Any twoparticleThere are forces attracting each other through the direction of their connecting lines.ThisgravitationThe size is proportional to the product of their mass, inversely proportional to the square of their distance, and related to the chemical composition of two objectsmediumCategory independent.
Galileo actually proposed in 1632centrifugal forceAnd centripetal force.Briard put forward the idea of gravitational square ratio in 1645. Newton proved the law of centrifugal force in his own way in his manuscripts from 1665 to 1666, but the word centripetal force first appeared in《On sports》In the first manuscript.It is generally believed that the law of centrifugal force isHuygensPublished in 1673《Pendulum clock》In a book.According to the manuscript "On the Motion of Rotating Objects" from August to October 1684, Newton may have proposed centripetal force and its definition for the first time in this manuscript.
The universal gravitation is proportional to the product of the mass of the interacting object, which is a necessary stage for the discovery of the inverse square law of gravitation to transition to the discovery of the law of universal gravitation·It took Newton 20 years from 1665 to 1685 to follow the centrifugal force - centripetal force -gravity- The evolution order of the concept of universal gravitation was finally proposed“Universal gravitation”This concept and vocabulary·Newton in《Mathematical Principles of Natural Philosophy》The third volume wrote: "Finally, if it is generally shown from experiments and astronomical observations that all celestial bodies around the Earth are attracted by the gravity of the Earth, and their gravity is proportional to the amount of material they contain, thenMoonSimilarly, according to the amount of substanceearthGravitation.On the other hand, it shows that our oceans are attracted by the gravity of the moon;And everythingplanetAttracted by gravity,cometAlso attracted by the gravity of the sun.Because of this rule, we must generally recognize that all objects, no matter what they are, are endowed with the principle of mutual gravitation.Because according to this representationUniversal gravitation(Universal gravity)... "
NewtonBetween 1665 and 1666, only the law of centrifugal force andKepler's third law, so it can only proveCircular orbitOn instead ofElliptical orbitThe inverse square of gravity on the.In 1679, he knew how to useKepler's second lawHowever, there was no breakthrough in the method of proof, which remained at the level of 1665-1666.Only in January 1684,Harley, RyanHookeBoth Newton and Newton can prove the inverse square relation of gravity on circular orbits, and both have known that the inverse square relation of gravity is observed on elliptical orbits. However, in the end, only Newton may be able to prove the inverse square relation of gravity on the basis of Kepler's third law, the centripetal force law evolved from the centrifugal force law and the mathematicalLimit conceptorCalculusConcept, only for usegeometryLaw proves this difficult problem.
hypothesis test
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Newton's conjecture
The attraction between the earth and the sun may be the same force as the earth's attraction to surrounding objects,followThe same rule.[2]
Basis of conjecture
（1）planetThe gravity between the sun and the planet cannot fly away from the sun, and the gravity between the object and the earth cannot make the object leave the earth;(2) At a high distance from the ground, gravity will not be significantly weakened, so this force must extend to a far place.
r: Distance between two objects (size)（rRepresents radial vector)
According to the International System of Units, the unit of F is Newton (N), moneAnd mtwoThe unit of is kilogram (kg), the unit of r is meter (m), and the constant G is approximately equal to
G=6.67 × 10 ⁻ ⁻ ⁻ N · m ²/kg ² (Newton square meter per square kilogram).
It can be seen that the repulsive force F will never exist, which means that the netaccelerationThe force of is absolute.(This symbol specification is made for compatibility with Coulomb's law. In Coulomb's law, absolute force represents twoElectronicsForce between.)
Internal formula of universal gravitation
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a=X/R X
External formula: X>=R
The external formula is consistent with the Newton formula, which means that the Newton formula is an approximation of the external formula.
Scope of application
The classical law of universal gravitation reflects human understanding of gravitation at a certain historical stage. It was discovered at the end of the 19th century,MercurystayPerihelionThe moving speed of is greater than the theoretical value, that is, it is found that the orbit of Mercury is tightened, and the actual value of the orbital tightening speed is 42.9 ″ per century.This phenomenon cannot be explained by the law of universal gravitation, but the tightening result calculated according to the general relativity is 43.0 ″ every century, within the allowable range of observation error.In addition, general relativity can better explain thered shiftAnd lightsunlightGravitationaldeflectionAnd so on.This shows that the gravity theory of general relativity is a step ahead of the classical gravity theory.
stayFaradayandmaxwellLater, people saw that the physical reality was not only particles but also electromagnetic fields.Electromagnetic field hasmomentumandenergyAnd can spreadelectromagnetic wave。This makes people think that the law of universal gravitation is also a physical reality, which can propagate gravitational waves. Many people try to detect it, but there is no good result yet.The propagation of electromagnetic waves can be explained by photons. Similarly, photons also lead to the introduction of the concept of graviton.Gravity is no longerAction at a distance, withGravitonIs the medium.But these are all areas that physicists are exploring.
Application scope of classical mechanics and introductionPlanck constantAnd vacuumlight speedTo define the territory of classical mechanics.Roughly speaking, the scope of application of the classical law of universal gravitation can also be expressed by a quantity.Now introduce the radius of gravity
, G and m represent the gravitational constant and the mass of the sphere generating the gravitational field respectively, and c is the speed of light.R represents the radius of the sphere generating the force field, if
There is no problem in applying Newton's law of gravity;Even for denseWhite dwarf，
Newton's law of universal gravitation can still be used;as forblack holeandBig Bang, should be applicationGeneral relativity。
Gravitational constant
When Newton deduced the law of universal gravitation, he failed to obtain the specific value of the gravitational constant G.The value of G was changed fromCavendishUse what he inventedTorsion scaleobtain.CavendishThe torsion balance test of, not only proved the law of universal gravitation by practice, but also let this law have a broader use value.
The basic principle of the torsion balance is to connect two weights of the same mass with a certain height at both ends of a rigid barweighbeamThe center of is suspended with a twisted wire.weighbeamIt can rotate freely around the twisted wire. When the gravity field is uneven, the gravity of the two masses is not parallel.The slight difference in this direction causes a small level in the two massesComponent forceAnd generate a torque to make the suspension system rotate around the twisted wire until it is balanced with the torque of the twisted wire.The small mirror on the twisted wire reflects the light onto the recording plate.When the twist wire rotates, the distance that the light moves on the phase plate marks the size of the twist angle.Balance positionIt is related to the torsion balance constant and the second derivative of gravity potential.Observe at least 3 directions at one measuring point, and determine 4 secondary derivative values,measurement accuracy How many is the averageErfou。
according toTorsionThe structural shape of the system is divided into z type, L type and diagonal arm type torsion scales.The z-type torsion balance is composed of a z-type balance arm made of light metal, two heavy loads with equal mass and a thin metal wire.The two heavy loads are respectively fixed at both ends of the z-shaped scale arm.The whole system is suspended by thin wires to form a torsion system.Since the two heavy loads are in different positions, when passing through the gravity equipotential planes Q ₁ and Q ₂ of the two heavy loads.When they are not parallel to each other or bent, the two heavy loads will befield of gravityThe role of the horizontal component.Whenfield of gravityWhen the horizontal components gH ₁ and gH ₂ are different in size and direction, the stalk arm will rotate around the twist wire until the weight of the horizontal rotation ismomentAnd twisted wireTorsional momentPhase equilibrium.The deflection angle of the scale arm is not only related to the structure of the torsion system and the torsion coefficient of the torsion wire, but also to the gravity change between two heavy loads.Therefore, the second derivative of gravitational potential can be obtained by accurately recording the deflection angle of the torsion system.Due to the high sensitivity of the torsion system, the balance arm will take a long time to stabilize.At the same time, photographic recording is required in 3-5 directions, so the instrument is attached withautomatic control systemAnd put it in a special small room to work.The operation of the instrument and the calculation of the measurement results are relatively cumbersome. It takes 2-3 hours to measure a point, and the workpiece efficiency is low.
The measurement result of the torsion scale is represented by a vector graph, with a short line representing the curvature. The vector direction corresponds to the orientation of the plane of minimum curvature, and the vector length represents the curvature difference of the equipotential surface.Draw the total gradient at the center of the short line with an arrow pointing to the direction of gravity increase.
The torsion balance has high sensitivity and can measure many parameters, but it also has its shortcomings.Due to its extremely high sensitivity and high requirements for the test environment, it is vulnerable to external interference, including temperature, ground vibrationAtmospheric pressureFluctuating, twistedAnelasticityEffect, etc.So foraccuracyGravimeter is generally used to measure gravity with low requirements.But for high-precision measurement, such asgravitationThe physical measurement, as well as the verification and calibration of high-precision instruments, all need to be completed using a torsion balance.Therefore, even today, torsion scales play an important role in the field of experimental physics.
G=6.67 × 10 ⁻ N · m ²/kg ² measured by Cavendish, which is very close to the current recognized value of 6.67 × 10 ⁻ N · m ²/kg ²;Until 1969, the measurement accuracy of G remained atCavendishOn the level of.[3]
Scientific significance
The discovery of the law of universal gravitation is one of the greatest achievements of natural science in the 17th century.It combines the motion law of objects on the ground withcelestial bodiesThe laws of movement have been unifiedphysicsandastronomyIts development has far-reaching implications.It explains for the first time (one of the four interactions in nature)Basic interactionThe law of human nature has set a milestone in the history of human understanding of nature.
The law of universal gravitation reveals the laws of celestial body movementastronomyIt has a wide range of applications in space navigation calculation.It provides a set of calculation methods for actual astronomical observation. Only a few observation data can be used to calculate the orbits of long-period celestial bodiesHalley's Comet、Neptune、PlutoThe discoveries of are all examples of great achievements in applying the law of universal gravitation.utilizeUniversal gravitation formula，Kepler's third lawIt can also calculate the mass of the sun, the earth and other celestial bodies that cannot be directly measured.Newton also explained that the gravitational pull of the moon and sunTidal phenomenon。According to the law of universal gravitation and other laws of mechanics, he also successfully explained the reasons for the flat shape of the earth's poles and the complex movement of the earth's axis.It overthrows the ancient human thought of the gravity of God.
It is of great significance for cultural development: it enables people to build confidence in understanding various things between heaven and earth, liberates people's minds, and plays a positive role in promoting the development of science and culture.
Please note that a in the above equationone, mass moneOfaccelerationIn fact, it does not depend on moneValue of.Therefore, it can be inferred that for any object, no matter what their mass is, they will fall to the ground at the same rate (ignoring the air resistance).
If there is only a very small change in r during the movement of an object, for example, near the groundfree falling movement——The gravitational acceleration will remain almost constant (see the entry Gravity).For a large object, the change of gravity at different sites due to the change of r will cause huge and considerable tidal force.
Order moneIs the earth mass 5.98 * 10 ² ≮ kg, mtwoIs 1kg, R is the earth radius of 6380000m, and F=9.8N is calculated by substituting the universal gravitation formula, which indicates that the gravity of a 1kg object on the earth surface is 9.8N.In other words, both sides of the equation are divided by mtwoThe result is the gravitational acceleration g.
Objects with spatial breadth:
If the object in question has a spatial breadth (far greater than the theoreticalparticle）The universal gravitation between them can be calculated by the sum of the universal gravitation of each equivalent particle of the object.In the limit, when the constituent particle approaches to "infinitesimal", it will be necessary to calculate the integration of the force between two objects (see the vector formula below) in the space range.
It can be concluded that if the mass distribution of an object is uniform and spherical, its gravitational attraction to the external object will be the same as that of all the masses concentrated on the objectGeometric centerThe principle is the same.(This does not apply to non spherically symmetric objects).
Schematic diagram of gravity in space near the earth: hereOrder of magnitudeThe curvature of the upper earth surface can be ignored, so the force lines can be approximately parallel to each other and point to the center of the earthNewton's law of universal gravitationIt can also be expressed in the form of vector equation to calculate the direction and size of gravity.In the following formula, the quantities shown in bold represent vectors.
Including:
F ₁ ₂: gravitation of object 1 to object 2
G: Universal gravitational constant
M ₁ and m ₂: respectively the mass of object 1 and object 2
R ₂ ₁=| r ₂ r ₁ |: distance between object 2 and object 1
R ₂ 1=r ₁+r ₂ distance between object 2 and object 1
: Unit vector from object 1 to object 2
It can be seen thatvectorThe form of the equation is similar to the scalar equation given previously, except that F in the vector equation is a vector, and the corresponding unit vector is multiplied on the right end of the vector equation.Moreover, we can see that: F ₂=F ₂ ₁
Similarly, the vector equation of gravity acceleration is similar to its scalar equation.
Gravity and Gravity
1. Gravity is generated by the attraction of the earth, but can we say that gravity is gravity?The analysis of this problem should start with the rotation of the earth.As the earth rotates, the objects on the earth move in a circle with it. The centripetal force F ₁=mr ω²=mR ω² cosa, F ₁ is provided by gravity F, which is a component of F, cosa is the cosine value of the angle between gravity F and the equatorial plane, and the other component of F ₂ is the weight force on the object, that is, F ₂=mg.
It can be seen that the universal gravitation of the earth on objects is the reason why objects are subject to gravity, but gravity is not completely equal to universal gravitation. This is because objects rotate with the earth and need a part of universal gravitation to provide centripetal force.
Meet mg=F-F direction on the equator (the object is subject to gravity and the ground supports the object FnAnd its resultant force acts as the centripetal force, FnIs equal to the gravity of the object).
At the two poles of the earth, since F direction=0, that is, mg=F, at other locations, mg, F and F direction are consistentParallelogram rule。The gravity of the same object is the smallest at the equator, and increases with the increase of latitude.
(2) Relation between gravity, gravity acceleration and height
At the height h above the ground, if the influence of earth rotation is not considered, mg '=F=GMm/(R+h) ²;On the ground, mg=GMm/R ².
The gravity acceleration g '=GM/(R+h) ² at the height h above the ground, and g=GM/R ² at the ground.
In the spaceship running in the orbit with a height of h from the ground, the gravity of the object with a mass of m is the universal gravity received there, that is, mg '=GmM/(R+h) ², but its gravity cannot be measured with a dynamometer.
Uniform circular motion
One celestial body orbits another central celestial bodyUniform circular motion。Its centripetal force is provided by gravity.That is, F=GMm/r ² ≈ mg=ma direction, while a direction=v ²/r=ω² r=v ω=(4 π²/T ²) r=4 π² f ² r. Therefore, applying the law of universal gravitation to solve problems related to celestial bodies, there are mainly the following metric relationships: F=GMm/r ² (r is the orbital radius)=mg=ma direction=mv ²/r=m ω² r=m (4 π²/T ²) r=m4 π² f ² r
field of gravity:
Globular clusterM13 proves the existence of gravity field.Gravity field is a vector field used to describe the universal gravitation of an object per unit mass at a certain point in any space.In fact, it is equal to the gravitational acceleration of the object at that point.
The following is a universalVectorial form, which can be applied to the calculation of more than two objects (such as the rocket traveling between the earth and the moon).For the case of two objects (for example, object 1 is a rocket and object 2 is the earth), we can use substitution and m instead of m ₁ to express the gravity field as:
So we can get:
This formula is not limited by the object that generates the gravity field.The unit of gravity field is the unit of force divided by mass;In the SI system, it is specified as N/kgㄢ(Newton per kilogram).
Astromechanics
oneCalculating the Mass of Celestial Bodies
(1) Calculate Earth Mass
If the rotation of the earth is not considered, the gravity of the object on the ground is the universal gravity of the earth on it
Mg=GmM/R ² From this, we can get the earth mass M=gR ²/G
(2) Calculate solar mass
Measuring the Earth's Orbit around the SunRevolution period, radius of revolution orbit, look at the orbit as a circle, the centripetal force of uniform circular motion is the universal gravitation
That is, GMm/R ²=m (2 π/T) ² R Earth mass is m, solar mass M=4 π² R ³/GT ²
Use similar methods to know the mass of artificial satellite, the period and orbit radius of satellite movement around a celestial body
The mass of celestial bodies can be calculated
twoEstimating the density of celestial bodies
If the radius of a celestial body is R, the orbit radius is R when the satellite moves around the surface of the celestial body,
It is also measured that the known operating cycle is T
If the satellite mass is m, GMm/R ²=m (2 π/T) ² R Celestial mass M=4 π² R ³/GT ²
Volume V=4 π R ³/3 ρ=M/V=3 π/GT ²
Existing problems
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brief introduction
Although Newton's description of gravity is very accurate for many practical applications, it also has several theoretical problems and has been proved to be incomplete.
Theoretical issues
There is no sign that the transmission medium of gravity can be identified, and Newton himself also said thatAction at a distanceNot satisfied (see the following entry "Limitations").
Newton's theory needs to define that gravity can propagate instantaneously.Therefore, the assumption of classical natural space-time concept is given, which can also makeJohannes Kepler The observed conservation of angular momentum holds.However, this is in direct conflict with Einstein's special theory of relativity, which defines the speed limit——vacuumSpeed of light in - At this speed signals can be transmitted.
Observation problems
Newton's theory does not fully explainMercuryAfter moving along its track toPerihelionThe precession phenomenon occurs when.Compared with the precession actually observed, the prediction of Newton's theory (generated by the gravitational drag of other planets) will have an error of 43 arc seconds every century.
Newton's theory predicted that the deflection of light under gravity is only half of the actual observation result.General relativity is closer to the observation.
Of all objectsGravity massAndInertial massThe same observation cannot be explained by Newton's system.General relativity regards it as a basic condition.See Item Equivalence Principle.
Theoretical limitations
When Newton's extraordinary work enabled the law of universal gravitation to be expressed in mathematical formulas, he was still dissatisfied with the implicit“Action at a distance”Views.He never "endowed the reason for this ability" in his words.In other cases, he uses the phenomenon of motion to explain why objects are affected by different forces, but in the case of gravity, he cannot use experimental methods to confirm that motion produces gravity.In addition, he even refused to put forward a hypothesis about the cause of the force generated by the ground, which violates the principle of scientific evidence.
Newton's classical mechanics only applies tolow speed、macroscopic, weak gravity, but not applicable tohigh speed、microcosmicAnd strong gravity.
Newton's discovery of gravity buried the so-called truth that "philosophers have his foolish attempts to explore the search of nature in vain", which is the same as his conviction that "there are various factors" that make "various unknown reasons" the basis of all "natural phenomena".These basic phenomena are still under study, and although there are many kinds of assumptions, the final answer is still not found.althoughEinsteinIt is true that Newton's hypothesis can more accurately explain the effect of gravity in determining the case, but he never gave a reason for this ability in his theory.In Einstein's equation, "matter tells space how to curve,and space tells matter how to move)，But this new idea, which is completely different from Newton's world, cannot make Einstein's "reason for producing this ability" more capable of distorting space than Newton's "reason for producing this ability".
Newton himself said: I have not yet been able to find the reason for these gravity characteristics from the phenomenon, and I can not speculate... the laws and abundantcelestial bodiesThe calculation of motion is enough to show that gravity does exist and can produce effects.One object can act on another object through the distance of real space without passing through any medium, on which their activities and forces can be transmitted from each other, which is a huge fallacy for me.Therefore, I believe that anyone with enough philosophical thinking ability will not indulge in it.I have not yet been able to discover the cause of these properties of gravity from phenomena and I feign no hypotheses... It is enough that gravity does really exist and acts according to the laws I have explained,and that it abundantly serves to account for all the motions of celestial bodies. That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe,no man who has in philosophic matters a competent faculty of thinking could ever fall into it.
It should be noted that the word "cause" used here does not mean "cause and effect" or "the defendant caused the death of the victim".Moreover, when Newton used the word "cause", he (obviously) meant an "explanation".In other words, the phrase "Newton's gravity is the cause of planetary motion" means that Newton's gravity explains the motion of planets.
evolutionary process
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Past theories
AristotleAccording to the theory of gravity, Aristotle believes that the speed of motion of an object is proportional to the resultant force of the outside world (or the gravity of the object itself), and inversely proportional to the viscosity of the moving medium of the object.
Nikola Tesla (Nikola Tesla) announced but never published gravitational dynamics theory;Part of the reason is that the details of the theory (if any) have not been disclosed and have not received the attention of physicists.
Induced GravityAndrei Sakharov (Andrei Sakharov) proposed that general relativity may originate from quantum field theory.
Le Sage's Theory of Gravity (also calledRasaki's theory of gravity）, proposed by George Louis Le Sage, this phenomenon is explained by a flow of light gas filled with the whole universe.
Nordstr ö m's Theory of Gravity, an early competitor of general relativity.
Whitehead's Theory of Gravity, another early competitor of general relativity.
Newton's law of gravitation
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The mutual attraction between any two objects caused by the mass. The line of action of the force is about the line between the centroids of the two objects. Its size is proportional to the mass of the two objects and inversely proportional to the square of the distance between the two objects.The law of universal gravitation was discovered by Newton after tracing the reason why objects on the ground were affected by gravity, and was officially published in 1687.withmone、mtwoRepresents the mass of two objects,rIndicates the distance between the two, then the force of mutual attractionFFor:
, whereGbe calledUniversal gravitational constant。This is the mathematical expression of the law of universal gravitation.Strictly speaking, the above formula is for two particles.Because the term "distance between two objects" refers to the distance between two particles.If one is a particle and the other is a finite body, the finite body can be divided into many particles, and the vector sum of their gravity can be obtained, so that the force of the entire finite body on the particle can be obtained.Newton once proved that a density is the distance from the center of the sphererThe gravity of a sphere with a function of 0 is the same as that of the whole sphere with its mass concentrated at the center of the sphere.Newton proved thatKepler's law, the movement of the moon around the earth, the causes of tides, the flat poles of the earth and other natural phenomena.Newton's law of universal gravitation is the basis of celestial mechanics.The orbits of artificial satellites, moon and planetary probes are calculated on the basis of this law.Experimental proof of the existence of universal gravitation and gravitational constantGCavendish made the measurement in 1798.At present, the recognized value of the gravitational constant is G=6.6732 × 10 ⁻ ¹ Nm ² kg ⁻ ²[4]。
General relativity
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In 1859, French astronomer Le Vie found that the value of the precession rate of Mercury's perihelion deviated from the value calculated by the law of universal gravitation by 38 ″ every hundred years (the value measured by American astronomer S. Newcomb was 43 ″).In 1915, Einstein created the general theory of relativity, which finally explained this problem and predicted the deflection of light in the gravitational field and the red shift of the spectrum.Astronomers also predicted the existence of black holes, making general relativity enter a new realm related to the evolution of the universe.
Einstein denied the existence of inertial coordinate system in space by the equivalence of acceleration coordinate system and gravitational field, and changed the characteristics of space by gravitational field.He believes that the motion of an object in the gravitational field is a geodesic along a four-dimensional curved Riemannian space.But in the case of weak gravitational field (such as the solar system), for many mechanical problems, it is much simpler to use Newton's law of universal gravitation than Einstein's general theory of relativity, and the difference between the two is very small.For simpleTwo body problemBecause the concept of "simultaneity" is mixed, it is difficult to deal with it mathematically with general relativity.
In the microcosmic world of particle interaction, universal gravitation is the weakest. The unification of universal gravitation, electromagnetic force and nuclear force needs further efforts of scientists.
reference
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1. Author: Wang Jiaxuan《Encyclopedia of China 》Volume 74 (first edition) Mechanical entry: gravitation: China Encyclopedia Press, 1987:490
