Gravity, Newton's laws of motion and the orbits of the planets Joe Wolfe School of Physics, The University of New South Wales. Flash animations by George Hatsidimitris An introduction to Newtonian gravity and planetary motion. Most of the mathematical detail has been omitted, but is available in links and references. Only Newtonian mechanics and gravity are considered here. We do, however, have a site introducing special relativity for the non specialist. Gravity in context in physics Forces, acceleration and Newton's laws of motion Circular motion requires centripetal acceleration Newton's law of gravity Cavendish measures the constant of gravitation (and thus 'weighs the earth, sun etc') Planetary motion, Kepler's laws, and Newton's laws Gravitational potential energy Escape velocity and black holes The limits to Newtonian gravity References and links Appendix: Relations among the forces of nature Appendix: Film clips illustrating Newton's second law Appendix: Centripetal accleration   The image shows the gravitational effect of a very massive cluster of galaxies in the right of the frame. This acts like a magnifiying but distorting lens, so that the light from a localised source, located beyond the cluster but on the same line, appears to us as arcs of a circlecentred on the cluster. The sketch is not to scale. Image by Warrick Couch, Physics, UNSW, using the Hubble Space Telescope. Click on image to enlarge.

Gravity is a puny force. You stand next to a tank containing many tons of water and you don't even notice its gravitational force on you. Conclusion: to produce substantial gravitational force, one of the objects has to approach planetary size. Let's be quantitative: A hydrogen atom is an electron and a proton. The electric attraction between them is bigger than the gravitational attraction by a factor of 5 x 10³⁹.

The attractive force that holds the nucleus together -- very aptly called the strong force -- is in some cases stronger than the electric force: if the nucleus is small enough, the strong force wins and it is stable. But really big nuclei fall apart. This appendix has a chart showing the known forces and some relations among them.

So how come gravity rules the universe?

• Nuclear forces only have a finite range ­ about the size of nucleus. (This finite range is determined by Heisenberg's Uncertainty Principle, using an argument that won a Nobel Prize for Hideki Yukawa.) Electromagnetism and gravity both have very large, perhaps infinite range.
• Electric charge comes in positive and negative, which cancel. Mass is always positive and, for gravity, every mass attracts every other mass. The earth contains almost exactly the same number of protons and electrons, so its electrical field is small.
So gravity runs the planets, the stars, the galaxies, galactic clusters... Let's see how.

Galileo and Newton: forces and accelerations

Before Galileo and Newton, Aristotle's writings dominated European thought about physics. Oversimplifying a little, it was thought that:
• things on earth fall or rise because they find their 'natural' place
• planets etc move due to various, usually supernatural forces: (in ancient Greek metaphor, they are pushed by the sirens and fates)
but there is no connection between the two: in fact, natural and supernatural are usually contrasted.

A link to an introduction to Newtonian mechanics and Galilean relativity

Physclips: an introduction to mechanics with film clips and animations.

Only acceleration requires force

Acceleration is the rate of change in velocity. Imagine you're driving at a fixed speed: you push the accelerator, and your speed increases: this is a positive forwards acceleration. But you also need brakes to stop: a force is required to decelerate. Forces cause accelerations, and also decelerations, which are just negative accelerations.

One of Galileo's and Newton's insights was that the 'natural' condition was zero acceleration. If there are no forces -- no friction, no air resistance etc -- then an object at rest stays at rest, and a moving object continues moving at a constant speed in a straight line. This is Newton's first law of motion.

Now a bigger force is required to accelerate a massive truck than to accelerate a bicycle. This is included in Newton's second law of motion, which states that the total force F applied to a mass m produces an acceleration a where

F = ma.
Force and accleration are vectors, meaning that they have direction, as well as magnitude. If the total force on a body is in the North direction, then it will accelerate North. Force is measured in newtons (abbreviation N). One newton is the force required to accelerate one kg at one metre per second per second. A newton is about the weight of a small apple.

Newton's law of motion says that a force is required to get an object moving (an acceleration in the direction of its motion), or to stop it moving (a negative acceleration in that direction). A force is also required to change the direction of its motion because, as we shall see, that is also an acceleration.

Note that mass is not the same as weight. If your mass is 70 kg, then your weight on earth is 70 kg multiplied by the earth's gravitational field, which is about 10 metres per second per second. So your weight is 700 newtons. Suppose you went to the moon, where the gravitational field is six times smaller. Your mass would still be 70 kg -- you would still be made of the same amount of material -- but your weight would be only 120 newtons. On the moon, you would fall more slowly and it would take less force to hold yourself upright, but it would still take the same force to accelerate you.

A change in motion requires a force You're standing in the bus when it starts moving forwards (when it accelerates forwards). You are holding onto the grab bar. The floor of the bus pulls your feet forward, the grab bar pulls your arm and body forward - you accelerate forward with the bus. Another passenger is not holding the bar. His feet accelerate forward with the bus, but not his body. The bus accelerates forwards, he falls backwards. When the bus driver brakes, the reverse applies. The bus and the rail apply a backwards force on you, and you decelerate (ie accelerate in the negative or backwards direction). The body of other passenger, however, continues forwards while the bus slows. The bus decelerates (ie accelerates backwards), he falls forwards.

Circular motion requires centripetal acceleration What happens when the bus turns to the right, at constant speed? According to Galileo and Newton, an unrestrained passenger would tend to continue travelling in a straight line. For the passenger not holding the grab bar, his feet turn the corner with the bus, but his body keeps going forwards. The result is that he falls to the left. In this case, the bus accelerates to the right, he falls to the left. (When you try this experiment, make sure that the person sitting on your left is simpatico.) There is an appendix below in which we derive an expression for the centripital acceleration: the acceleration towards the centre in circular motion. For an object moving at speed v around a circle with radius r, the acceleration is ac = v2/r. The period T is the time taken for one circle, so v = 2πr/T. So we can also write ac = 4π2r/T2. (Without maths, we can still explain in an inexact way why it has this form: the bigger v is, the larger the acceleration, because it goes from forwards at v to backwards at v twice per cycle. But the cycle repeats at a rate proportional to v/r, so it is proportional to both v and to v/r. For a real explanation, you need the maths.) So an object travelling in a circle at constant speed is always accelerating but, because the acceleration is towards the centre of the circle (it's called a centripital acceleration) it doesn't add to or subtract from the speed. A force that causes a centripetal acceleration is called a centripetal force. Let's now look at some accelerations and forces.

Demonstrations of forces and acclerations

These film clips use a spring to accelerate a bowling ball. You can tell whether the spring applies a force from whether it is stretched or not, as in these photos.

    

         

See more of these film clips below.

In 1687, Isaac Newton wrote "I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve; and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly."

Here, translated into modern units, is Newton's calculation of the centripital acceleration of the moon due to its circular orbit around the earth. Its direction is towards the earth. We can work out its magnitude from our equation ac = 4π2r/T2, using the distance from the earth to the moon (380,000 km) and its period of 27.3 days. I don't know whether Newton ever thought about falling apples, but if he did, he would have known that (in today's units) they fall at 9.8 ms-2. So the ratio of the accelerations is Now we might expect that the effect of the earth's gravity on the moon would be less, because it is further away than the apple. For the apple, the distance from the earth's centre is the earth's radius: Re = 6370 km. So the ratio of the distances is     so

Hence Newton's idea: what if the apple and the moon accelerate according to the same law? What if every body in the universe attracts every other, via an inverse square law? Note the contrast with the Aristotelean ideas mentioned above: for Newton, the same laws govern the 'heavens' as govern familiar objects. (One could see an extension of this idea as the ambit of contemporary physics: to find a simple set of laws that explain everything, from the biggest to the smallest, over the whole history of the universe.)

Newton's law of gravity

In modern notation, we write it thus: if two bodies with masses m₁ and m₂ are separated by a distance r, then the attraction between them is
There are several things to note:
• The negative sign is used to mean an attraction. If m₁, m₂ and the separation between them are positive (and they always are -- there are no negative masses), then the gravitational force between them is attractive. As we saw above, this is why gravity always wins on the large scale.
• The force is proportional to each of the masses. This is not surprising: experimentally we find that two apples, side by side, fall with the same acceleration as one apple, so the force on twice the mass is twice the force on one.
• There is a constant of proportionality, G, the universal constant of gravitation, which we can measure -- see below.
• Gravitational forces, like all forces ever discovered, come in pairs that are equal and opposite. If m₁ attracts m₂ with a force F, then m₂ attracts m₁ with a force - F. The observation that all forces come in such pairs is Newton's third law of motion.
• The r² in the denominator means that the attraction gets weaker in proportion as the square of the separation.
Why is it inverse square? is an interesting question. One explanation is that it comes from the three dimensional, approximately flat geometry of our universe. It comes from the area of a sphere, which is 4πr², where the factor (1/4π) is absorbed in the constant G. In electricity, the analogous relation between geometry and the strength of the field is called Gauss' Law.

Another interesting point is that the same quantity, mass, that appears in Newton's equations of motion, also appears in his law of gravitation. This is not due to a lack of imagination on Newton's part: he thought carefully about this coincidence. Why is the quantity that determines inertia equal to (or at least proportional to) that which determines gravity? Perhaps it is not a coincidence, which is Mach's Principle. Or perhaps they really are the same thing, which is a starting point for Einstein's theory of General Theory of Relativity.

It is difficult to measure G simply because gravity is so weak: the forces between masses than can be manipulated are small. Of course one can easily measure the gravitational force between the earth and another object -- that is what we call the other object's weight -- but to get G we need to know the mass of the earth, and we do not know that without G. This diagram shows, schematically, the technique used by Henry Cavendish in 1798 to obtain the first accurate measurement of G. Two large masses are mounted on a rod, which is suspended by a thin wire. When the wire is twisted, it tends to untwist itself (technically, it exerts a restoring torque proportional to the angle of twist). This can be calibrated: one can determine the forces required at each end of the rod to twist the wire by a given angle (for instance, this can be determined by rotating the rod, letting it go, and measuring the frequency of its oscillation). Once the system is stable, two large, known masses are positioned near the masses on the rod. The tiny gravitational attraction between the pairs of masses twists the wire slightly, and the force is calculated from the new equilibrium position. In the diagram, the angle has been exaggerated. In practice, the deflection may be very small, but can be measured by mounting a mirror on the rod and measuring the deflection of a beam of light projected onto a distant screen. Here is a link to a Do it yourself Cavendish experiment. Cavendish used Newton's equation for gravitational force: From the deflection and the calbration of wire he calculated F. He already knew m1 and m2, so he calculated G. In modern units, its value is G = 6.67 x 10-11 Nm2kg-2. In other words, two one kg masses separated by a metre attract each other with a force of 0.00000000007 newtons. Cavendish could thus measure the mass of the earth, Me. An object of mass m at the earth's surface is Re from the earth's centre. Its weight is approximately* mg, where g is the acceleration it has in free fall.

Knowing the mass of the earth and its orbit around the sun, one knows the centripital force exerted by the sun on the earth, and so one can calculate the mass of the sun. (The orbits of moons around most of the planets gives the masses of the planets.)
* "Approximately" is there because the earth rotates. See the discussion of the Foucault pendulum, which introduces inertial frames of reference. Because of the earth's rotation, the centripital acceleration of a point on the surface of the earth must be added to the acceleration of a falling body, measured with respect to the earth at that point, to obtain its 'real' acceleration, ie the acceleration in an inertial frame. A consequence is that, measured with a spring balance, an object would seem to weigh more at the poles than at the equator. There is also an introduction to inertial frames of reference on our relativity site.

Gravity is a puny force

Although measurements of the constant of gravitation are now much more accurate than that made by Cavendish, G is still the least well known of the fundamental constants, simply because gravity is so weak. Let's illustrate this with some values.
• Oil tankers are among the largest objects we can easily move. Two 100,000 tonne tankers, anchored 100 m apart would exert a gravitational attraction on each other of approximately Gm₁m₂/r² = 70 newtons. This force (roughly equal to the weight of a case of wine) is tiny compared to the forces of wind, waves and current. (Further it is equal to the gravitational attraction of the volumes of water that they have displaced and so would be harder to measure.)
• What about the gravitational attraction between two 70 kg people sitting 0.4 m apart? Here, Gm₁m₂/r² is 2 micronewtons, which is too small to feel, even with the tip of the finger.
Two conclusions follow. One is that we can usually neglect gravity unless at least one of the bodies approaches planetary size. The second conclusion is a quotation attributed to Albert Einstein "Gravitation cannot be held responsible for people falling in love".

However, as we mentioned above, the cancellation of positive and negative electrical charge and the limited range of nuclear forces means that, on the large scale, gravity wins. Which brings us to

and what has been known, after a metaphor of Plato, as "the music of the spheres".

As long ago as the fifth century BC, Leucippus and Democritus proposed a heliocentric universe, ie one in which the planets orbit the sun. On the mistaken assumption that the earth's motion ought to be noticeable, Hipparchus (second century BC) and Ptolemy (second century AD) proposed a universe, in which the sun and planets executed complicated motions around the earth.

The Danish astronomer Tycho Brahe (1546-1601) made very many, very careful, naked eye observations of the positions of the planets. He was joined by Johannes Kepler, a tireless calculator. After a long time trying to fit circular orbits and even musical harmonies to the data, Kepler eventually discovered that the data were all well fitted by the following empirical laws.

Kepler's laws

• 1. All planets move in elliptical orbits, with the sun at one focus.
  Except for Pluto, the elliptical orbits of the recognised planets are approximately circles. The mass of the sun is so much greater than that of the planets, that the sun is the centre of mass of the solar system.
• 2. A line joining the planet to the sun sweeps out equal areas in equal time.
  ie Planets move more slowly at apogee (distant), more rapidly at perigee (close)
• 3. The square of the orbital period (ie the planet's year) is proportional to the cube of the semi-major axis of its orbit.
  Slow orbits for distant planets, fast for close.
Why is it so? What underlying reasons are there for Kepler's laws. It is difficult for a modern person to have a feeling for the magnitude of this puzzle. For two millenia, this was the riddle of riddles. Christopher Marlowe created Doctor Faustus (perhaps based on the historical character Paracelsus) -- a philosopher/scientist who sells his soul for knowledge. Here is an extract from the climax of the play, in Act VI:
Faustus:    I am resolv'd: Faustus shall ne'er repent.
   Come, Mephistophiles, let us dispute again,
   And argue of divine astrology.
   Tell me, are there many heavens above the moon?
   Are all celestial bodies but one globe,
   As is the substance of this centric earth?
Mephistopheles:      As are the elements, such are the spheres
   Mutually folded in each other's orb,
   And, Faustus,
   All jointly move upon one axletree
   Whose terminine is termed the world's wide pole;
   Nor are the names of Saturn, Mars, or Jupiter
   Feign'd but are erring stars.
Faustus:    But tell me, have they all one motion, both situ et tempore? 
Mephistopheles:    All jointly move from East to West in twenty-four hours upon the poles of the world; but differ in their motion upon the poles of the zodiac.
Faustus:    Tush!
   These slender trifles Wagner can decide;
   Hath Mephistophiles no greater skill?
   Who knows not the double motion of the planets?
   The first is finish'd in a natural day;
   The second thus: as Saturn in thirty years; Jupiter in twelve; Mars in four; the Sun, Venus, and Mercury in a year; the moon in twenty-eight days. Tush, these are freshmen's suppositions. But tell me, hath every sphere a dominion or intelligentia?
Mephistopheles:      Ay.
Faustus:    How many heavens, or spheres, are there?
Mephistopheles:      Nine: the seven planets, the firmament, and the empyreal heaven.
Faustus:    Well, resolve me in this question: Why have we not conjunctions, oppositions, aspects, eclipses, all at one time, but in some years we have more, in some less?
Mephistopheles:      Per inæqualem motum respectu totius*.          *On account of their unequal motion in relation to the whole
Faustus:    Well, I am answered. Tell me who made the world.
Mephistopheles:     I will not.
Are we ready now for the Age of Enlightenment? As a simple example, let's apply Newton's laws to the motion of most of the planets. For circular motion, the centripital acceleration is, as before
a_c = 4πr²/T².
Now add a force whose magnitude is GmM/r² and set F = ma:
GmM/r² = F_gravity = ma_c = m4π²r/T²,   so:

T² = (4π²/MG)r³

Kepler's third law. The same force that makes the apple fall rules the heavens. In these notes, we see that Kepler's second is also a simple consequence of Newton's laws of motion and gravitation.

When you lift something up, you expend energy -- you do work on it. This energy is stored, in the sense that you can get it back again. It is called gravitational potential energy. For instance, during periods of low electricity demand, energy authorities use electrical energy from coal-fired power stations to pump water up the Snowy Mountains into high dams. In periods of high demand, they can then let it flow back downhill to get back the gravitational potential energy of that water via the hydroelectric generators.

The gravitational potential energy U of two objects, masses M and m, at separation r, is obtained from Newton's equation by integration (details in the notes):

The zero of potential energy is taken, by astronomical convention at infinite separation, so it is always negative at finite separation. This does not affect the main point that work must be done to lift something: gravitational potential energy increases with separation. (If M is the earth and m a small object, we speak of the U of m in the field of M. However U is a property of the pair of objects and we could equally talk of the potential eneryg of M in the field of m.)

So far, we have only mentioned Newton's theory of gravitation. Although it is an excellent theory, it does not agree with experiment if one investigates extremely large fields, or moderately large fields with very high precision. In other words, it is wrong. However, it is such an excellent approximation that Newtonian gravity is what we use to calculate in almost all circumstances, while recognising that it is just a very convenient approximation to more exact theories.

To be accepted, any new theory of gravity must give virtually the same answer as Newton's in all of the many cases where Newton's theory works, but must also give the right answer for the cases where we know that Newton's theory fails, such as the slight deflection of starlight passing very close to the sun (observable during a solar eclipse) and certain aspects of the orbit of Mercury that cannot be accounted for by considering the effects of the other planets.

Several other theories do this, of which one of the simplest and by far the most widely used is Einstein's General Theory of Relativity. Philosophically, General Relativity is very different from Newtonian gravity, in that it doesn't use forces. Rather, the effect of large masses is to curve space, and the curvature of space determines the motion of objects. It is often summarised thus: "matter tells space how to curve, and space tells matter how to move".

So, if there are several competing theories that all do better than Newton's, what theory of gravitation should we use? The mission of Gravity Probe B, recently launched, was to conduct experiments to distinguish among some current theories of gravitation, so we should know soon.

However, both Einstein's and Newton's theories of gravity have a problem when they encounter quantum mechanics, and that problem involves the very nature of space and time. We explain this further on Gravity, relativity and quantum mechanics.

Fortunately, the scale of the problem is that of the Planck length, which is 1.6 x 10^-35 metres. This is immeasurably small, and it is perhaps not too surprising that our ideas about space would need to be revised on this scale.

(Yes, small: the Planck length is 0.000000000000000000000000000000000016 metres. Let's compare it with the size of an atom, which is already about 100,000 times smaller than anything you can see with your unaided eye. Suppose that you measured the diameter of an atom in Planck lengths, and that you counted off one Planck length each second. To measure the atomic diameter in Planck lengths would take you 10,000,000 times the current age of the universe.)

• Some notes about gravity from an introductory lecture course in mechanics.
• Resources for high school physics including the FAQ in high school physics.
• An interesting paradox: does this calculation show that the moon should fall into the sun?
• Physclips: an introduction to mechanics with film clips and animations.

Appendix: Film clips illustrating Newton's second law

The film clips below use a spring to accelerate a bowling ball. You can tell whether the spring applies a force from whether it is stretched or not, as in these photos. Use the step frame button on the film clips.
    

Physclips has a section on circular motion.

Happy birthday, theory of relativity! As of June 2005, relativity is 100 years old. Our contribution is Einstein Light: relativity in brief... or in detail. It explains the key ideas in a short multimedia presentation, which is supported by links to broader and deeper explanations.