How can the properties of the Earth be determined? And how is the gravitational constant then determined from them?
A satellite can determine the G.M.. But, how they are individually determined nowadays ?
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3$\begingroup$ For question 3: en.wikipedia.org/wiki/Cavendish_experiment $\endgroup$– BernhardCommented Sep 25, 2012 at 7:41
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2$\begingroup$ also for classical ways of determining the radius en.wikipedia.org/wiki/Radius_of_earth $\endgroup$– anna vCommented Sep 25, 2012 at 16:17
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$\begingroup$ If you are interested in this, you definitely need to read Terence Tao's Cosmis Ladder presentation, powerpoint available here: terrytao.wordpress.com/2007/05/31/the-cosmic-distance-ladder $\endgroup$– recipriversexclusionCommented Sep 25, 2012 at 17:06
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A simple googling would've provided your answer. OK Let's see..!
Radius (History): The dimensions of the Earth was first measured by Greek philosopher Eratosthenes in 200 B.C. by using simple geometric methods. He dug a narrow well at Syene (now Aswan). The well would reflect the sunlight at about 90° on the first day of summer. At Alexandria, about 5,000 stadia north of Syene (one stadia was about 100 yards or 0.16 km), he placed a stick and measured the length of the shadow at noon of the first day of summer. As the sun is too large compared to the size of the earth, the sun's rays are practically assumed parallel to each other as they hit the earth.
By comparing the angle made by the shadows at Alexandria with the angle formed by connecting the center of the earth with Alexandria and Syene, he obtained a displacement angle of 7.2°. The fraction of the Earth's circumference lies within Alexandria and Syene, and is proportional to this displacement angle. Now, 7.2° is $\frac{1}{50^{}th}$ of a full circle, which means that the distance between that two cities is 1/50 of the Earth's circumference. $$\frac{360°}{\theta}=\frac{2\pi r}{l}$$
To get the size of the earth, he multiplied 5,000 stadia with 50. To get the radius of the earth, he divided the earth's circumference by $2\pi$. He got the radius to be 6,366 km. The radius of the Earth as determined using satellite equipments by our guys is 6378 km. (Old-timer proved himself only with a stick..!) Here's a geographic view of those places:
Gravitational Constant (History): Newton's constant $G$ was not measured until 71 years after Newton's death by Henry Cavendish with his Cavendish Experiment, performed in 1798 (by using torsion balance). It was determined to be $6.74×10^{-11}m^3 kg^{-1}s^{-2}$ which which differs by only 1% from the currently accepted value: $6.67428 × 10^{−11} m^3 kg^{−1} s^{−2}$
Mass (History): I googled out. But, I can't find any, better than this one...
Acceleration due to gravity $g=9.8ms^{-2}$
Radius of Earth $R=6.38×10^6m$
Gravitational Constant $G=6.67×10^{-11}m^3kg^{-1}s^{-2}$
By relating $F=ma$ and $F=\frac{GMm}{r^2}$, Mass of earth is determined to be $M=5.98×10^{24}kg$
Here's a paper explaining these somewhat brief.
