Newton's theory of Gravity

1. Newton's laws of motion

English natural philosopher Isaac Newton unified Italian natural philosopher Galileo Galilei's theory of falling bodies with German astronomer Johannes Kepler's laws of planetary motion.

Newton published his laws of motion and universal gravitation in The Mathematical Principles of Natural Philosophy, commonly known as the Principia, in 1687.

1.1 Newton's first law

Newton's first law of motion states that objects continue to move in a state of constant velocity, which can be zero, unless acted upon by an external force. The tendency of an object to resist a change in motion is known as inertia.

Diagram showing a stationary ball. Image states ‘with no outside forces, this object will never move’.
Diagram showing a moving ball. Image states ‘with no outside forces, this object will never stop’.

Newton's first law. Image credit: Helen Klus/CC-NC-SA.

Galileo had first suggested this law, but it was still not universally accepted because it contradicted Ancient Greek philosopher Aristotle's laws of physics.

1.2 Newton's second law

Newton's second law shows how an object will be affected if an external force does act upon it. This law states that the rate of change of momentum of a body is proportional to the resultant force acting on it, and will be in the same direction.

This means that:

Force =
Momentum/Time
.

Here, Force = F, Momentum = p, and Time = t, so that:

F =
p/t
.
Momentum = Mass × Velocity, and Acceleration =
Velocity/Time
and so Newton's second law can be stated as:

Force = Mass × Acceleration.

Here, Mass = m, and Acceleration = a, so that:

F = ma.

This shows that less force is needed to push something lighter - which means that less massive objects have less inertia.

Diagram showing two balls, one is accelerating faster than the other. Image states ‘the more force, the more acceleration’.

Newton's second law. Image credit: Helen Klus/CC-NC-SA.

1.3 Newton's third law

Newton's third law states that the force on an object is always due to another object; all forces act in pairs that are equal in magnitude and opposite in direction. This is why you feel recoil when you strike an object, and why you do not fall through the Earth due to the pull of gravity.

Diagram showing a balloon moving up as air rushes out. Image states ‘every action has an equal and opposite reaction’.

Newton's third law. Image credit: Helen Klus/CC-NC-SA.

1.4 The conservation of momentum

The combination of Newton's second and third laws shows that momentum must be conserved. This means that the total momentum of two objects will remain the same before and after a collision.

This is because if F = ma and F = -F, then ma = -ma, a =
v/t
and so
mv/t
=
-mv/t
.
Here, v = Velocity.

2. Newton's law of universal gravitation

Newton's law of universal gravitation states that every mass attracts every other mass in the universe, and the gravitational force between two bodies is proportional to the product of their masses, and inversely proportional to the square of the distance between them.

Spherical objects like planets and stars act as if all of their mass is concentrated at their centre, and so the distance between objects should include the distance to the centre of both objects.

Newton stated that the force of gravity is always attractive, works instantaneously at a distance, and has an infinite range. Most importantly, it affects everything with mass - and has nothing to do with an object's charge or chemical composition.

This means that it can account for both the downwards force caused by the pull of the Earth - as described by Galileo - and the force that causes the planets to orbit the Sun - as described by Kepler.

Newton's law of gravitation shows that objects with different masses fall at the same rate when combined with his second law of motion. This is because an object's acceleration due to the force of gravity only depends on the mass of the object that is pulling it:

if F = m1a and F = G
m1m2/r2
, then m1a = G
m1m2/r2
, and so a =
Gm2/r2
.
Diagram showing two masses attracted by mutual gravitation.

Newton's law of gravitation. Image credit: modified by Helen Klus, original image by Dennis Nilsson/CC-A.

Here, G is a constant that's the same for everything in the universe, r is the radius, m1 is the mass of the less massive object (e.g. a feather or hammer), and m2 is the mass of the more massive object (e.g. a planet). This means that a feather and hammer will all fall at the same rate if they are dropped in the same place and there is no air resistance.

Gif showing a feather and hammer being dropped on the Moon and landing at the same time.

Commander David Scott dropping a feather and hammer on the Moon in 1971. Image credit: NASA/Public domain.

2.1 Newton's cannon

In 1728, Newton demonstrated the universality of the force of gravity with his cannonball thought-experiment[1]. Here Newton imagined a cannon on top of a mountain. Without gravity, the cannon ball should move in a straight line. If gravity is present then its path will depend on its speed. If it is slow, then it will fall straight down. If it reaches the orbital speed, then it will orbit the Earth in a circle or ellipse, and if it is faster than the escape velocity, which is about 11 km/s on Earth, then it will leave the Earth's orbit.

Diagram of Newton's cannon thought experiment, this shows how gravity describes the motion of objects on Earth and in space.

Newton's cannon thought-experiment. Image credit: Brian Brondel/CC-SA.

2.2 Newton's religion and absolute space

There is evidence that Newton was motivated to universalise his theory of gravitation because of his religious beliefs[2a]. In the 1600s, members of the Royal Society typically believed that science investigated the same truth as the bible, and that knowledge came in two forms: revealed truth, which comes from studying scripture, and natural theology, which seeks to learn about God by studying their creation. The conclusions of natural theology were only accepted if they agreed with revealed truth.

In the second edition of the Principia Newton stated that nature reveals a creator. He claimed that this was evident firstly, from the fact that the universe formed at all. Secondly, from the fact that the masses that did form are placed so that they do not fall together under their own mutual gravitation, and thirdly, Newton found evidence of design in the specifics of our Solar System, such as the fact that the orbits of the planets are all in the same direction and plane. Newton claimed that the eccentric orbits of comets alone reveal the existence of a creator and described God as "an intelligent and powerful being"[3a].

In private correspondence, Newton expressed his belief in the Hebrew God of the Old Testament[2b]. He considered himself to be Arian, believing that scripture had been wrongly interpreted at the first Council of Nicaea, in 325, when the Trinity first became an important part of Christen theology. Newton's belief that the Holy Trinity are not three persons remained illegal in England throughout his lifetime.

Newton stated that because a singular God "exists always and every where"[3b], space and time must be absolute. This means they provide a background in which things take place, and would continue to exist even if the universe were devoid of all physical matter. Newton argued that God's eternal nature implies absolute time, and God's infinite duration corresponds to absolute space.

The belief that space and time are absolute is known as spacetime substantivalism because it implies that space is composed of some kind of pseudo-substance, like Aristotle's aether.

2.3 Leibniz and relative space

German mathematician Gottfried Leibniz accepted that the laws of physics could be universalised, but objected to Newton's concept of absolute space. Firstly, he argued that Galileo had shown there's no such thing as absolute velocity, and so there can't be any such thing as absolute space, from which it's derived. Secondly, Leibniz objected to Newton's description of absolute space as a kind of physical entity because it has no causal powers or independent existence. Leibniz claimed that space is purely a mental entity[4][5].

The view that space only exists when physical objects are present is known as relationism. Relationism can be countered by the idea that although there is no absolute velocity, there is absolute acceleration, and absolute space can be derived from this[6].

This argument was reassessed in the first half of the 20th century, after the publication of German-Swiss-American physicist Albert Einstein's theory of general relativity[7].

2.4 Olbers' paradox

Newton's infinite, eternal, universe posed problems for astronomers as well as philosophers. In 1720, English astronomer Edmond Halley stated that if the universe is eternal, and the stars are infinitely old, then the sky should be as bright as the surface of the Sun in all directions[8][9]. This is because the starlight from an infinite amount of stars would have reached us by now, filling every part of the sky. This view was first considered by English astronomer Thomas Digges in the 1500s, and then by Kepler, German natural philosopher Otto von Guericke, French natural philosopher Bernard de Fontenelle, and Dutch natural philosopher Christiaan Huygens[10]. It was popularised by German astronomer Heinrich Olbers in 1823, and is known as Olbers' paradox[11].

Olbers' paradox was not resolved until American astronomer Edwin Hubble provided evidence of the big bang in the first half of the 20th century.

2.5 Weighing the Earth

In 1797, British natural philosopher Henry Cavendish used Newton's equations in order to weigh the Earth. He did this by measuring the faint gravitational attraction of a number of objects in the laboratory[12].

Once Cavendish could compare the different forces exerted by different masses, he could calculate Newton's proportionality constant G, and use it to work out the mass of other objects. The mass of the Earth was calculated to be about 6×1024 kg (this is six million, billion, billion), the value accepted today.

2.6 The precession of Mercury

Newton's theory was first questioned in 1859, when French mathematician Urbain Le Verrier showed that Mercury's orbit could not be explained by Newton's equations[13].

Diagram showing how Mercury orbits the Sun in an open, rather than closed, ellipse.

The precession of Mercury's orbit. Image credit: Mpfiz/Public domain.

Mercury does not form a closed ellipse when it orbits the Sun. Instead, the ellipse rotates. This sort of movement is known as precession, and was later explained by Einstein's theory of general relativity.

3. References

  1. Newton, I., 1728, 'A Treatise of the System of the World', F. Fayram.

  2. (a, b) Snobelen, S. D., 2001, '"God of gods, and Lord of Lords": The Theology of Isaac Newton's General Scholium to the Principia ' in 'Science in Theistic Contexts: Cognitive Dimensions', University of Chicago Press.

  3. (a, b) Newton, I. and Motte, A. (trans), 1846 (1726), 'General Scholium' in 'The Mathematical Principles of Natural Philosophy', Daniel Adee.

  4. Leibnitz, G. W. and Clarke, S., 2006 (1715), 'A Collection of Papers, Which passed between the late Learned Mr. Leibnitz, and Dr. Clarke, In the Years 1715 and 1716', The Newton Project.

  5. McDonough, J. K., 'Leibniz's Philosophy of Physics', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  6. Hugget, N. and Hoefer, C. 'Absolute and Relational Theories of Space and Motion', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  7. Norton, J. D., 'The Hole Argument', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  8. Halley, E., 1720, 'Of the Infinity of the Sphere of Fix'd Stars', Philosophical Transactions of the Royal Society of London, 31, pp.22-24.

  9. Halley, E., 1720, 'Of the Number, Order, and Light of the Fix'd Stars', Philosophical Transactions of the Royal Society of London, 31, pp.24-26.

  10. Harrison, E., 1990, 'The Dark Night Sky Riddle-Olbers's Paradox', The Galactic and Extragalactic Background Radiation, 139, pp.3.

  11. Olbers, H. W. M., 1823, 'Uber die Durchsichtigkeit des Weltraumes (On the Transparency of Space)' in 'Astronomisches Jahrbuch', C. F. E. Spaethen.

  12. Cavendish, H., 1798, 'Experiments to determine the Density of the Earth', Philosophical Transactions of the Royal Society of London, 88, pp.469-526.

  13. Le Verrier, U., 1859, 'Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète' ('Letter from Le Verrier to Faye on Mercury's theory and the motion of the perihelion of this planet'), Comptes rendus hebdomadaires des séances de l'Académie des sciences, 49, pp.379–383.

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