Acceleration of gravity

**Acceleration of free fall on the surface ^{[1] of} some celestial bodies, m / s ² and g**
Land	9.81 m / s ²	1.00 g	The sun	273.1 m / s ²	27.85 g
Moon	1.62 m / s ²	0.165 g	Mercury	3.68—3.74 m / s ²	0.375-0.381 g
Venus	8.88 m / s ²	0,906 g	Mars	3.86 m / s ²	0.394 g
Jupiter	23.95 m / s ²	2,442 g	Saturn	10.44 m / s ²	1,065 g
Uranus	8.86 m / s ²	0.903 g	Neptune	11.09 m / s ²	1,131 g

Acceleration of a free fall ( acceleration of gravity ) is the acceleration given to a body by gravity , with the exception of other forces from consideration. In accordance with the equation of motion of bodies in non - inertial reference systems ^{[2], the} acceleration of gravity is numerically equal to the force of gravity acting on an object of unit mass .

The acceleration of gravity on the Earth's surface g (usually pronounced “same” ) varies from 9.780 m / s² at the equator to 9.82 m / s² at the poles ^[3] . The standard (“normal”) value adopted in the construction of systems of units is g = 9.80666 m / s² ^[4] ^[5] . The standard value of g was defined as "average" in a sense on the whole Earth, it is approximately equal to the acceleration of gravity at a latitude of 45.5 ° at sea level . In approximate calculations, it is usually taken equal to 9.81, 9.8 or, roughly, 10 m / s².

Physical Entity

Two components of the acceleration of gravity on Earth g : gravitational (in the approximation of homogeneity of the Earth) is equal to GM / r ² and centrifugal, equal to ω ² a , where a is the distance to the earth's axis, ω is the angular velocity of the Earth.

For definiteness, we assume that we are talking about accelerating free fall on Earth. This value can be represented as the vector sum of two terms: gravitational acceleration caused by gravity, and centripetal acceleration associated with the rotation of the Earth .

Centripetal Acceleration

Centripetal acceleration is a consequence of the rotation of the Earth around its axis. It is the centripetal acceleration caused by the rotation of the Earth around its axis that makes the largest contribution to the non-inertia of the reference frame associated with the Earth. At a point located at a distance a from the axis of rotation, the centripetal acceleration is ω ² a , where ω is the angular velocity of the Earth's rotation, defined by the expression ω = 2π / T , in which T is the time of one revolution around its axis ( stellar day ), equal to for Earth 86164 seconds. Centripetal acceleration is directed normal to the axis of rotation of the Earth. It can be estimated that at the equator it is 3.39636 cm / s ² , and at other points its vector does not coincide with gravitational acceleration, which is directed to the region of the center of the Earth.

Gravity Acceleration

**Gravitational acceleration at various heights h above sea level**
h , km	g , m / s ²	h , km	g , m / s ²
0	9.8066	20	9,7452
one	9.8036	50	9.6542
2	9,8005	80	9,5644
3	9.7974	100	9,505
four	9.7943	120	9,447
five	9.7912	500	8.45
6	9,7882	1000	7.36
eight	9.7820	10,000	1,50
ten	9,7759	50,000	0.125
15	9,7605	400,000	0.0025

In accordance with the law of universal gravitation , the value of gravitational acceleration on the surface of the Earth or another planet is associated with the mass of the planet M by the following relation:

g=G{\frac {M}{r^{2}}}

{\ displaystyle g = G {\ frac {M} {r ^ {2}}}}

,

where G is the gravitational constant (6.67408 (31) · 10 ⁻¹¹ m ³ · s ⁻² · kg ⁻¹ ) ^[6] , and r is the radius of the planet. This relation is valid under the assumption that the distribution of mass over the volume of the planet is spherically symmetric. The above ratio allows you to determine the mass of any planet, including the Earth, knowing its radius and gravitational acceleration on its surface. Historically, the mass of the Earth was first determined by Henry Cavendish , who made the first measurements of the gravitational constant.

Gravitational acceleration at a height h above the surface of the Earth (or another planet) can be calculated by the formula:

g(h)={\frac {GM}{(r+h)^{2}}}

{\ displaystyle g (h) = {\ frac {GM} {(r + h) ^ {2}}}}

,

where M is the mass of the planet.

Acceleration of free fall on Earth

The acceleration of gravity at the Earth's surface depends on latitude, time of day, and other factors. It can be approximately calculated (in m / s²) according to the empirical formula ^[7] ^[8] :

g=9{,}780318(1+0{,}005302\sin ^{2}\varphi -0{,}000006\sin ^{2}2\varphi )-0{,}000003086h,

{\ displaystyle g = 9 {,} 780318 (1 + 0 {,} 005302 \ sin ^ {2} \ varphi -0 {,} 000006 \ sin ^ {2} 2 \ varphi) -0 {,} 000003086h,}

Where $\varphi$ ${\ displaystyle \ varphi}$ - latitude of the place in question, $h$ ${\ displaystyle h}$ - height above sea level in meters . The obtained value only approximately coincides with the acceleration of gravity at a given place. For more accurate calculations, it is necessary to use one of the models of the Earth's gravitational field , supplementing it with corrections related to the rotation of the Earth, tidal influences and other factors.

Spatial changes in the Earth's gravitational field ( gravitational anomalies ) are associated with the heterogeneity of its structure, which can be used to search for minerals ( gravity exploration).

On average, the acceleration of gravity at the equator is lower than at the poles, due to the centrifugal forces arising from the rotation of the planet, and also because the radius r at the poles is smaller than at the equator due to the flattened shape of the planet. However, places of record low and high g values are somewhat different from this simplified model. Thus, the lowest g value was recorded on Mount Huascaran in Peru (9.7639 m / s²) 1000 km south of the equator, and the largest (9.8337 m / s²) 100 km from the north pole ^[9] .

Acceleration of free fall for some cities
City	Longitude	Latitude	Altitude, m	Acceleration of gravity, m / s ²
Berlin	13.40 east	52.50 N	40	9.81280
Budapest	19.06 east	47.48 N	108	9.80852
Washington	77.01 west	38.89 N	14	9,80188
Vein	16.36 east	48.21 N	183	9,80860
Vladivostok	131.53 east	43.06 N	50	9,80424
Greenwich	0.0 east	51.48 N	48	9.81188
Cairo	31.28 east	30.07 N	thirty	9,79317
Kiev	30.30 east	50.27 N	179	9,81054
Madrid	3.69 east	40.41 N	667	9,79981
Minsk	27.55 east	53.92 N	220	9,81347
Moscow	37.61 east	55.75 N	151	9.8154
New York	73.96 W	40.81 N	38	9.80247
Odessa	30.73 east	46.47 N	54	9.80735
Oslo	10.72 east	59.91 N	28	9,81927
Paris	2.34 East	48.84 N	61	9,80943
Prague	14.39 east	50.09 N	297	9.81014
Rome	12.99 east	41.54 N	37	9,80312
Stockholm	18.06 east	59.34 N	45	9.81843
Tokyo	139.80 east	35.71 N	18	9,79801

Measurement

The acceleration of gravity at the Earth's surface can be measured using a gravimeter . There are two types of gravimeters: absolute and relative. Absolute gravimeters measure the acceleration of gravity directly. Relative gravimeters operating on the principle of spring weights determine the increment of gravitational acceleration relative to the value at a certain starting point.

The acceleration of gravity on the surface of the Earth or another planet can also be calculated on the basis of data on the rotation of the planet and its gravitational field. The latter can be determined by observing the orbits of satellites and other celestial bodies near the planet in question.

Notes

↑ For gas giants and stars, “surface” is understood as a region of lower altitudes in the atmosphere, where the pressure is equal to the atmospheric pressure on Earth at sea level ( 1.013 × 10 ⁵ Pa ).
↑ An analogue of the equation of Newton’s second law , which holds for non-inertial reference frames.
↑ Free fall tel. Acceleration of free fall .
↑ Declaration of the III General Conference on Weights and Measures (1901) (English) . International Bureau of Weights and Measures . Date of treatment April 9, 2013.
↑ Dengub V.M., Smirnov V.G. Units of quantities. Dictionary dictionary. - M.: Publishing house of standards, 1990. - S. 237.
↑ CODATA Value: Newtonian constant of gravitation (neopr.) . physics.nist.gov. Date of treatment February 23, 2016.
↑ Grushinsky N.P. Gravimetry // Physical Encyclopedia : [in 5 volumes] / Ch. ed. A.M. Prokhorov . - M .: Soviet Encyclopedia, 1988. - T. 1: Aaronova - Bohm effect - Long lines. - S. 521. - 707 p. - 100,000 copies.
↑ Acceleration of free fall // Physical Encyclopedia : [in 5 vol.] / Ch. ed. A.M. Prokhorov . - M .: Great Russian Encyclopedia, 1994. - T. 4: Poynting - Robertson - Streamers. - S. 245-246. - 704 s. - 40,000 copies. - ISBN 5-85270-087-8 .
↑ Peruvians live easier than polar explorers?

Literature

Enokhovich A. S. A Brief Guide to Physics. - M .: "Higher School", 1976. - 288 p.

[1] For gas giants and stars, “surface” is understood as a region of lower altitudes in the atmosphere, where the pressure is equal to the atmospheric pressure on Earth at sea level ( 1.013 × 10 ⁵ Pa ).

[2] An analogue of the equation of Newton’s second law , which holds for non-inertial reference frames.

[3] Free fall tel. Acceleration of free fall .

[4] Declaration of the III General Conference on Weights and Measures (1901) (English) . International Bureau of Weights and Measures . Date of treatment April 9, 2013.

[5] Dengub V.M., Smirnov V.G. Units of quantities. Dictionary dictionary. - M.: Publishing house of standards, 1990. - S. 237.

[6] CODATA Value: Newtonian constant of gravitation (neopr.) . physics.nist.gov. Date of treatment February 23, 2016.

[7] Grushinsky N.P. Gravimetry // Physical Encyclopedia : [in 5 volumes] / Ch. ed. A.M. Prokhorov . - M .: Soviet Encyclopedia, 1988. - T. 1: Aaronova - Bohm effect - Long lines. - S. 521. - 707 p. - 100,000 copies.

[8] Acceleration of free fall // Physical Encyclopedia : [in 5 vol.] / Ch. ed. A.M. Prokhorov . - M .: Great Russian Encyclopedia, 1994. - T. 4: Poynting - Robertson - Streamers. - S. 245-246. - 704 s. - 40,000 copies. - ISBN 5-85270-087-8 .

[9] Peruvians live easier than polar explorers?