Gravity Constant

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[/caption] The constant of gravity, or gravity constant, has two meanings: the constant in Newton's universal law of gravitation (so is commonly called the gravitational constant, it also occurs in Einstein's general theory of relativity); and the acceleration due to gravity at the Earth's surface. The symbol for the first is G (big G), and the second g (little g).

Newton's universal law of gravitation in words is something like "

the gravitational force between two objects is proportional to the mass of each and inversely proportional to the square of the distance between them

". Or something like F (the gravitational force between two objects) is m

1

(the mass of one of the objects) times m

2

(the mass of one of the other object) divided by r

2

(the square of the distance between them). The "

is proportional to

" means all you need to make an equation is a constant … which is G.

In other words: F = Gm

1

m

2

/r

2

The equation for little g is simpler; from Newton we have F = ma (a force F acting on a mass m produces an acceleration a), so the force F on a mass m at the surface of the Earth, due to the gravitational attraction between the m and the Earth is F = mg.

Little g has been known from at least the time of Galileo, and is approximately 9.8 m/s

2

  • meters per second squared – it varies somewhat, depending on how high you are (altitude) and where on Earth you are (principally latitude).

Obviously, big G and little g are closely related; the force on a mass m at the surface of the Earth is both mg and GmM/r

2

, where M is the mass of the Earth and r is its radius (in Newton's law of universal gravitation, the distance is measured between the centers of mass of each object) … so g is just GM/r

2

.

The radius of the Earth has been known for a very long time – the ancient Greeks had worked it out (albeit not very accurately!) - but the mass of the Earth was essentially unknown until Newton described gravity … and even afterwards too, because neither G nor M could be estimated independently! And that didn't change until well after Newton's death (in 1727), when Cavendish 'weighed the Earth' using a torsion balance and two pairs of lead spheres, in 1798.

Big G is extremely hard to measure accurately (to 1 part in a thousand, say);

today's best estimate

is 6.674 28 (+/- 0.000 67) x 10

-11

m

3

kg

-1

s

-2

.

The Constant Pull of Gravity: How Does It Work?

is a good NASA webpage for students, on gravity; and

the ESA's GOCE mission webpage

describes how satellites are being used to measure variations in little g (GOCE stands for Gravity field and steady-state Ocean Circulation Explorer).

The Pioneer Anomaly: A Deviation from Einstein's Gravity?

is a Universe Today story related to big G, as is

Is the Kuiper Belt Slowing the Pioneer Spacecraft?

;

GOCE Satellite Begins Mapping Earth's Gravity in Lower Orbit Than Expected

is one about little g.

No surprise that the Astronomy Cast episode

Gravity

covers both big G and little g!