Ask A Scientist© Physics Archive

name         Meredith
status       student
age          30s

Question -   I am embarrassed to ask this, but I cannot think of one
physics equation that would say that gravity is anyway related to the
rotation of the earth. Yet all my science students think that the earth
has gravity because it spins. I want to correct them, but then in the
back of my mind, I get nervous, and think maybe I'm wrong. It's driving
me crazy. Would the earth's gravity still be 9.8 m/s per second if the
earth kept revolving around the sun, but stopped spinning?
--------------------------------------------------------------------------
    Why are you embarassed?  You are correct.  You can't think of any
physics experiment that would demonstrate that gravity is related to
rotational motion because it isn't.  If the Earth were not rotating
relative to the stellar background, its gravitational field would be
unchanged.  Mind you, it IS true that the acceleration ``due to
gravity'' would be very slightly more than 9.8 m/s^2 anywhere but at
the poles because there would no longer be any centrifugal force
reducing the effect of gravity.  But that isn't what your students are
saying anyway.  (And we are ignoring any issues of general relativity
here, as they are weensy corrections.)
    You probably want a counter-argument to throw their way.  Let's
see.  You need first of all to convince them that the gravitational
attraction between two bodies depends on the properties of both
(e.g. the mass of both bodies appears in in Newton's law of
gravitation).  Point out to them that the weight of any small body,
which is the gravitational attraction between it and the Earth, varies
with its size.  Having two students step on the nearest scale will
verify this.
    Now note that they have assumed that the gravitational attraction
between a large body (the Earth) and a small body (a person or object
on the Earth's surface) depends on the spin of the LARGE
body. (Because if the Earth were not spinning, there would be no such
force, according to them.)
    And ask them why, in that case, the gravitational force between the
small body and the large body does NOT depend on the spin of the SMALL
body.  (You can easily demonstrate that the time it takes a spinning
object to fall is identical to the time it takes a nonspinning object
to fall.)
    You can also perform the thought experiment of considering two
bodies near each other, one spinning planet, one nonspinning small
object.  Gradually you transfer mass from one to the other, keeping
the angular momentum constant (which means the planet starts spinning
faster and faster, incidentally, but perhaps irrelevantly).  After a
while the object will become as large as the planet was, and the
planet as small as the object was.  Is the gravitational attraction
still due to the (very quickly) spinning ``micro''planet left over?
If so, then we should see that two spinning objects -- if necessary
very quickly spinning objects -- should attract each other more
strongly then they do when not spinning.  And this is not, of course,
observed.
    If their arguments are vaguely based on centrifugal forces, ask
them why gravity does not vary near the poles, where you are going
around the Sun once a year, but not being flung around in a circle
every 24 hours.
    I hazard that your students are suffering from a peculiarly
American malady -- the tendency to confuse proximity to causation.
The Earth does two interesting things as far as human beings are
concerned: it spins, giving us day and night, and it exerts gravity,
giving us up and down.  (I suppose a case can be made for a third: it
goes around the Sun tilted, which gives us summer and winter.)  It is
human, or perhaps more correctly American human nature, to assume that
because the two phenomena occur on the same object one must cause the
other.  Gravity is the more mysterious fact, more in need of
explanation, so it is assigned to be the effect, and spin assigned to
be the cause.
    The root cause is that American students, in particular, find it
difficult to cope with the state of ``not knowing''.  They need to
feel they know, even if they can't possibly given the evidence
available to them.  They are happier feeling they know, and later
finding themselves wrong, than in feeling ignorant, and later being
enlightened.  It's an unfortunate tendency, which tends to hamper
their scientific abilities, since a good scientist needs to cherish
the feeling of ``not knowing'' as long as possible.  It is only in
this mental silence, free of unjustifiable hypothesizing, that the
truth, which always starts out as a tiny voice indeed, can be heard.
    Hence you might do them a favor if you explore with them first WHY
they feel there is a connection between spin and gravity.  If the
answer is ``just because'' or ``it seems sensible'' or something
equally data- and logic-free, then you can do them some good by
discussing how very dangerous trusting that kind of unsupported
evidence-free ``hunch'' can be, not only scientifically, but in other
areas.  The tendency to judge too hastily and on too little actual
data is a particularly unfortunate American vice.  You can point out
to them that the very best and most successful scientists and
inventors are those who are the least inclined to think they know the
answer before they are absolutely, positively, convinced-by-mountains-
of-data sure they know.  In science plausible hypotheses are a dime
a dozen, but good, fact-supported theories are rare gems indeed. 
Grayce
=====================================

Well first of all there is no need for embarrassement, your question is
quite good and shows insight into gravity, due to mass and due to
rotation.
Having said all of that, lets analyze the question.
The acceleration due to gravity on the is as you not 9.8 m/s2,
this is from the mass of the earth and the rotation.
Gravity due to the earth's mass is
G(Gravitational Constant) x Mass of Earth/(radius of earth)squared.
Sorry, I can't get sub or superscripts on e-mail.
The gravity due to the rotation is simply the centripetal acceleration,
v squared/Radius, we will do this at the equator-since I don't have to
mess with sines or cosines. The velocity is ( 2 pi Radius)squared/time
or about 2500 meters/sec--square this and divide by the Radius of the
earth, 2500/(21,000 meters) or around 0.1 m/s2. This is around 100 times
smaller than the acceleration due to the masses...
I hope that I didn't mess up any numbers, it's Sunday.
H. Myron
===================================

Dear Meredith,
You are correct. Gravity is a universal property
of all objects and is not considered in
ordinary
Newtonian mechanics to have anything to do with the
state of motion. So the number 9.8 applies whether
the earth is spinning or standing still.

Best regards,
prof. topper
======================================
The acceleration due to gravity is indeed 9.8 m/s/s (g) as a result of the
size of the mass of the earth pulling equally on every kg of mass on its
surface.  This number becomes smaller as the distance increases between the
center of the earth and an object.  It actually decreases by the square of
the increase in distance meaning that the effect of gravity falls quickly.
Even Mars does indeed pull on you and me - gravity never drops to zero
because you can never reach zero by dividing a number - but the force is so
small as to never be measure.

Back to the earth... Although I have never seen the calculation, I expect we
could prove (and maybe another reply will do it for you) that the effect of
the rotation of the earth lessens g at the equator and causes it to be
greater at the poles.

If anything, the spin of the earth lessens the effect of gravity and it
definitely does not cause it.  Perhaps using the pull of the moon as an
example would help make this clear.  It is removed from the surface of the
earth and yet has a considerable effect on the oceans.  You might also look
up the work of Henry Cavendish and Philipp von Jolly, both of whom used a
very sensitive balance and a large lead mass to measure gravity.

Good luck with your young and impressionable minds.

Larry Krengel
=============================================

        A common misconception is that anything spinning has gravity due to its
spin.  Newtonian Gravitation has to do with a field that is created by the
mass that is in it.  The greater the mass, the greater the attractive
force.  Sir Isaac Newton found:
        F = G(m1 * m2)/(r^2)
to be the gravitation force of attraction between two bodies.  He used
calculus to find his results for a uniform spheroid.  The earth and moon
are good approximations of sphereoids.  The constant of proportionality in
the equation, G, was measured years later by Lord Cavendish.  Cavendish, in
an ingenious experiment, found:
        G = 6.67 * 10^-11 N m^2/s^2
This is a very tiny number, but is very important in astronomical
calculations where the masses of the bodies are very large.  You may wish
to tell your class about this experiment.  The experiment does not involve
any spinning of bodies.

        The gravitational field intesity at any point in space is found by the
slope of a Force versus Mass graph.  Using a spring scale, measure the
amount of force different masses exert at a given point in space.  At the
earth's surface, the gravitational field intensity is about 9.8 N/kg.  By
using Newton's second law, we find that the downward acceleration due to
gravity near the earth's surface is about 9.8 m/s^2.  Go elsewhere in space
and this number will differ.  If you go to a region of lower mass, say the
moon, the gravitational field intensity is lower (1.6 m/s^2).  Measuring
the local gravitational field intensity is easy to do in the classroom with
spring scales (callibrated in Newtons) and a set of hooked masses.


        Another common misconception about gravity is that there is no gravity in
space.  They see that the astronauts are weightless and think that gravity
is not present.  The Space Shuttle would continue in a straight line
forever if it were not for the gravitational field of the Earth pulling the
craft into orbit.  (see Newton's First Law of Motion)  Your weight is the
force of support pushing up on you.  If you are not supported by anything,
you will be accelerated in the gravitational field.  Stand on a table with
a spring scale hand.  Hang a mass from it.  Note that there is a
measureable support force on the object at rest.  Now, while holding this
system, jump off the table.  While you are falling, you are weightless (and
so is the mass on the scale).  Clearly, the force due to gravity was
working on you (you accelerated), but you were weightless due to the lack
of a support force.  Please note that your mass never changed.
        Throw a ball, lightly, horizontally.  It falls a meter away.  Throw the
ball a bit harder, and it strikes the ground a bit farther away.  Is the
ball weightless while in flight?  (Yes.)  Is the ball under the influence
of the Earth's gravitational field? (Yes, it is falling)  What would happen
if you could throw the ball fast enough that its path would match the
curvature of the earth?  The ball would be constantly falling . . . around
the earth.  That is what the astronauts are doing:  falling around the
earth.  They are high enough in the atmosphere where air friction is so
extremely small that they do not burn up.  They are weightless since they
have no support force.  They are in the earth's gravitational field since
they are always falling.

Good question!  Thanks for using NEWTON BBS.
---Nathan A. Unterman
================================================

The force of gravity is related to the earth's MASS, not its rotation.  Also,
the value of the force of gravity is related to the separation between the
two masses used to measure the force (the earth is usually one of the
masses).  The gravitational acceleration constant depends on where on the
earth's surface it is measured but the average value of 9.8 m/(second
squared) is accurate enough for most work.

In addition, the force of gravity is slightly offset by the 'centripetal'
force or the inertial effect of the rotation.  This effect would be greatest
at the equator and zero at the poles.  If the earth were not rotating the
measured gravitational acceleration constant would be slightly larger because
this compensating inertial action would not be present.
Bradburn
================================================

Your students have part of a right idea, but it is leading them in entirely
the wrong direction.  They are probably thinking about science fiction
books and movies in which space stations generate "artificial gravity" by
spinning.  This is due to the commonly misunderstood centripetal
acceleration, which is perceived as a "centrifugal force."  The centripetal
acceleration is what makes objects move in curved paths, instead of in
straight lines.  when an object moves in a cirle, the centripetal
acceleration is constantly pulling it toward the center of the circle.
This is misinterpreted as a force pulling away from the center of the
circle, but that "force" is actually just the tendency of an object to keep
moving in a straight line.

If it weren't for gravity, the spinning of the earth would cause us all to
fly off into space, not stick to the earth.  This is just like swinging a
weight on a string in a circle;  if you let go, it sails off in the
direction it was moving when you let go.  In fact, the spinning of the
earth works against gravity.  This effect is small everywhere, greatest at
the equator, and zero at the poles.  So if the earth stopped spinning,
gravitational acceleration would increase everywhere on earth except at the
poles.

So, go ahead and correct your students.  Maybe you should even have them do
the experiment of swinging a weight on a string. (outside.)  Let them see
for themselves that the spinning of the earth does not pull us inward.
Then maybe they'll be convinced.


                Richard E. Barrans Jr.
====================================================

The Earth does not have gravity because it spins.  The gravitational
attraction between the Earth and a person on the Earth depends only on
the mass of the Earth, the mass of the person, and the distance between
the Earth's center of mass and the person's center of mass.

However, the force a person's feet exert on the ground (i.e., the
person's weight) does depend on the Earth's spin.  What if the Earth
started spinning faster and faster?  At some speed, the Earth's
gravitational force would not be enough to keep us on the surface.
We'd fly off into space because the acceleration (v^2/r) required to keep
us moving in a circle of the Earth's radius at the rotational speed of the
Earth would be greater than the acceleration of Earth's gravity.

The figure 9.8 m/s/s is a measured value, and it includes the effect of
the Earth's spin as a small correction.  If the Earth were not spinning,
the measured value would be .034 m/s/s larger.  If the Earth were spinning
so fast that a day took only five minutes, then the gravitational acceleration
would be spent entirely to keep us moving in a circle, and we'd all feel
weightless.

Here are the numbers and equation I used:
radius of Earth: 6.4x10^6 meters
speed of object at the equator: 465 m/s
acceleration required to keep an object moving at speed v in a circle of 
radius r:
v^2/r

Tim Mooney
Beamline Controls & Data Acquisition Group
Advanced Photon Source, Argonne National Lab
=======================================================