how the weight of earth has been arrived at?

Answer 1

Once you know the value of G, the gravitational constant, you can find the mass of Earth in several ways. Pretzels gave you one good way. Another is to measure the acceleration due to gravity of an object falling near earth's surface. You need a setup where friction and air resistance are negligible, but that's not really too hard to accomplish (or correct for). Then use the equation

g = GM/R^2

where g is the acceleration, M is the mass of Earth, and R is the radius of Earth (the ancient Greeks figured that out). (Do some algebra and solve for M first, of course.)

Another way is by watching a satellite orbit Earth (either the Moon or a man-made satellite). Then the equation to use is

v^2 = GM/r

where v is the speed of the satellite, and r is the radius of the satellite's orbit around Earth (measured from the center of the satellite to the center of Earth). Solve for M to find the mass of Earth.

These equations come about from fairly basic physics (I teach this in my freshman physics class in college).

Answer 2

Newton's law of gravity gives the gravitational force (F) between two objects as a function of mass (m1, m2), distance (r) and the universal gravitational constant (G) as:

F = G*m1*m2/r^2

In 1798 Cavendish performed his famous experiment to determine G. He used some 350 lb lead spheres and a very sensitive torsion balance. The value of G is well known as:

G = 6.6742 x 10^-11 newton*m^2/kg^2

Once you know this, it is easy. A scale shows that a 1 kg mass at the surface of the Earth weighs 9.8 newtons. The distance between the surface and center of the Earth is 6.378 x 10^6 m.

Put this into a rearragned form of Newtons gravity:

m1 = (F*r^2)/(m2*G) = (9.8 newton * (6.378 x 10^6 m)^2)/(1 kg * 6.6742 x 10^-11 newton*m^2/kg^2) = 5.97 x 10^24 kg.