Gee

From Academic Kids

This article is about the unit of acceleration. GEE is also the name of a WWII radio navigation device built and implemented by the RAF for use in night bombing. For the Latin alphabet letter, see G.

g (also gee, g-force or g-load) is a non-SI unit of acceleration defined as exactly 9.806 65 m/s², which is approximately equal to the acceleration due to gravity on the Earth's surface.

This conventional value was established by the 3rd CGPM (1901, CR 70). The total acceleration is found by vector addition of the opposite of the actual acceleration (in the sense of rate of change of velocity) and a vector of 1 g downward for the ordinary gravity (or in space, the gravity there). For example, being accelerated upward with an acceleration of 1 g doubles the experienced gravity. Conversely, weightlessness means a zero acceleration in an inertial reference frame.

The symbol g is properly written in lowercase and italic, to distinguish it from the symbol G, the gravitational constant, which is always written in uppercase and italic.

The value of g defined above is an arbitrary midrange value on the Earth, approximately equal to the sea level acceleration of free fall at a geodetic latitude of about 45.5; it is larger in magnitude than the average sea level acceleration on Earth, which is about 9.797 645 m/s². The standard acceleration of free fall is properly written as gn (sometimes g0) to distinguish it from the local value of g that varies with position.

The units of acceleration due to gravity, meters per second squared, are interchangeable with newtons per kilogram. The quantity, 9.806 65, stays the same. These alternate units may be more helpful when considering problems involving pressure due to gravity, or weight.

Variations of Earth's gravity

The actual acceleration of a body at the Earth's surface depends on the location at which it is measured, smaller at lower latitudes, for two reasons.

The first is that the rotation of the Earth imposes an additional acceleration on the body that opposes gravitational acceleration. The net downward force on the body is therefore offset by a centrifugal force that acts upwards, reducing its weight. This effect on its own would result in a range of values of g from 9.789 m/s² at the equator to 9.823 m/s² at the poles.

The second reason is the Earth's equatorial bulge, which causes objects at the equator to be further from the planet's centre than objects at the poles. Because the force due to gravitational attraction between two bodies (the Earth and the object being weighed) varies inversely with the square of the distance between them, objects at the equator experience a weaker gravitational pull than objects at the poles.

The combined result of these two effects is that g is 0.052 m/s² more, hence the force due to gravity of an object is 0.5% more, at the poles than at the equator.

If the terrain is at sea level, we can estimate g, at a height h in the air above it:

<math>g=9.780 318 4 \left( 1+A {\sin}^2 L-B {\sin}^2 2L \right) -3.086 \times 10^{-6}h<math>

where

g = acceleration in m/s²
A = 0.005 302 4
B = 0.000 005 9
L = latitude
h = height in metres above sea level.

The last term, 3.086 × 10-6 s-2 (0.3086 mGal/m in non-SI units; to use those units in the formula above the other constants need to be modified as well), is the free air correction: gravity decreases with height, at a rate which near the surface of the Earth is such that linear extrapolation would give zero gravity at a height of one half the radius of the Earth, i.e. the rate is 9.8 m/s² per 3200 km.

For flat terrain above sea level a term is added, for the gravity due to the extra mass; for this purpose the extra mass can be approximated by an infinite horizontal slab, and we get 2πG times the mass per unit area, i.e. 4.2 × 10-10 m3 s-2 kg-1 (0.000,042 mGal/(kg/m²)) (the Bouguer correction). For a mean rock density of 2.67 kg/cm this gives 1.1 × 10-6 s-2 (0.11 mGal/m). Combined with the free-air correction this means a reduction of gravity at the surface of ca. 2 µm/s2 (0.20 mGal) for every metre of elevation of the terrain. (The two effects would cancel at a surface rock density of 4/3 times the average density of the whole Earth.)

For the gravity below the surface we have to apply the free-air correction as well as a double Bouguer correction. With the infinite slab model this is because moving the point of observation below the slab changes the gravity due to it to its opposite. Alternatively, we can consider a spherically symmetrical Earth (http://en.wikipedia.org/wiki/Divergence_theorem#Spherically_symmetric_mass_distribution) and subtract from the mass of the Earth that of the shell outside the point of observation, because that does not cause gravity inside. This gives the same result.

Local variations in both the terrain and the subsurface cause further variations; the gravitational geophysical methods are based on these: the small variations are measured, the effect of the topography and other known factors is subtracted, and from the resulting variations conclusions are drawn. See also physical geodesy and gravity anomaly.

Calculated value of g

Given the law of universal gravitation, g is merely a collection of terms in that equation:

<math>F = G \frac{m_1 m_2}{r^2}=(G \frac{m_1}{r^2}) m_2 <math> where g is the bracketed terms and thus:
<math>g=G \frac {m_1}{r^2}<math>

We can plug in values of <math>G<math> and the mass and radius of the Earth to obtain the calculated value of g:

<math>g=G \frac {m_1}{r^2}=(6.6742 \times 10^{-11}) \frac{5.9736 \times 10^{24}}{(6.37101 \times 10^6)^2}=9.822 \mbox{m/s}^2<math>

This agrees approximately with the measured value of g. The difference may be attributed to several factors:

  • The Earth is not homogeneous
  • The Earth is not a sphere
  • The choice of a value for the radius of the Earth (an average value is used above).
  • The normal measured g also includes the centrifugal force effects due to the rotation of the Earth.

There are significant uncertainties in the values of G and of m1 as used in this calculation. However, the value of g can be measured precisely and in fact, the reverse calculation may be done to compute the mass of the Earth.

Usage of the unit

The g is used primarily in aerospace fields, where it is a convenient magnitude when discussing the loads on aircraft and spacecraft (and their pilots or passengers). For instance, most civilian aircraft are capable of being stressed to 4.33 g (42.5 m/s²; 139 ft/s²), which is considered a safe value. The g is also used in automotive engineering, mainly in relation to cornering forces and collision analysis.

One often hears the term being applied to the limits that the human body can withstand without blacking out, sometimes referred to as g-loc (loc stands for loss of consciousness). A typical person can handle about 5 g (50 m/s²) before this occurs, but through the combination of special g-suits and efforts to strain muscles —both of which act to force blood back into the brain— modern pilots can typically handle 9 g (90 m/s²). Resistance to "negative" or upward gees, which drive blood to the head, is much less; typically in the 2-3 g (20 to 30 m/s²) range the vision goes red, probably due to capillaries in the eyes bursting under the increased blood pressure.cs:Gravitační zrychlení de:Erdbeschleunigung es:Intensidad de la gravedad he:תאוצת הכובד ja:重力加速度 pl:Przyspieszenie ziemskie pt:Acelerao da gravidade sl:Težni pospešek

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