Terms: Gravitation (or gravity), inverse-square law, solar constant.

Stories: The Story of Newton's apple is traced to its source. Also, mention is made of the black hole at the center of our galaxy, of Cavendish and Eötvös (and in this lesson plan, of Coulomb).

    Starting the lesson:

    Today we will learn about Newton's theory of universal gravitation. The story is often told that Newton was inspired to this by a falling apple--with some going so far as to suggest it bonked him on the head as he was sitting under a tree. We will come back to that story.

    Of course, Newton did not "discover" gravity. People in his time were well aware that the Earth attracted all objects near its surface, and Galileo, as you learned some time ago, experimented with falling objects. Other people studied the trajectories of cannonballs, which depend on gravity. But all these observations had to do with what happened close to Earth, in a region accessible to experimenters. The objects in the sky--planets orbiting in space, the Moon circling around the Earth--seemed to belong to a different class of phenomena, with separate rules, such as Kepler's laws. Astronomers were only beginning to understand those phenomena.

    Newton's achievement was in making the connection, in proposing that the same force that pulled apples down also pulled the Moon.

    Suppose, he said, that the force decreased like the square of the distance from the center of Earth. The Moon is 60 times more distant than the surface of the Earth, so the force of gravity should be 60 times 60, or 3600 times weaker. Would that equal the centripetal force mv²/R or 4π²N²R needed to hold the Moon in its orbit?

    Newton could calculate that--he knew R from Aristarchus and Hipparchus, and he knew N = 1/T, where T was the orbital period of the Moon. As you will see, the result fit well, and Newton concluded that it was indeed the gravity of the Earth that held the Moon to its orbit. He also guessed (correctly) that the Sun had a force of gravity much like the Earth's, and that was what kept the planets going around it. That is why he named his idea the theory of universal gravitation.

    Was the inverse squares law just a lucky guess? It was more. Imagine a source of light--say, the Sun--shining at equal intensity in all directions, sending out each second sunlight carrying an energy of I joules--that is, radiating a power of I watts. What is the density at which solar power flows at a distance of R meters, across an area of, say, 1 square meter (1 meter²)?

    A sphere of radius R centered on the Sun has an area 4πR² meters². Since all parts of the sphere receive equal illuminaton, each square meter receives a power I/4πR² watts, and obviously, that intensity decreases like 1/R². At Earth's orbit it equals 1360 watt/m², a quantity known as the solar constant. At the orbit of Jupiter, R is about 5 times larger, and the light intensity is only 1/ 5² = 1/25 of the solar constant, which is why Pioneers 10 and 11, and Voyagers 1 and 2, preferred a power source other than solar cells. And although a lightbulb does not shine with equal intensity in all directions, your reading illumination, too, will decrease approximately like the square of the distance from it.

    [By the way: for historical reasons, phenomena which spread energy and obey the 1/R² law are often classified as radiation. Microwaves used to heat food in the kitchen are called radiation, fast ions and electrons emitted by radioactive substances are also called radiation--even though the two phenomena are different and unrelated. Microwaves are similar to light and radio, ions and electrons are fast-flying pieces of matter.

    Students should be made aware that the damaging kinds of radiation are the ones also known as ionizing radiation, "ionizing" meaning they deliver enough energy to atoms and molecules to tear off electrons. These can be fast ions and electrons from radioactivity, or high-energy "electromagnetic" radiation similar to microwaves, radio waves and light, but of the types known as x-rays and gamma rays.]

    Newton's "universal" gravitation led to a principle which today is all too easily taken for granted: that the same physical laws hold anywhere in the universe. All the astronomical evidence since Newton's time seems to confirm it: not just the laws of gravity, but the chemical elements, the speed of light, the way light is produced and other physical processes seem to be the same on Earth and in distant stars, or even in other galaxies.

Now let us go back to that apple. (Continue with the original story of Newton's apple as given in "Stargazers", which a student--with clear voice--might read aloud in class).

(Since the entire lesson focuses on a single calculation, not much can be asked beyond its details.)

--That the force of gravity which causes objects to fall was also the force that kept the Moon in its orbit around Earth.

(1) That the gravitational attraction of a sphere is the same as what we would get, if we let its entire mass be concentrated at the center. By symmetry, of course, the force is directed to the center.

(2) That the strength of the force of gravity decreased with distance R from the center of attration like 1/R² ("An inverse square law of attraction").

    The French military engineer Augustine Coulomb lived a century after Newton. In 1777 he won a prize for a new method to measure the Earth's magnetism--by suspending a horizontal bar magnet by its middle, from a long string, with a pointer attached--or better, a small mirror (draw on the board). When a beam of light is reflected from the mirror, even small magnetic variations can be measured, even those that twist the magnet by just a tiny amount from its quiet position.

    Coulomb also adapted this "torsion balance" instrument to measure the force between two magnetic poles--either repelling or attracting. You bring the pole end of a second bar magnet near one of the poles of the suspended magnet. The force between them can then be measured by the twisting of the suspension string, which exerts an elastic force. Coulomb found that magnetic poles, like gravity, attracted and repelled with a force that decreased like the inverse of the square of the distance.

    Coulomb's experiment was so sensitive that static electricity, which generated its own forces, sometimes interfered with its observations. That gave Coulomb the idea of measuring electric forces the same way. He replaced the magnet with a little dumb-bell shaped stick, with small spheres at the end. He then charged one sphere with static electricity, brought close to it another charged sphere, and measured the force. Guess what? Electric forces also decreased like 1/R².

    A reclusive English gentleman named Henry Cavendish set out to use the same method to measure the force of gravity directly. Instead of two lightweight spheres at the end of the stick, he used two heavy ones, and attracted one of them by a third sphere, a big one. The force was tiny and because of the big masses, the instrument reacted agonizingly slowly. But in 1796, he finally succeeded--showing that not just the sphere of the Earth, but even spheres of lead in the lab, exerted a measurable force of gravity.

    Thus for a while it seemed that three major forces in Nature--magnetic, electric and gravity--all obeyed the inverse squares law. Some differences existed--magnetic poles always came in pairs, and unlike the other two forces, which could both attract and repel, gravity could only attract. Nature seemed remarkably symmetric--until 1820, when Oersted and Ampére showed that magnetism was really quite, quite different.

G ("big gee") is known as the gravitational constant, but Newton did not know that number. All he knew was the acceleration g ("little gee") at the Earth's surface, about 9.81 m/s^{2     What is the relation between G and g ?}

--The weight of mass m is mg Newtons, and that should be equal to the pull of the Earth, if its mass M were concentrated in its center. Let R be the radius of the Earth. Then
mg = G M m / r² so g = G M / r²

Problem: If the mass density of Earth is D kg/meter³, express g (1 meter³ of water weighs 1000 kg, so D will amount to a few thousands).

The volume of a sphere of radius R is (4π/3)R³, so its mass M is (4π/3)DR³, and we get
g = G (4π/3)DR³ / R² = (4π/3) G D R

    There would be less mass pulling down objects on the surface--but the surface would also be closer to the center of attraction! Still, as the above formula shows, the first factor outweighs the second.

By the formula, if R → R/4, then g → g/4

Gravity would decrease to 1/4 of its existing strength.

    How does g', the gravitational acceleration on the surface of the Moon, compare to g, the acceleration at the surface of Earth? In other words, how would the weight mg' of an object on the Moon compare to its weight mg on Earth? (Use a calculator, of course.)

The weight of mass m on the Moon is
                mg' =   G (0.012307 M) m /(0.2725 R)²
                      =   [0.012307/(0.2725)²] G M m / R²
                      =   (0.012307/0.74256) mg   =   0.16574 mg

Divide both sides by 0.16574

6.034 g' = g

Note to the teacher: Variations of this problem can be given as homework or on tests, with numbers adjusted to apply to other planets or moons of the solar system (see Bill Arnett's "nineplanets" web page), or you can invent your own. For instance: what is the gravitational acceleration on the surface of a planet which has twice the radius of Earth and 8 times the Earth's mass? (No need for a calculator with this one.)

    Since the Moon's radius is about 1/4 that of Earth (0.2725, more exactly), if both had the same density, then by a previous calculation, g' would be about 4 times weaker than g. Actually, however, g' is smaller than that, about 6 times weaker than g, suggesting the Moon has less density and therefore less attraction.

    Part of the reason may be the iron core of the Earth, since iron is denser than rock. Also, the Earth is much larger, pressing down with greater weight on its innermost layers and perhaps compressing them to greater density.