## Derivation Of The Gravitational Force

Newton derived the gravitational force law from his three laws of motion, together with certain thought experiments and data reflecting the motion of the Moon relative to the Earth. As they are illuminating, we reproduce some of Newton's arguments, phrasing them in modern language.

The actual motion of the Moon around the Earth is quite complicated due to gravitational interactions with the Sun and other nearby planets. As a reasonable first approximation, however, the Moon's orbit is roughly circular. In uniform circular motion, the centripetal acceleration (centripetal acceleration is the rate of change of velocity in the direction tangent to the circle) has magnitude where v is the speed of the Moon in its orbit, and r is the orbital radius measured from the center of the Earth to the center of the Moon. The average speed of the Moon in its orbit is where T is the orbital period of the Moon. Thus the magnitude of the centripetal acceleration is

Both r and T may be determined from astronomical observations and long-distance surveying techniques.

The radius of the lunar orbit is about 3.84 x 108m, and the orbital period of revolution is 27.3 days, or about 2.36 x 106 seconds. Substituting these numerical values into the preceding formula, we find v =

Let ae denote the acceleration due to gravity on the surface of the Earth, and let am denote the acceleration due to Earth's gravity at the distance of the lunar orbit. Then

### The Lunar Calendar

Calendars based on the lunar orbital period of 27.3 days are still in use in most areas of the world. The Islamic calendar is the calendar used to date events in predominantly Muslim countries and used by Muslims everywhere to determine the proper day on which to celebrate Muslim holy days. It is a purely lunar calendar having 12 lunar months in a year of about 354 days. By contrast, The Chinese calendar is a lunisolar calendar formed by combining a purely lunar calendar with a solar calendar. This combination is performed by inserting an extra month every second or third year, so that the same months approximately correspond to the same seasons.

On the other hand, radius(lunar orbit) 3.84 x 108 m radius(Earth) 637 x 106 m

Since 602 = 3600, one may conclude that the acceleration caused by the gravitational force evidently decreases proportionally to the inverse square of the distance. By the second law, F is proportional to a, so the gravitational force must also vary inversely to the square of the distance. In other words, where r is the distance between the Earth's center and the center of the Moon, and the symbol "<*" means "proportional to."

Newton then reasoned that the strength of the gravitational interaction between the Earth and the Moon depends strongly on the masses of the two objects, and not strongly on other physical properties, such as chemical composition. It certainly didn't have to be this way. Nevertheless, Newton's intuition was correct: we live in a universe where nongravitational interactions between the Earth and the Moon are negligible, and where the strength of the interaction is governed by the mass.

Let M be the mass of the Earth and m be that of the Moon. With these assumptions and using the third law of motion, the Earth exerts a force FM on m on the Moon that is equal and opposite to the force Fm on M exerted by the Moon on the Earth. More precisely, these forces are equal in magnitude but opposite in direction:

In particular, the length (or magnitude) of these two vectors is the same. Let's call their common magnitude F (M,m). Then our argument shows that this is a symmetric function:

The symmetry of Equation 3.6 implies that the function F is a sum of terms that take the form of a sum or product of the two masses, that is,

for some constants A, B. We can in fact rule out all possibilities except the first power of the product of the masses by appealing to the second law and to experiment.

We now describe some very important experiments, originally due to Galileo, that help to fix the functional form of F. Consider the hypothetical action of dropping an object off of the top of the Empire State Building. It has been observed that if we neglect the effects of air resistance, then any object you could drop would reach the ground at the same time, regardless of the mass of the object! In other words, if we fix a mass M, and consider dropping other objects of masses m1,m2, . . . onto M, then the acceleration experienced by these objects is actually independent of m1,m2, . . . and so on. This has the disturbing consequence that if we dropped a chicken and a pound of lead off of a high building at the same time on a planet with no atmosphere, they would hit the ground at exactly the same time.

By the second law applied to system m, we have where the magnitude \agrav\ cannot depend on m by the above observation. Therefore, Fgrav/m cannot depend on m, either. The only way to reconcile this with Equation 3.7 is to take A = 0 and n = 1, which implies Fgrav <^Mm, with a constant of proportionality that, as we have already seen, must depend on r and (to some good approximation) the r-dependence takes the form r ~2. Let's then factor out the r-dependence and call the remaining constant (which now really is constant) G, the gravitational constant. This yields the result:

The constant G can be measured by experiments, and is found to have the approximate value G ~ 6.67 x 10-11 N m2/kg2. Equation 3.8 is called Newton's law of universal gravitation, though even in Newtonian mechanics, it is not a fundamental law; rather it is an analytic expression derived from the other laws and certain measurements.

The meaning of the word universal in the law of universal gravitation is questionable. The equation does seem universal in the sense that we expect it to also hold in some other part of the universe, such as the Andromeda galaxy. This property of a physical theory is called translation invariance, and it means that fundamental physics is the same here as in some other part of the universe. As intuitive as this property may sound, it is probably not correct in all cases. Some theories imply that parts of the universe might have been in contact early in the universe, but as the universe expanded they lost contact and their local laws of physics went separate ways.

Parallel universes can have different values of the fundamental constants of cosmology, such as the Newton constant, G, that we have become familiar with. It is not known whether some more fundamental theory fixes the value of G to be what it is in our universe, though we can say that had G (and other constants) been outside of a certain range, then structure such as galaxies could not have formed (and we could not exist). Imposing the condition that the universe must facilitate galaxy formation is actually a quite strong condition, and restricts one to a small corner of the space of possible values for parameters such as G. Inquiry and debate about these issues continues today; see in particular the work of Max Tegmark on parallel universes.