*© Fernando Caracena* 27 Oct 2012

*The musical theme for this section is "La Forza del Destino", by Giuseppi Verdi.*

**Newtonian mechanics and gravity**

**Newtonian mechanics and gravity**

At the age of 23, Sir Isaac Newton experienced a miracle year (Annus Mirabilis) between 1665 and 1666. He had left Trinity College in Cambridge to escape the Black plague at his mother's farm in Woolsthorpe, Lincolnshire. There (according to legend) as he sat under an apple tree, he realized that the same force of gravity that pulls an apple to the ground also keeps the moon in its orbit about the earth. Postulating that gravity is a universal attractive force that acts between any two material objects in proportion to the product of their masses and inversely with square of the of the distance between their centers, he was able to reproduce Kepler's laws of planetary motion. In his own words in a letter to Pierre Des Maizeaux, this is what he had to say:

*.....I began to think of gravity extending to the orb of the Moon & (having found out how to estimate the force with which a globe revolving within a sphere presses the surface of the sphere) from Keplers rule of the periodical times of the Planets being in sesquialterate proportion of their distances from the centers of their Orbs, I deduced that the forces which keep the Planets in their Orbs must be reciprocally as the squares of their distances from the centers 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... *

Although he developed his theories of mechanics and gravitation early in his professional life (in the 1660s), it was not until 1687 when Newton published his ideas in his *Philosophiae **Principia Mathematica.*

Mathematically, Newton expressed the form of the gravitational force acting between two objects, having the respective masses, m_{1} and m_{2}, as follows:

**F**_{2} = - G * m_{1 *} m_{2 * }**u**_{r12 * r12-2}, (1)

where **u**_{r12} is a unit vector that point from point 1 to point 2(the locations of m_{1} and m_{2}), r_{12 }is the distance between the centers of the masses_{,} and G is the universal gravitational constant, which is determined experimentally. The negative sign means that the force is attractive—it draws objects together. Using the above formula (1), Newton was able to account for all the quantitative observations of planetary motion as summed up elegantly in Kepler's laws of planetary motion^{1}; but the final proof of concept remained to be tested by actually measuring the very feeble force of attraction between two ordinary objects in the laboratory, where the formula predicts that ordinary sized objects exert forces on each other, which are imperceptible to our raw senses.

**Experimental determination of the Universal Gravitational Constant**

Henry Cavendish used an ultra-sensitive^{2} torsion balance (Fig. 1) that not only responded to the very weak pull of gravity between ordinary sized objects, but was also sensitive enough to measure it to several decimal places. Cavendish using measured values of gravitational force, masses and their distance of separation and Newton's formula (1) could have computed the value of the universal gravitational constant, G; but he did not determine G itself. Rather, he computed the mass of the earth. but the numbers he measured given the value of *G* as 6.754x10^{11 }m^{3}s^{-2}kg^{-1}. (*Philosophical Transactions of the Royal Society*, 1798 p469 )

Charles Vernon Boys, Assistant Professor of Physics at the Royal College of Science, having improved the balance almost a century after Cavendish, determined the value of *G* to be 6.658x10^{11}m^{3}s^{-2}kg^{-1}.

[*Philosophical Transactions of the Royal Society*, **186**, 1, (1895)]

The current Google™ search value is G = 6.67384(±80) × 10^{-11} m^{3} kg^{-1} s^{-2}.

We call the gravitational force that the earth exerts on an object its weight. Unlike mass, weight is not an intrinsic property of an object but varies from place to place, especially from planet to planet. For example, compute the ratio of the weight of a test object on the moon and the weight of the same object on the earth by plugging in the following numbers into Newton's formula for gravitational force (1) :

Moon: mass=7.36x10^{22} kg; radius=1.74x10^{6} m

Earth: mass=5.97x10^{24} kg; radius=6.38x10^{6} m.

In this example, the mass of the test object will cancel out, as will the universal gravitational constant; show that the ratio turns out to be 0.166. A 100 lb ball on the earth will weigh only 16.6 lbs on the moon!

*Project: Do an Internet search and determine your weight on different planets of the solar system. You can also look up the values of the fundamental constants and read about the difficulties in obtaining precise values of G.*

Even on the earth, the acceleration of gravity will vary slightly from place to place because of small variations in height above sea level and because of the distribution of mass in mountains and valleys and variations of density below the surface. The deflection of a plumb bob from the true vertical is measurable effect. Nevil Maskelyne the English Astronomer Royal proposed measuring G and the earth's mass in 1772 by measuring the plumb bob deflections near a mountain. Maskelyne himself made such measurements on an isolated mountain, Schiehallion, in Scotland during four months in the summer of 1774 and presented his results confirming Newton's theory of gravitation to the Royal Society on July 6, 1775.

As space travel becomes commonplace and people properly understand the concepts of mass and weight, the expression “I have to lose weight” will have to be more accurately stated as “I have to lose mass.”

When the mass of an object remains constant, a force on the object produces an acceleration given by Newton's equation

**F** = m **a**. (2)

If this force is supplied by gravity it is just the weight of the object

**F** = m **g**. (3)

Equating the first mass (m_{1}) to the mass of the Earth in (1) M_{e} = m_{1 }and the second to m=m_{2} , which is a small test mass, and using the radius of the Earth (r_{e}) as the distance, we get

m **g = - **G M_{e} m** u**_{r} r_{e}^{-2} ,** ** (4a)

**where ****u**_{r} is a unit vector pointing up from the center of the Earth toward the test mass. Note that the value of the test mass cancels out on both sides of (4a), so that we can write it as

**g**_{e}** =** - G M_{e}** u**_{r}** **r** _{e}^{-2} . ** (4b)

One finds athe rather surprising result in (4b) that any object dropped toward the earth from a place near its surface will accelerate at the rate of G times the mass of the earth divided by the radius of the earth squared. Observe that this is independent of the mass of the object being dropped. Thus, an object dropped near the surface of the earth accelerates at a certain rate independent of its mass. This rate is (9.8± 0.02 meters/sec)/sec as experimentally verified.

When you form the ratio of the acceleration of gravity on the moon's surface to that on the earth's surface, everything cancels out except for terms involving the masses and radii of the planets:

g_{m}/g_{e}= (M_{m}/ M_{e}) r_{e}^{2}/ r_{m}^{2 } . (5)

Using the masses and radii for the astronomical bodies involved here, we can solve for the ratio as follows:

g_{m}/g_{e}= (7.36x10^{22}/ 5.97x10^{24}) * (6.38x10^{6}/ 1.74x10^{6})^{2}

or

g_{m}/g_{e}= 0.166 .

Galileo maintained that in the absence of air resistance the acceleration of gravity is the same for all objects regardless of their masses. In his time a vacuum was not realizable so that this prediction could not be checks. This prediction of Galileo’s is also a feature of Newton’s theory on gravitation. Recently Galileo’s hypothesis and Newton’s prediction were tested in vacuum by an astronaut before television cameras. He dropped a hammer and a feather on the moon and they fell exactly together to the moon’s surface.

1I. The orbits of the planets are ellipses, each orbit having the Sun at one of the ellipse's focus.

II. As a planet orbits, the line joining the centers of the planet and the Sun sweeps out equal areas in equal times.

III. The ratio of the squares of the orbital periods of any two planets is equal to the ratio of the cubes of their semimajor axes.

2 The Cavendish balance had a history of development that dated to a torsion balance proposed by Charles Coulomb in 1784 (not published) for measuring very feeble forces. It was independently developed and built by the Rev. John Michell who died in 1793 before he could use it. Henry Cavendish acquired the instrument and used it to determine the mean density of the earth. The Cavendish balance was ultra-sensitive for its time; but in modern times we have much more sensitive instruments; nevertheless, the universal gravitational constant is the least precisely measured fundamental constant. Large discrepancies in its value have been observed by various, contemporary experimenters.

(1) :

Moon: mass=7.36x10^{22} kg; radius=1.74x10^{6} m

Earth: mass=5.97x10^{24} kg; radius=6.38x10^{6} m.

In this example, the mass of the test object will cancel out, as will the universal gravitational constant; show that the ratio turns out to be 0.166. A 100 lb ball on the earth will weigh only 16.6 lbs on the moon!

*Project: Do an Internet search and determine your weight on different planets of the solar system. You can also look up the values of the fundamental constants and read about the difficulties in obtaining precise values of G.*

Even on the earth, the weight of anything will vary slightly from place to place because of small variations in height above sea level and because of the distribution of mass in mountains and valleys and variations of density below the surface. The deflection of a plumb bob from the true vertical is measurable effect. Nevil Maskelyne the English Astronomer Royal proposed measuring G and the earth's mass in 1772 by measuring the plumb bob deflections near a mountain. Maskelyne himself made such measurements on an isolated mountain, Schiehallion, in Scotland during four months in the summer of 1774 and presented his results confirming Newton's theory of gravitation to the Royal Society on July 6, 1775.

As space travel becomes commonplace and people properly understand the concepts of mass and weight, the expression “I have to lose weight” will have to be more accurately stated as “I have to lose mass.”

When the mass of an object remains constant, a force on the object produces an acceleration given by Newton's equation

**F** = m **a**. (2)

If this force is supplied by gravity it is just the weight of the object

**F** = m **g**. (3)

Equating the first mass (m_{1}) to the mass of the Earth in (1) M_{e} = m_{1 }and the second to m=m_{2} , which is a small test mass, and using the radius of the Earth (r_{e}) as the distance, we get

m **g = - **G M_{e} m** u**_{r}, r_{e}^{-2} (4a)

where **u**_{r} is a unit vector pointing up from the center of the Earth toward the test mass. Note that the value of the test mass cancels out on both sides of (4a), so that we can write it as

**g**_{e}** = - **G M_{e}** u**_{r}** r _{e}^{-2}. ** (4b)

One finds the rather surprising result in (2b) that any object dropped toward the earth from a place near its surface will accelerate at the rate of G times the mass of the earth divided by the radius of the earth squared. Observe that this is independent of the mass of the object being dropped. Thus, an object dropped near the surface of the earth accelerates at a certain rate independent of its mass. This rate is (9.8± 0.02 meters/sec)/sec as experimentally verified.

When you form the ratio of the acceleration of gravity on the moon's surface to that on the earth's surface, everything cancels out except for terms involving the masses and radii of th e planets:

g_{m}/g_{e}= (M_{m}/ M_{e}) r_{e}^{2}/ r_{m}^{2 } . (5)

Using the masses and radii for the astronomical bodies involved here, we can solve for the ratio as follows:

g_{m}/g_{e}= (7.36x10^{22}/ 5.97x10^{24}) * (6.38x10^{6}/ 1.74x10^{6})^{2}

or

g_{m}/g_{e}= 0.166 .