DETERMINATION OF THE EARTH’S RADIUS In 200 B.C., the size of the Earth was calculated to within 1% accuracy! Eratosthenes used Aristotle's idea that, if the Earth was round, distant stars in the night sky would appear at different positions to observers at different latitudes. Eratosthenes knew that on the first day of summer, the Sun passed directly overhead at Syene, Egypt. At midday of the same day, he measured the angular displacement of the Sun from overhead at the city of Alexandria - 5000 stadia away from Syene. He found that the angular displacement was 7.2 degrees - there are 360 degrees in a circle, making 7.2 degrees equivalent to 1/50 of a circle. Geometry tells us that the ratio of 1/50 is the same as the ratio of the distance between Syene and Alexandria to the total circumference of the Earth. Thus, the circumference can be estimated by multiplying the distance between the two cities, 5000 stadia, by 50, equaling 250,000 stadia. C equals circumference (5000 times 50 or 250,000) the unit of the "stadium" was about 0.15 km. This means that Eratosthenes estimated the circumference of the Earth to about 40,000 km. He also knew that the circumference of a circle was equal to 2 times π (3.1415...) times the radius of the circle. (C = 2πr) With this information, Eratosthenes inferred that the Earth's radius was 6366 km. Both values are very close to the accepted modern values for the Earth's circumference and radius, 40,070 km, and 6378 km respectively, which have since been measured by orbiting spacecraft. The diameter of a circle is twice the radius, giving us a diameter for Earth of 12,756 km. Note: Eratosthenes was measuring the polar radius, and his value (using the 0.15 km/stadium conversion) lies between the polar and equatorial values. DETERMINATION OF THE GRAVITIONAL CONSTANT Isaac Newton expressed the Universal Gravitation Equation in 1687: where F is the force of attraction between objects G is the Universal Gravitational Constant M is the mass of the larger object m is the mass of the smaller object R is the separation between the centers of mass of the object After that, there really wasn't much interest in G. Most scientists simply considered it a proportionality constant. they were more interested in gravity than gravitation. In 1798, Henry Cavendish made an experiment to determine the Earth’s density. He used a torsion balance to measure the force of attraction between the two masses. The Cavendish experiment uses a torsion equilibrium to measure the weak gravitational force between lead balls. A torsion equilibrium consists of a bar suspended at its middle by a thin wire. Twisting the wire requires a torque. The way it works is that the gravitational force attracting the balls together turns the bar, overcoming torque resistance from the wire. That resistance is a function of angle turned and the torsion coefficient of the wire.

1. PHYSICAL GEODESY
REPORT ON
DETERMINATION OF THE EARTH’S RADIUS, MASS,
AND GRAVITIONAL CONSTANT
BY
AHMED YASSER AHMED MOHAMED NASSAR
SECTION: 1
1. DETERMINATION OF THE EARTH’S RADIUS

2. Page 1 of 6
In 200 B.C., the size of the Earth was calculated to within 1%
accuracy! Eratosthenes used Aristotle's idea that, if the Earth was
round, distant stars in the night sky would appear at different
positions to observers at different latitudes. Eratosthenes knew
that on the first day of summer, the Sun passed directly overhead
at Syene, Egypt. At midday of the same day, he measured the
angular displacement of the Sun from overhead at the city of
Alexandria - 5000 stadia away from Syene. He found that the
angular displacement was 7.2 degrees - there are 360 degrees in a
circle, making 7.2 degrees equivalent to 1/50 of a circle. Geometry
tells us that the ratio of 1/50 is the same as the ratio of the
distance between Syene and Alexandria to the total circumference
of the Earth. Thus, the circumference can be estimated by
multiplying the distance between the two cities, 5000 stadia, by
50, equaling 250,000 stadia.
C equals circumference
(5000 times 50 or 250,000)
the unit of the "stadium" was about 0.15 km. This means that
Eratosthenes estimated the circumference of the Earth to about
40,000 km. He also knew that the circumference of a circle was
equal to 2 times π (3.1415...) times the radius of the circle. (C =
2πr) With this information, Eratosthenes inferred that the Earth's
radius was 6366 km. Both values are very close to the accepted
modern values for the Earth's circumference and radius, 40,070

3. Page 2 of 6
km, and 6378 km respectively, which have since been measured
by orbiting spacecraft.
The diameter of a circle is twice the radius, giving us a diameter
for Earth of 12,756 km.
Note: Eratosthenes was measuring the polar radius, and his value
(using the 0.15 km/stadium conversion) lies between the polar
and equatorial values.
2. DETERMINATION OF THE GRAVITIONAL
CONSTANT
Isaac Newton expressed the Universal Gravitation Equation in
1687:
where
F is the force of attraction between objects
G is the Universal Gravitational Constant
M is the mass of the larger object
m is the mass of the smaller object
R is the separation between the centers of mass of the object
After that, there really wasn't much interest in G. Most scientists
simply considered it a proportionality constant. they were more
interested in gravity than gravitation.

4. Page 3 of 6
In 1798, Henry Cavendish made an experiment to determine
the Earth’s density. He used a torsion balance to measure the
force of attraction between the two masses.
The Cavendish experiment uses a torsion equilibrium to measure
the weak gravitational force between lead balls. A torsion
equilibrium consists of a bar suspended at its middle by a thin
wire. Twisting the wire requires a torque.
The way it works is that the gravitational force attracting the balls
together turns the bar, overcoming torque resistance from the
wire. That resistance is a function of angle turned and the torsion
coefficient of the wire. At some angle, the torque resistance
equals the gravitational force.
However, the inertia of the balls causes them to go beyond the
equilibrium point and thus create a harmonic oscillation around
that point. The rate of oscillation is then used to determine the
spring constant or torsion coefficient of the wire, which is
necessary in the final calculation of G.

5. Page 4 of 6
In Cavendish's original experiment, used the following values:
Mass of large ball M = 158 kg (348 lbs.)
Diameter of large ball dM = 30.5 cm (12 in)
Mass of small ball m = 0.73 kg (1.6 lbs.)
Diameter of small ball dm = 5 cm (2 in)
Length of rod separating small balls L = 1.86 m (73.3 in)
Separation of large balls L = 1.86 m (73.3 in)
Distance between the centers of the large and small balls R =
0.225 m (8.85 in)
The derived equation for G is:
where
G is the Universal Gravitation Constant
L is the length of the torsion bar
θ is the angle the bar turns
Re is the equilibrium point distance between M and m
T is the oscillation frequency
M is the mass of the larger object
The calculated value of G from this experiment is:

6. Page 5 of 6
3. DETERMINATION OF THE EARTH’S MASS
Once Universal Gravitation Constant (G) has been found, the
attraction of an object at the Earth's surface to the Earth itself can
be used to calculate the Earth's mass and density:
Acceleration due to gravity g=9.8ms−2
Radius of Earth R=6.38×106m
Gravitational Constant G=6.67×10−11m3kg−1s−2
Mass of earth is determined to be M=5.98×1024kg