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HOW IS THE MASS OF THE
EARTH
DETERMINED?
How is the mass of the Earth determined?
Newton, Galileo, Henry Cavendish, and Eratosthenes contributed
to this amazing calculation.
In Newton's equation, F is the gravitational force, G is a
constant of proportionality, M and m are the two masses exerting
the forces, and r is the distance between the two objects.
-
G was calculated by Henry Cavendish in 1798, and was
determined to be 6.67 x 10-11
m3/(kg sec2).
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Also needed is
Newton's second law of motion, F=ma, where F is the
force applied to an object, m is the mass of the object, and a
is its acceleration due to the force.
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Galileo determined that the acceleration due to the force of
gravity of Earth was a constant equal to 9.8 m/sec2
near the surface of the Earth.
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Lastly, you need to know the radius of the Earth; this was
first calculated by the Greek Eratosthenes thousands of years
ago (by comparing shadows in wells during the summer solstice
about 230 B.C.).
CALCULATING THE
MASS OF THE EARTH
1. F = GmM/r2 = ma,
where F
is the gravitational force, G is the gravitational constant, M is
the mass of the Earth, r is the radius of the Earth, and m is the
mass of another object (near the surface of the Earth).
2. GM/r2= a
(The m's
canceled out.) Now solve for M, the mass of the Earth.
3. M = ar2/G,
where a=9.8m/sec2, r=6.4 x 10
6m, and G=6.67 x 10-11m3/(kg
sec2).
4. M = 9.8 x (6.7 x 106)2/6.7
x 10-11 = 6.0 x 1024
kg
THE PEOPLE BEHIND THE
CALCULATION
CAVENDISH, HENRY
Henry Cavendish (1731-1810) was an English chemist and
physicist. Cavendish discovered that hydrogen gas was a
substance different from ordinary air (whose components he
analyzed), described the composition of water (hydrogen and
oxygen) and other important works. Cavendish was the first
person to determine Newton's gravitational constant
gravitational constant and accurately measured of the
Earth's mass and density. |
ERATOSTHENES
Eratosthenes (276-194 BC) was a Greek scholar who was the
first person to determine the circumference of the Earth. He
compared the midsummer's noon shadow in deep wells in Syene
(now Aswan on the Nile in Egypt) and Alexandria. He properly
assumed that the Sun's rays are virtually parallel (since
the Sun is so far away ). Knowing the distance between the
two locations, he calculated the circumference of the Earth
to be 250,000 stadia. Exactly how long a stasia is is
unknown, so his accuracy is uncertain. He also accurately
measured the tilt of the Earth's axis and the distance to
the sun and moon, and devised a method for determining the
prime numbers up to a given number (the Sieve of
Eratosthenes). Eratosthenes made numerous contributions to
the sciences and arts in many fields, including geography,
mathematics, astronomy, chronography (calendars), music, and
literature. Eratosthenes was a brilliant all-around scholar;
although not the top expert on any topic, he was well-versed
in all subjects, and therefore nicknamed "Beta" (which is
the second letter of the Greek alphabet). |
GALILEI, GALILEO
Galileo Galilei (1564-1642) was an Italian mathematician,
astronomer, and physicist. He was the first person to use a
telescope to observe the skies (in 1609). Galileo discovered
four moons of
Jupiter, observed the phases of
Venus, studied sunspots, and discovered many other
important phenomena. After publishing the many discoveries
he made using his telescope, including the motion of the
Earth around the Sun (the
Copernican System), Galileo was accused of heresy by the
Inquisition (in 1633). |
NEWTON, ISAAC
Isaac Newton (1642-1727) was an English mathematician and
physicist who invented calculus (simultaneously, but
independently of Leibniz), formulated the laws of
gravitation and mechanics, investigated the nature of light
(he discovered that sunlight is made of light of different
colors). |
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