spinning
round and round if free or bouncing back and forth off the stops if
constrained. To avoid this we need to damp the system so
kinetic energy acquired by the bar is more rapidly dissipated. Well,
nothing's more damp than water, so we add a water brake to
the arm which turns in a fixed reservoir. The resulting drag as the
balance arm moves is much greater than air resistance and frictional
losses in the fibre, and reduces the oscillations to a tolerable
degree.
The Gravitational Balance
“The time has come,” the Hacker said,
“To talk of many things:
Of plastic foam—and tuna cans—
Of chunks of lead—and string—
And how the force of gravity—
Will make the balance swing.”
So here's the sophisticated, high-tech, big science apparatus we'll
use to observe the subtle curvature of spacetime. An aluminium ladder
serves as the support from which the balance arm is suspended. Nylon
monofilament fishing line, as shown above,
is knotted to the middle of the third cross-beam at the back of the
ladder, one above the brace bearing the little white box, about which
more later. Using a ladder or similar movable support frame allows
setting up the balance in the middle of the room. This is important
because we are bending spacetime in the basement, in this
case an underground storage room at Fourmilab. Ground level is about
even with the ceiling of this room, about 45 cm above the top of the
ventilation window at the upper right of the picture. An underground
room is ideal because it minimises temperature variations and
vibration which might perturb the balance arm. Both walls shown in
this picture are sunk into solid limestone rock—if you set up the
balance near one of these walls, the gravitational field from all that
rock will mask that of the test masses, and the balance will assume a
“gravity gradient” position with one of the ends of the bar pointing
toward the wall, and will budge only slightly under the influence of
the test masses. With the bar in the middle of the room, the tidal
influence of the mass of the wall and the rock behind it is reduced to
a negligible value. The pipe on the wall at the right is part of the
serpentine pressurised hot water heating system; it was disabled to
prevent air currents from disrupting the balance arm. In fact, since
the room is underground, the heating system is rarely engaged, and
only in the depths of winter, never in June.
The Balance Arm and Cradle
The balance arm is a 5 × 5 × 30 cm bar of plastic foam, hacked from a
5 cm thick slab of packing material with a
Swiss Army knife. The
bar is suspended in a cradle made of insulated telephone wire.
The bar is held in its cradle by friction and the indentation made
in the soft plastic foam due to the weights at either end of the
bar; it's easier to adjust the bar for proper alignment this way
than if it were glued to the cradle.
The Support Fibre
The nylon monofilament that suspends the cradle is barely visible at
the top of the picture—it is fastened by a knot to a loop formed
into the cradle wires by twisting them. The monofilament is a very
fine “six pound test” (about 3 kg capacity) fishing line
manufactured in Japan; a 300 metre spool of it costs about US$9. The
masses which cause the bar to turn when a gravitational force acts
upon them are lead “sinkers” used by fishermen, each
weighing 169 grams. Two are placed on each end of the balance beam,
giving it a total weight of 676 grams. Be sure to place the weights
on both ends of the beam simultaneously so it doesn't topple, then
adjust the placement so the beam is horizontal. Nylon monofilament is
very elastic: when you put the weights on the beam the support line
will stretch and the beam will end up closer to the ground. You may
have to adjust the attachment of the line to the ladder (or other
support) or, as I did, twist the cradle wires to restore the beam to
the desired height. Finally, when you first hang the beam, it may take
some time to release stresses in the fibre remaining from the
manufacturing process and from its having been rolled onto a spool.
It's best to let the arm hang for a couple of days, free to turn, to
allow these initial stresses to equalise before attempting any
experiments with gravitation.
The Water Brake
The height of the beam is important because of the need for it to fit
properly with the water brake. If the beam is allowed to swing
freely, it will be terribly underdamped—once it starts to swing,
only air friction and the minuscule losses in the fibre will act to
stop it. This causes the beam to bounce around incessantly, masking
the steady influence of gravitation. The water brake dissipates the
energy of these unwanted oscillations precisely as an automotive shock
absorber does; the flap's motion does work on a viscous fluid, water
in this case, and deposits its energy in heating it.
The water brake consists of a flap which projects downward from the
balance arm (in this case, a piece of aluminium cut with scissors from
the tray of a “heat and eat” meal, fixed with white glue
into a slot cut into the bottom of the balance beam). The flap
projects into a reservoir (a tuna fish can) filled with water. A more
viscous fluid such as salad oil would provide greater damping and less
bouncing than water, but I opted for water since it's less icky to
clean up when the inevitable spill occurs and can be disposed of when
the experiment's done without a visit to the village recycling barrel.
If I were rebuilding the balance beam, I would use a longer and
narrower flap and/or a larger and deeper water reservoir. If
the flap is only slightly smaller than the inside diameter of the
reservoir, you have to be very careful that the flap and reservoir are
centred on the beam. Otherwise, the flap will touch the edge of
the reservoir and freeze the beam in place, as that frictional force is many
orders of magnitude greater than the gravitational force we wish
the beam to respond to. The water reservoir can be as large as
you like, as long as it doesn't interfere with placing the
test masses; the larger it is, the less you have to worry
about its being precisely centred.
Test Masses and Supports
Blocks of plastic foam support the test masses so their centre of
gravity is at the same height as the masses at the ends of the balance
beam, maximising the attraction. The foam also keeps the balls from
tending to roll away. The black rectangle, actually an inverted mouse
pad, serves as a background for the time display superimposed by the
video camera, rendering it more readable when images are reduced in
scale so movies download more rapidly.
Use the densest objects you can obtain for the ends of the balance
beam and as test masses: lead sinkers, steel balls, plutonium
hemispheres, etc. Density is important because the gravitational
force varies as the inverse square of the distance between the
centres of mass of two objects. With a dense substance, the
centre of mass is closer to the surface, so you can get the centres of
mass closer together and enhance the gravitational force. For
example, consider two pairs of one-kilogram spheres, the first made of
lead (density 11.3 g/cm³), the second of pine wood (density about
0.43 g/cm³), placed so the surfaces of each pair of spheres are
1 cm apart. A one kilogram lead sphere has a radius of 2.76 cm,
so the centres of mass are separated by 1+2.76×2, or 6.52 cm. A
one kilogram sphere of pine has a radius of 8.22 cm, by comparison,
so the centres of mass of the two pine spheres are 1+8.22×2 = 17.44 cm
apart. Taking the square of the ratio of these distances shows that
the gravitational force between the lead spheres is more than 7
times that of the pine spheres. Since attraction is linear by mass but
inverse square in distance, you're better off with a modest mass of
high-density material than a large mass of a substance with lesser density.
It's best to use a nonmagnetic material like lead for the weights
on the ends of the balance arm. The forces we're working with are
so small that if you use, for example, steel ball bearings on the
arms, you may end up accidentally reinventing the compass instead of
detecting the force of gravity.
The Spy Cam
“So what's the little white box on the back of the
ladder?”, you ask. Okay…. It's a BSR Model 500
surveillance camera which lets me observe the state of the experiment
as it runs. The Sony camcorder I use to make movies doesn't generate
video output while recording, so I can't use its video feed to monitor
what's happening. Popping into the room where the experiment's
running is a no-no—air currents from opening and closing the
door, not to mention walking around in the room could seriously
disrupt things. The BSR camera and accompanying 13 cm (diagonal)
monitor allows keeping tabs on what's happening in a non-intrusive
manner. I made a custom interface of the BSR camera/monitor cable to
Fourmilab's ubiquitous RJ-45 cabling, so I can place two BSR cameras
anywhere on the site and monitor either from anywhere else. At the
right is an image from the surveillance camera taken at the end of an
experiment, confirming that the balance beam has come to a stop
against the foam block supporting the mass at the top. The camera is
sensitive to infrared and includes infrared LEDs to illuminate nearby
objects, and has a microphone linked to a speaker in the monitor.
This makes it ideal for anxious parents who wish to monitor their
sleeping baby; spacetime hackers can use the infrared illumination to
view the balance beam without the thermal disruption of incandescent
lamps or direct sunlight. The storage room where I ran this
experiment has fluorescent strip lighting on the ceiling, and I
observed no detrimental effects from its being illuminated. Of
course, if the room you're using is equipped with that low-tech
miracle called a window, you can dispense with all this complexity.
Gravitation in Action
The following time-lapse movies (about 30 seconds per frame) show the
torsion balance responding to the gravitational field generated by two
740 gram competition pétanque balls. The picture at left shows the
camera angle employed in both movies. In each, the movie begins with
the bar stationary, in contact with one of the balls or the foam
supporting it. The balls are then shifted to the opposite corners,
where they attract the lead weights on the ends of the bar. The bar
then turns, slowly at first and then with increasing speed as it is
accelerated by the gravitational force growing as the inverse square
of the decreasing distance between the masses. The bar bounces when
it hits the stop on the other end, and finally, after a series of
smaller and smaller bounces as the water brake dissipates its kinetic
energy, comes to rest in contact with the closer ball or support.
This is the lowest energy state, at which the bar will always arrive
at the end of the experiment.
Movie 1
Movie 2
Pay no attention to the plastic robot ant—she's just curious.
It's a long story.
A Tide in the Affairs of Man
What we've demonstrated by these experiments is the universality of
gravitation; there is nothing special about the Earth that makes
objects fall toward it. Everything attracts everything else;
the Earth's attraction is greater simply because the Earth is
so much more massive than the objects we encounter in everyday
life. Only by canceling out the Earth's gravitation by means of
a torsion balance were we able to observe the gravitational
attraction between masses of less than a kilogram.
The universality of gravitation means that every object in the
universe is interlinked in a web of mutual attraction; the universe is
transparent to gravitation. The most distant galaxies exert a pull on
you, as you do upon them—immeasurably tiny to be sure, but
present just the same. From a practical standpoint, universality
means there's no way to shield your torsion balance from the
gravitational attraction of masses in its vicinity; you can only set
it up sufficiently far from other massive objects so the attraction of
the test masses predominates. One interesting massive object to
consider is yourself (I use “massive” only in the sense of
"possessing mass", not pejoratively; if you took it that way, perhaps
you should check out my on-line
diet book).
Using the gravitational force calculator
earlier in this document, we can compute the gravitational attraction
between the 338 gram mass at the end of the balance beam and the 740
gram test mass at the 14 cm distance when the beam is at the midpoint
between the masses to be 0.000085 dynes. Now suppose you're crouching
down in order to move the test masses, with your centre of gravity one
metre from the closer test mass, and that you weigh 65 kg. Plugging
these numbers into the calculator shows that your own gravitational
attraction on the nearer end of the beam is 0.000147 dynes, 1.7 times
as great as that of the test mass. Your actual influence on the
motion of the balance arm is less, however, since what matters is the
difference in force exerted on the masses at the two ends of
the balance arm. Since your centre of gravity is more distant than
the test masses, the difference is less.
Let's work it out. Assume the centres of gravity of the two masses on
the balance arm are 25 cm apart, and that you're crouching so the arm
makes a 45° angle with your centre of gravity, one metre from the
centre of the arm. The nearer mass is then 17.68 cm closer than the
more distant one and the difference in gravitational attraction (or tidal
force) on the two masses is the difference in attraction on a
mass 91.16 cm distant and one 108.84 cm away. The calculator gives
the attraction on the near end of the arm as 0.0001764 dynes and the
far end as 0.0001238 dyne, with a difference of 0.0000527 dynes.
Now recall that the force exerted by the test mass was 0.000085 dynes,
only 1.6 times as large, so even taking into account the reduced tidal
influence due to your greater distance, the force you exert on the
balance cannot be neglected. This makes it essential to remotely monitor
the experiment so your own mass doesn't disrupt it.
In practice, air currents due to your motion and resulting from convection
driven by your body's temperature being above room temperature may exert
greater forces on the balance arm than the gravitational field generated
by your mass. In any case, it's best to let the experiment evolve
on its own, observed from elsewhere.
Enlightenment Deferred: An Historical Speculation
Nineteen centuries elapsed between the death of Archimedes in
212 B.C. and the publication of
Newton's Principia in 1687. Given the philosophical
implications of Newton's theory, it's interesting to speculate what
might have happened had Archimedes discovered the universal nature of
gravitation.
To do this, he would have had to suspect that attraction was
universal, suggest an experiment to confirm this, and perform that
experiment, with results validating the hypothesis. Here is
information in Archimedes' possession which might have suggested the
universality of gravitation.
The Earth is a sphere. The shape of its shadow on the Moon
during a lunar eclipse demonstrated this, and was
confirmed by the next item. The assertion that “the
ancients thought the world was flat” is
nonsense—Columbus didn't “discover the world
was round”: he discovered that his own estimate of
the diameter of the world was wrong by a factor of two
compared to that available to Archimedes; if he hadn't
inadvertently discovered the New World, he and his
unfortunate crew would have died of starvation far from
the coast of China.
The approximate radius of the Earth. Around
250 B.C., by measuring the
difference in the angle of sunlight at noon on the June
Solstice, which illuminated the bottom of a well at Syene
(now Aswan) Egypt near the Tropic of Cancer, with the
length of the shadow cast on the same date and time by a
vertical pillar in Alexandria, a known distance to the
North, Eratosthenes determined the Earth's circumference.
Archimedes corresponded extensively with Eratosthenes and
other scholars in Alexandria, and knew of this result.
Archimedes himself calculated the value of π as between 3
10/71 and 3 1/7, with a mean value of 3.14185, allowing
accurate computation of the Earth's radius from the
circumference.
The approximate mass of the Earth. Assuming the Earth
to have the same density as common rocks such as
limestone (2.7 g/cm³) gives an estimate within a
factor of two of the correct value. The actual density of
the Earth is 5.52 g/cm³.
How to calculate with very large numbers.
In The Sand Reckoner
(Ψαμμίτης)
in 215 B.C., Archimedes
invented a positional number system which allowed writing
and calculating with arbitrarily large quantities, which he
demonstrated by calculating not only how many grains of
sand would fill the volume of the Earth, but how many
grains of sand would fill the entire universe (which the
Greeks estimated to be about one light year in diameter).
The latter number, about 1063, is comfortably
larger than any of the quantities associated with
gravitation.
The existence of electrostatic and magnetic forces which
appeared to act at a distance. The inverse square
behaviour of these forces was not
known in antiquity, however.
Suppose then that, given these facts, Archimedes embarked upon the
following chain of reasoning.
Objects fall, not in a fixed direction, but toward the
centre of our world. If they fell in a fixed direction,
if I dropped a rock down a well in the south of
Egypt and a well in Syracuse, separated by a
substantial percentage of the world's circumference,
one would hit the wall of the well before striking the
bottom. This doesn't happen, so objects fall everywhere
toward the centre of the world.
Why does the world attract falling objects? Is there
something special which endows it with
this property? Yet the world seems to be made of the same
substances as everything else. What is the difference
between the world and a rock?
Εὕρηκα,
εὕρηκα!
The world is much larger than
a rock! Perhaps every object attracts
every other. We only feel the world's attraction
because it is so large.
But if this is so, might the celestial bodies be objects
no different from the world and its inhabitants, and subject
to the same forces?
If attraction is universal, might an artificer
be able to build a device to show it?
Such a device must be isolated from falling down. Perhaps
a horizontal balance, free to turn in either direction,
with weights at each end to be attracted to objects
in their vicinity….
The Archimedes Apparatus
It seems plausible, then, given the knowledge at hand and a chain of
inference which, in retrospect at least, appears straightforward, that
Archimedes could have suspected the universality of gravitation. But
could he have demonstrated it? Unlike many scholars in
ancient Greece who contented themselves with philosophical arguments,
Archimedes was an intensely practical man, renowned as a military
engineer as well as a mathematician and philosopher. His laws of the
lever and buoyancy were tested experimentally, and so we should expect
he would subject any inference about gravitation to experimental
confirmation. Now that we've succeeded in bending spacetime in the
basement with common household materials of the late 20th century,
let's see if the experiment can be done using only materials
Archimedes might have employed.
Let's try to redesign the torsion balance using only materials
available in antiquity.
The Balance Arm
Instead of plastic foam, we use a strip of pine wood,
2 cm wide, ½ cm high, and 30 cm long.
Notches are cut in the edges of the beam near each end to
secure the support cradle. For masses at the ends of the
balance beam we may continue to use lead, which was produced
in Egypt in the 2nd millennium B.C.
and in Europe no later than the 6th century B.C. As the discoverer of specific
gravity, Archimedes would understand the merit in using the
densest substance available. (Gold, almost twice as dense as
lead, would be an even better choice. Perhaps King Hiero of
Syracuse, grateful to Archimedes for exposing the goldsmith
who adulterated the gold in his crown with silver, might have
contributed gold weights for the balance beam, thereby taking
the first small step down the road to government-funded Big
Science. Wishing to remain in the domain of basement science,
we shall forgo royal subsidies and soldier on with lead.)
The Cradle
To support the balance arm, we substitute twine
made of vegetable fibre for telephone wire. Actually,
since copper was known for thousands of years before
the Greeks, a lightweight copper cradle could have been
made, but it would have been more work to fabricate
and has no advantage compared to the twine. Thread,
string, and rope were made from a variety of natural
fibres by all ancient cultures.
The Support Fibre
The support fibre is the most difficult component to
replace with a 3rd century B.C. analogue. Nylon
monofilament so closely approaches the ideal of a
massless support free of torsional resistance that
doing without it requires experimenting with a variety
of alternatives and compromising with the shortcomings
of whatever is selected. I finally settled on a very thin
vegetable fibre support “peeled” from a piece of
twine by unwinding it. The fibres you find in rope or
twine are a variety of lengths—you have to separate them
and then select individual fibres long enough to support
the balance arm. To obtain sufficient strength, I used
four separate fibres selected from the twine. If the
rope or twine has been twisted or braided, you'll have
to let the fibre hang for an extended period of time (three
or four days at least) to release its internal stresses.
In choosing and using any natural fibre support, you have to
approach the project with a willingness to learn by trial and
error and a great deal of patience. Each kind of fibre has
its own “personality”, and the quirks can take some time to
understand. For example, many plant and animal fibres are
sensitive to moisture—if a summer thunderstorm increases the
relative humidity from 50% to 99% in the space of an hour,
your balance arm may start to swing wildly as the fibre
absorbs moisture from the air. Further,
plant fibres tend to tear, both under tensile stress and
when twisted. This can cause your balance beam to
“spontaneously” shift to a different equilibrium point or,
after having been displaced, return to a different location
than the starting point.
Would Archimedes have appreciated the importance of choosing a
supple and well-behaved support fibre? I think so. From the
radius of the Earth, which he knew, and assuming its density
to be the same as rocks such as limestone (about half the
actual density of the Earth), the ratio of the Earth's
mass to that of whatever test masses were employed could be
estimated within a factor of two. Making the simplest
assumption (which has the additional merit of being correct)
that attraction is proportional to mass, it is clear that
the force acting on the masses at the ends of the balance
arm is minuscule, so a fibre which offers the least possible resistance
to twisting should be employed.
Test Masses
Lead or gold (Monarch! Archimedes is doing natural
philosophy in the bathtub again!) test masses would be
preferable, but to show how robust this experiment is I opted
for a couple of rocks—two kilogram paving stones like those
which border every highway in Switzerland, where roads are so
built to last that Julius Cæsar would shake his head in
admiration.
No Water Brake
After experimenting with a variety of vegetable fibres,
I decided to proceed without a water brake for this
experiment. The balance arm is more prone to oscillation,
but the friction in the support fibre, much greater than
in synthetic nylon monofilament, damps the oscillations
adequately. Allowing the lead weights on the balance arm
to collide with the stone test masses also dissipates
substantial energy, further reducing the need for a brake.
Unreconciled Residua
I didn't bother to replace the aluminium ladder with a support
Archimedes might have used. Any carpenter could fashion a
more suitable replacement for the ladder. In fact, a wooden
saw-horse would have been better for all these experiments,
but I don't have one and didn't feel like making one from
various pallets and spare lumber in the High Bay.
The concrete floor would also seem strange to Archimedes, but
it is irrelevant to the experiment. A smooth stone floor, as
existed for millennia before, would produce identical results.
The Archimedes Experiments
The following movies demonstrate universal gravitation with an apparatus
which, as argued above, could have been conceived by Archimedes and
built from materials he could readily obtain.
Movie 3
Movie 4
The Archimedes Enlightenment?
Suppose this had happened. Consider how easily it could have.
Would such a discovery in Archimedes' time have had an impact
comparable to Newton's or, occurring in a very different social
and intellectual milieu, would it have been regarded as no more than
a curiosity? How might human history have played out had the
Enlightenment begun 1900 years before Newton?
References
Click on titles to order books on-line from
Archimedes. The Sand Reckoner. English
translation in Newman, James R.
The World of Mathematics.
Mineola, NY: Dover, 2003.
ISBN 978-0-486-43268-7.
Cavendish, Henry. “Experiments to determine the density
of the Earth”. Philosophical Transactions of the Royal
Society of London, Part II (1798), pp. 469–526.
Eötvös, Loránd von.
“Über die Anziehung der Erde auf verschiedene
Substanzen.”
Math. Naturw. Ber. aus Ungarn 8, 65–68 (1889).
Eötvös (his Hungarian surname is pronounced like
“ut-vush” in English) improved the original Cavendish
torsion balance to its modern form and used it to
test the equivalence principle (in his final publication on
the topic in 1922) to better than one part in 5×1010.
Feynman, Richard P., Robert B. Leighton, and Matthew Sands.
The Feynman Lectures on Physics Vol. 1 (Chapter 7).
New York: Basic Books, [1963] 2011.
ISBN 978-0-465-02382-0.
Hogben, Lancelot.
Mathematics for the Million (Chapter 12).
New York: W.W. Norton, [1937, 1967], 1993.
ISBN 978-0-393-31071-9.
Icikovics, Jean-Pierre and Nicolas Journet, eds.
“Archimède.”
Les Cahiers de Science&Vie: Les
pères fondateurs de la science 18 (December 1993).
ISSN 1157-4887.
Newton, Isaac. Philosophiæ naturalis principia
mathematica. London: Streater, 1687. English
translation by A. Motte, revised by A. Cajori,
Sir Isaac Newton's Mathematical Principles of Natural
Philosophy and His System of the World, 1729. A
modern edition is published by the University
of California Press as
ISBN 978-0-520-29088-4.
Rucker, Rudy.
Mind Tools.
Boston: Houghton Mifflin, 1987.
ISBN 978-0-395-46810-4.
Trifonov, D. N., and V. D. Trifonov.
Chemical Elements: How they Were Discovered.
Translated from the Russian by O. A. Glebov and I. V. Poluyan.
Moscow: Mir Publishers, 1982.
The stark difference in the strength of the electromagnetic and
gravitational forces is evident in the picture to the left. The
bright square in the jaws of the pliers is a 4 mm cubical magnet. It
is lifting a spherical steel pétanque (a lawn bowling game popular in
southern France and Switzerland) ball which weighs 550 grams.
Consider this picture in the following way: we're pitting our valiant
little magnet, with a volume of 0.064 cubic centimetres, weighing
less than one gram, pulling up with the electromagnetic force,
against the entire Earth, pulling down with gravity. And the
winner is…the magnet. A one gram magnet (I'm being generous: I
don't have a scale on which its weight reads other than zero)
out-pulls the Earth, which weighs 5.9736×1027 grams and has
a volume of more than 1027 cubic centimetres.
The balance arm is a 5 × 5 × 30 cm bar of plastic foam, hacked from a
5 cm thick slab of packing material with a
The height of the beam is important because of the need for it to fit
properly with the water brake. If the beam is allowed to swing
freely, it will be terribly underdamped—once it starts to swing,
only air friction and the minuscule losses in the fibre will act to
stop it. This causes the beam to bounce around incessantly, masking
the steady influence of gravitation. The water brake dissipates the
energy of these unwanted oscillations precisely as an automotive shock
absorber does; the flap's motion does work on a viscous fluid, water
in this case, and deposits its energy in heating it.
The water brake consists of a flap which projects downward from the
balance arm (in this case, a piece of aluminium cut with scissors from
the tray of a “heat and eat” meal, fixed with white glue
into a slot cut into the bottom of the balance beam). The flap
projects into a reservoir (a tuna fish can) filled with water. A more
viscous fluid such as salad oil would provide greater damping and less
bouncing than water, but I opted for water since it's less icky to
clean up when the inevitable spill occurs and can be disposed of when
the experiment's done without a visit to the village recycling barrel.
“So what's the little white box on the back of the
ladder?”, you ask. Okay…. It's a BSR Model 500
surveillance camera which lets me observe the state of the experiment
as it runs. The Sony camcorder I use to make movies doesn't generate
video output while recording, so I can't use its video feed to monitor
what's happening. Popping into the room where the experiment's
running is a no-no—air currents from opening and closing the
door, not to mention walking around in the room could seriously
disrupt things. The BSR camera and accompanying 13 cm (diagonal)
monitor allows keeping tabs on what's happening in a non-intrusive
manner. I made a custom interface of the BSR camera/monitor cable to
Fourmilab's ubiquitous RJ-45 cabling, so I can place two BSR cameras
anywhere on the site and monitor either from anywhere else. At the
right is an image from the surveillance camera taken at the end of an
experiment, confirming that the balance beam has come to a stop
against the foam block supporting the mass at the top. The camera is
sensitive to infrared and includes infrared LEDs to illuminate nearby
objects, and has a microphone linked to a speaker in the monitor.
This makes it ideal for anxious parents who wish to monitor their
sleeping baby; spacetime hackers can use the infrared illumination to
view the balance beam without the thermal disruption of incandescent
lamps or direct sunlight. The storage room where I ran this
experiment has fluorescent strip lighting on the ceiling, and I
observed no detrimental effects from its being illuminated. Of
course, if the room you're using is equipped with that low-tech
miracle called a window, you can dispense with all this complexity.
The following time-lapse movies (about 30 seconds per frame) show the
torsion balance responding to the gravitational field generated by two
740 gram competition pétanque balls. The picture at left shows the
camera angle employed in both movies. In each, the movie begins with
the bar stationary, in contact with one of the balls or the foam
supporting it. The balls are then shifted to the opposite corners,
where they attract the lead weights on the ends of the bar. The bar
then turns, slowly at first and then with increasing speed as it is
accelerated by the gravitational force growing as the inverse square
of the decreasing distance between the masses. The bar bounces when
it hits the stop on the other end, and finally, after a series of
smaller and smaller bounces as the water brake dissipates its kinetic
energy, comes to rest in contact with the closer ball or support.
This is the lowest energy state, at which the bar will always arrive
at the end of the experiment.
It seems plausible, then, given the knowledge at hand and a chain of
inference which, in retrospect at least, appears straightforward, that
Archimedes could have suspected the universality of gravitation. But
could he have demonstrated it? Unlike many scholars in
ancient Greece who contented themselves with philosophical arguments,
Archimedes was an intensely practical man, renowned as a military
engineer as well as a mathematician and philosopher. His laws of the
lever and buoyancy were tested experimentally, and so we should expect
he would subject any inference about gravitation to experimental
confirmation. Now that we've succeeded in bending spacetime in the
basement with common household materials of the late 20th century,
let's see if the experiment can be done using only materials
Archimedes might have employed.
