One last estimate of the geometry of
the whole Earth, and its implications
for the composition of the planet.
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Remember that the current
estimate for the Earth’s meridional
(through the poles) circumference
is 40,008km, close to what we
estimated.
The equatorial circumference is
40,074km.
The average circumference is
approximately 40,060km. We will
use this value.
The circumference of a circle (or
sphere) is its diameter x ∏ , so
Earth’s diameter is ~12,752km.
The radius of a circle (or sphere) is
half the diameter, so Earth’s radius
is 6,376km.
The volume of a sphere is 4/3∏r3.
Our goal in this exercise is to estimate Earth’s
density, which is usually expressed as g/cm3.
Therefore we need to convert the km to cm. This
makes for doing arithmetic with very big
numbers now, but makes our final step MUCH
easier.
1km = 1000m, and 1m = 100cm, so our
conversion requires us to multiply the radius in
km by 100000 (1000x100)
This gives us a radius of about 6.376 x 108cm
Plugging this value into the sphere-volume
equation above tells us that Earth’s volume is
about 1.08575832413598 x 1027 cm3.
This gives us the volume we need for a density
calculation, but we still need a mass – a number
of grams.
Obviously there are
some options that are
not open to us.
Isaac Newton realized
that he could determine
the mass of Earth by
seeing how it interacted
gravitationally with
other bodies.
This is presented on the
next 2 pages if you are
interested, but the math
is a bit more involved
than what we’ve done
before, so if you prefer
to skip it, then just take
professor Newton’s
word for it.
Xg
Newton didn’t really discover gravity. The first guy to drop a big rock on his toe discovered gravity. What Newton
discovered was a couple of very important details about how gravity works. At least at big scales.
The first is what we call the gravitational constant (usually signified by capital G). This is a number that enters into
every calculation of the gravitational attraction between objects and lets us calculate what the strength of their
attraction is. The current estimate of the value of that constant is: G = [(6.673x10-11m3)/(kg*s2)].
The force of a gravitational attraction (F) is [(G x the mass of one object x the mass of the other object)/(the distance
between them2), or F=[G(m1*m2)]/d2.
Newton also had the great insight that gravity is an acceleration of objects. Things fall with an ever-increasing
speed, at least in a vacuum. It was well known even before Newton that a force generated during motion is equal to
the mass of the moving object x its acceleration. The acceleration of gravity is usually represented by a small g, and
mass by a small m. On Earth, g = ~9.8m/s2.
So the force of gravity affecting a ball, for example, as it falls to Earth is F=mg. Notice that we now have two
equations for the value of F. If we substitute mg for F in the first equation, we get: mg = [G(m1*m2)]/d2.
Either m1 or m2 is the mass of our hypothetical ball [call it m(b)], the other is the mass of the earth [call it m(e)] to
which the ball is falling. So if we re-label, we get m(b) g = [G(m(b)*m(e))]/d2.
We are interested in the value of m(e) , so we can rearrange and get m(e)=[(m(b)gr2)/(G m(b))]. Cancelling we find (to
our relief) that the other mass (m(b)) cancels and we get: m(e) = gr2/G. We already know what g and G are.
Newton took this distance to be between the centers of mass of the two objects. This is not perfectly accurate and
we will see how to use that inaccuracy in the second geology course, but for now this will give us a good enough
value for the distance we need. Remember that the distance between a falling ball (just before it hits the ground)
and the center of the Earth is, on average, about 6,376km – the radius of the Earth.
Newton’s work lets us derive a formula for the mass of the Earth, if we know a few things.
First, we need Newton’s gravitational constant: G = [(6.673x10-11m3)/(kg*s2)].
Second, we need to know the local force of gravity, which is g = ~9.8m/s2
Finally we need to know the radius of Earth, which we’ve already calculated to be ~6,376km (which we
must convert to meters to match the units in the other values). 6,376km = 6,376,000m or 6.376 x 106 m.
The equation we need is:
m(e) = gr2/G.
(m(e) is read “mass of earth”.)
Substituting values for variables gives us:
m(e) = [(9.8m/s2)*(6.376 x 106 m)2]/[(6.673x10-11m3)/(kg*s2)]
= [(9.8m/s2)*4.0653376 x 10 13 m2*(kg*s2)]/ (6.673x10-11m3)
(Notice that we can cancel the s2 terms. Soon we’ll get rid of all the m terms too.)
= (3.984030848m3kg)/ (6.673x10-11m3)
(As promised, we now have m3 terms in numerator and denominator, and can cancel them.)
= ~5.97 x 1024 kg. This is about what you’ll find if you look it up in a book.
If you are just rejoining us, the short version of Newton’s insight is that he can mathematically
determine the mass of earth to be about 5.97 x 1024 kg.
Density is a material’s mass divided by a unit volume, generally expressed as g/cm3 so we need to
multiply by 1000 to convert kg to g. This is easy with scientific notation: the Earth’s mass is 5.97 x
1027 g.
Now recall that we determined a few pages back that the volume of Earth is about
1.08575832413598 x 1027 cm3. Notice that both the mass and the volume have “1027“ tacked
onto the end. Great news -- these will cancel!
Earth’s overall density is therefore roughly :
(5.97 x 1027 g)/(1.08575832413598 x 1027 cm3) = ~5.5 g/cm3, a value you can
check in an astronomy book.
Now we have a problem. What is it? (HINT: What is Egypt made of?)
Egypt is made, for the most part, of “granite”, sedimentary rocks
(sandstone, limestone, and mudstone), loose sediment (sand and
mud), and a little basalt.
GRANITE:
SANDSTONE:
LIMESTONE:
BASALT:
Very hard
Very hard and brittle
Soft
Very hard
Light color
Light color
Light color
Dark color
Coarse Crystals
Medium Crystals
Crystals variable
Very Fine Crystals
Uncommon or
inaccessible.
Common
Common in most areas.
Rare.
Moderately dense:
~2.5-2.9 g/cc
Lower density:
~1.6-2.6 g/cc
Lower density:
~1.5-2.5 g/cc
Very dense:
~3.0 g/cc
Used in Egypt for
construction/decoration
of important buildings
and statues.
Used in Egypt for
construction when
limestone was not
available.
Used in Egypt for
construction of
buildings and for
statues.
Used in Egypt for
decoration of important
buildings and for
statues.
Density of Whole Earth = ~5.5 g/cm3. Absolutely nothing in Egypt is as dense as that!
How can the Earth be denser than
any of the rocks we know?