Gravitational constant G?

Question

what is Gravitational constant G and how did they get it

Answer 1

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal gravitational constant, Newton's constant, and colloquially Big G.[1] It should not be confused with "little g" (g), which is the local gravitational field (equivalent to the local acceleration due to gravity), especially that at the Earth's surface; see Earth's gravity and Standard gravity.



According to the law of universal gravitation, the attractive force (F) between two bodies is proportional to the product of their masses (m1 and m2), and inversely proportional to the square of the distance (r) between them:



F = G m1 m2/r^2



The constant of proportionality, G, is the gravitational constant.



The gravitational constant is perhaps the most difficult physical constant to measure.[2] In SI units, the 2006 CODATA-recommended value of the gravitational constant is:[3]



G = 6.67428+or- 0.00067 )* 10^-11 m^3 kg^-1 s^-2



The gravitational constant appears in Newton's law of universal gravitation, but it was not measured until 1798 — 71 years after Newton's death — by Henry Cavendish (Philosophical Transactions 1798). Cavendish measured G implicitly, using a torsion balance invented by the geologist Rev. John Michell. He used a horizontal torsion beam with lead balls whose inertia (in relation to the torsion constant) he could tell by timing the beam's oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused. However, it is worth mentioning that the aim of Cavendish was not to measure the gravitational constant but rather to measure the mass and density relative to water of the Earth through the precise knowledge of the gravitational interaction. The value that he calculated, in SI units, was 6.754 × 10−11 m3/kg/s2[4]

The accuracy of the measured value of G has increased only modestly since the original experiment of Cavendish. G is quite difficult to measure, as gravity is much weaker than other fundamental forces, and an experimental apparatus cannot be separated from the gravitational influence of other bodies. Furthermore, gravity has no established relation to other fundamental forces, so it does not appear possible to measure it indirectly. Published values of G have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive.[2][5]



In the January 5, 2007 issue of Science (page 74), the report "Atom Interferometer Measurement of the Newtonian Constant of Gravity" (J. B. Fixler, G. T. Foster, J. M. McGuirk, and M. A. Kasevich) describes a new measurement of the gravitational constant. According to the abstract: "Here, we report a value of G = 6.693 × 10−11 cubic meters per kilogram second squared, with a standard error of the mean of ±0.027 × 10−11 and a systematic error of ±0.021 × 10−11 cubic meters per kilogram second squared."

Answer 2

To show where the gravitational constant 'G' came from, it is necessary to consider how Newton derived his equation of universal gravitation from Kepler's laws of planetary motion. Kepler's laws are empirical (equations fitted to the data) and they are based upon the observational tables obtained by the astronomer Tycho Brahe.



Newton's law of universal gravitation may be derived from Kepler's laws. It is just this approach, that Newton adopted - starting with Kepler's second law (k2 for short). The element of area dA swept out by the radius vector, of an orbiting planet, in time dt is approximately: -



dA = ½r(r + dr)dθ



Hence, in the limit (calculus): -



dA = ½r²dθ



If k2 is written as: -



r²dθ/dt = constant = C or d(r²dθ/dt)/dt = 0



In this approximation the planets and the sun are considered to be point masses.



Now, from k1. The equation of an ellipse in polar coordinates with one focus as the origin is: -



1 - e.cosθ = (a(1 - r²))/r



Where a is the semi major axis and e is the eccentricity of the ellipse.



If we differentiate twice with respect to time and use K2 (constant form above) to eliminate dθ/dt, we obtain: -



(C²/r²).e.cosθ = - a(1 -e²).d²rdt²





A central conservative field of force has the potential equation: -



F = -dV/dr



From Newton's second law of motion it may be shown that the equation may be written down as: -



F(r) = m((d²r/dt²) - r.(dθ/dt)²)



.......= -(mC²/(a.(1 - e²)r²).(ecosθ +(a.(1 - e²)/r))



Hence, using K1 (from the above equation form): -



F(r) = -mC²/(a(1 - e²)r²)



If we now relate the constant C to the orbital period of the planet: -



dA/dt = 0.5C



With



A =√(πa²(1 - e²))



and this gives



√(πa²(1 - e²)) = 0.5CT



Hence, from the F(r) equation: -



F(r) = -4π²a³m/(T²r²) = -Bm/r²



Where B = 4π²a³/T²



Newton's great contribution was to realise that the forces between the earth and the sun were the same. Thus, for the sun (mass M): -



B/M = G



and for the earth (mass m constant B’)



B'/m = G



So that for the Sun, B = GM



Or



F = -GMm/r²



This is how Newton derived his famous equation using Kepler's empirical laws and the equation of a central conservative force field and found the gravitational constant G.



I hope this brief proof is of some assistance.

Answer 3

It's Saturday night you sad git. put the physics down and step away from the books.



there are things called "girls" out there. go to a pub and find one. don't ask them this question though. not unless you want laughing at.



seriously man - get a life :)

Answer 4

rude dude blokeyfella...