cphil_mech2
MECHANICS
CONTINUED
by Ray Shelton
Isaac Newton was born on Christmas day, 1642, the same year that Galileo died eleven months earlier, in a manor house in Woolsthorpe in Lincolnshire near Grantham, England. His father, a small farmer, had died a few months before Isaac was born. After his mother married in 1645 the rector of North Witham, she left Isaac with his maternal grandmother at Woolsthorpe. Having acquired the rudiments of education at nearby schools, at an age of twelve she sent Newton to a grammar school at Grantham, where he lived in the house of an apocecary. After the death of her second husband in 1656, Newton’s mother returned to Woolsthrope and removed her eldest son from school so that he could prepare himself to manage the farm. But it became obvious that Newton was not interested in farming. He was a quiet and frail boy but he was ingenious at constructing mechanical devices, such as windmills, water-clocks, kites, and sun-dials, and he did well enough at his studies that, through the intervention of an uncle, the rector of Burton College, he was allowed to go to the university instead of becoming farmer; in 1661 at 18 he entered Trinity College at Cambridge as one of boys who did menial services for expenses. Although there is no record of his formal progress as a student, Newton read widely in mathematics and mechanics. He was noticed by Isaac Barrow, the Lucasian Professor of Mathematics, who encouraged him in the studies of mathematics and optics. About the time he took the Bachelor’s degree, Newton discovered the binomial theorem and made his first notes on his discovery of the “method of fluxions,” which was later called differential calculus. In 1665 he graduated and, when the university was closed because of the Great bubonic Plague, he returned to his home in Woolsthorpe where he spent in isolation the next two years.
These two years of uninterrupted work were the most productive of his life. He conceived the first two laws of motion, the formula for centripetal acceleration, and, in 1666, when he was 24, the law of universal gravitation. This must have been the time of the famous incident of the falling apple. One of the sources of this story is a biography of Newton written by his friend William Stukeley in 1752; Stukeley records that on one occasion when he was having tea with Newton in a garden under some apple trees, Newton told him that “he was just in the same situation, as when formerly, the notion of gravitation came to mind. It was occasion’d by the fall of an apple, as he sat in a contemplative mood…” This does not mean that he was the first to have the conception of gravity; for this concept had been developing for nearly a century. But it was probably when he saw the connection between the motion of the moon (and the planets) and the falling of the apple; that he saw that gravity was the cause of both celestial and natural terrestrial motion. Newton dates his discovery to 1666;
“In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing the dignity of any Binomial into such a Series. The same year, in May, I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions, and the next year in January had the theory of Colours, and in May following I had entrance into the inverse method of Fluxions. In the same year I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which a globe revolving within a sphere presses the surface of the sphere… from Kepler’s Rule of the periodical times of the planets being in a sesquialterate proportion of their distances from of their orbs, I deduced that the forces which keep the planets in their orbs must be reciprocally as the square of their distances from the centers about which they revolve and thereby compared the forces requisite to keep the Moon in her orb with the force of gravity at the surface of the earth and found them to answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since.”
He did not publish any of these ideas for many years, because he could not prove mathematically that the attractive force between two spherical masses could be considered as acting at the geometrical centers of the sphere. He supplied this proof later to his friend, Edmond Halley (1656-1742 A.D.), for whom the comet was named.
In 1667 after the plague, Newton returned to Cambridge, where he remained until 1692. After he successively became a Minor Fellow, a Major Fellow, a Senior Fellow, and following Barrow’s resignation in favor of Newton, he was appointed Lucasian Professor of Mathematics at the age of 27. Newton did much of his work on optics then, and at the request of the Royal Society, he sent them in December, 1671, a reflecting telescope that he had designed and constructed; two months later he was elected a Fellow of the Society. He published in 1672 in the Transactions of the Royal Society a paper entitled, “Theory about Light and Color,” in which he described and explained the operation of his telescope; but it involved him in such a bitter controversy about the nature of light with Robert Hooke (1635-1703 A.D.) and others, that he resolved never to publish again.
In August of 1684, Halley traveled to Cambridge, to ask Newton’s advice on the problem of the nature of the orbit of a planet; that is, what is the shape of its orbit, if its acceleration was always toward the sun and inversely proportional to the square of its distance from the sun. Newton said to his friend’s surprise that it was an ellipse. Halley asked him how did he know that. Newton replied that he had calculated it. Halley asked him for the calculation but he could not find it. He had solved the problem of the orbit of a planet around the sun many years before but had misplaced the solution and could not find it at the moment, but he said that he would send it to him later. Three months later Halley received a short treatise with the title On the Motion of Bodies in Orbit, known as De motu, containing nine theorems and seven problems. Halley was so impressed with its importance that he went to Cambridge again to urged Newton to publish it. After some seventeen or eighteen months of labor during 1685 and 1686, the result was the Principia Mathematica Philosophiae Naturalis (Mathematical Principles of Natural Philosophy) or, as it is usually known, the Principia, which was published in 1687, establishing Newton’s reputation as one of the all time greatest scientists.
In 1689 and again in 1701 Newton was elected a Member of Parliament by the University. In 1696 he was made Warden of the Mint and he helped materially in re-establishing the currency of the country. In 1699 became Master of the Mint, which position he held until his death. He moved to London, and his home was cared by his niece Catherine Barton. In 1703 he became president of the Royal Society, a position which he held until his death in 1727. In 1704 with the controversy about the nature of light lessened by the death of Hooke, he published his Opticks. This work was written in English instead of the usual Latin, in which the Principia was written. By this time Newton had been become renowned throughout the scientific world and was knighted by Queen Anne in 1705 for his contributions to science. In the last two decades Newton worked on theological subjects and biblical prophecies. From an early period of his life Newton had been much interested in theological studies. Earlier in 1690 he wrote, in the form of a letter to Locke, an Historical Account of Two Notable Corruptions of the Scripture, regarding two passages on the Trinity. He left in manuscript a work entitled Observations on the Prophecies of Daniel and Apocalypse, and other works of biblical exegesis. At the time of his death on March 20, 1727, in his eighty-fifth year, Newton was revered throughout Europe and the honors bestowed upon him at burial in Westminster Abbey, after lying in state in the Jerusalem Chamber, were greater than those accorded most nobility.
Newton’s Mathematical Principles of Natural Philosophy, usually called the Principia, published 1687, is his greatest work. The Principia is divided into three books:
Book I, Definitions, Axioms and Procedures;
Book II, Demonstration of propositions of forces in motion;
Book III, System of the World.
In Book I the three laws of motion and their consequences are established. The law of gravitation is set forth as a mathematical hypothesis from which Kepler’s three laws of planetary motion is deduced. The orbits of two attracting bodies revolving about their common center of mass is calculated. The more complicated problem of the interaction between three bodies, each attracting the other two (the famous “three body” problem), is analyzed and, although the analysis is not complete, makes a good beginning in developing the modern form of the theory of perturbations. The motion of the simple pendulum and of the cycloidal pendulum is completely discussed.
In Book II the effect of resisting medium upon motion is discussed in detail. The science of fluid dynamics, including streamline effects, is established. The calculus of variations and the mathematical treatment of wave motion is introduced.
In Book III the system of the world is set forth. The motion of the planets, the satellites of Jupiter, and the orbits of the comets are studied in detail and are shown to be explained by the law of gravitation. The complicated motion of the moon, the precession of the equinoxes, and the tides is examined in detail and is systematically explained. The equatorial bulge of the earth is predicted quantitatively, which was later found to be extremely accurate. The density of the earth is estimated as between five and six times that of water, which was later found to be about five and one half times that of water.
The Principia is presented on the model of Euclid’s Elements, but does not approach the rigor of the Elements. The book was not written for popular consumption and it is difficult to understand without any background in mathematics, physics or astronomy. The title indicates the level and scope of the book; the words in the title, “Mathematical Principles,” sets the level and the words “Natural Philosophy” sets the scope and subject of the book (Natural Philosophy included the sciences of physics and astronomy). Only rarely does the Principia resort to the calculus, preferring to rely on geometrical arguments and demonstrations. And the formulation of the principles of mechanics in the book is not that which is learned today; the present formulation is that of Joseph Louis Lagrange, W. R. Hamilton, and a host of other writers since Newton, rather than of Newton himself. Although the formulation is not that of Newton, the principles are “equivalent” to Newton’s, even though covering topics that Newton could not have anticipated.
The foundations of the Newtonian mechanics is given in the three Axioms, or Laws of Motion introduced in Book I of the Principia. We repeat them here with his comments.
“Law I. Every body continues in its state of rest, or of uniform of motion in a right line, unless it is compelled to change that state by forces impressed upon it.
Projectiles continue in their motions, so far as they are not retarded by the resistance of the air, or impelled downward by the force of gravity. A top, whose parts by their cohesion are continually drawn aside from rectilinear motion, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in freer spaces, preserve their motions both progressive and circular for a much longer time.”
Expressing Newton’s First Law (law of inertia) in symbols:
If F = 0, then a = 0,
where F is the net impressed force and a is the acceleration.
This First Law of Motion is sometimes called the Law of Inertia, where inertia is defined as the property of a body by virtue of which a body resists change in its motion. The Dutch scientist, Christian Huygens (1629-1695 A.D.), anticipated Newton’s statement here 14 years earlier in his greatest work published in 1673, the Horologium oscillatorium,
“If gravity did not exist, nor atmosphere obstruct motion of bodies, a body would maintain forever a motion once impressed upon it, with uniform velocity in a straight line.”
“Law II. The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body was moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.”
Newton had already defined “quantity of motion” as the product of mass times velocity, which product is called today “momentum.” In symbols,
p = mv,
where p is momentum of the body, m is its mass and v is its velocity. Therefore, Newton’s phrase “change of motion” is the same as the change of momentum, that is,
Δp = p2 – p1,
where the Greek letter Δ means “the change of”, p1 is the initial momentum at the beginning of change of momentum, and p2 is the final momentum at end of change of momentum. The phrase “motive force” designates what is now called impulse which is defined as the product of the impressed force and the elapsed time the force is applied, that is,
Impulse = FΔt,
where Δt is the elapsed time the force F is applied. This meaning of the phrase is indicated by Newton’s words, “whether that force be impressed at once, or gradually and successively.” Thus Newton’s Second Law may be restated in modern terms: “the change in momentum is proportional to the impulse applied;” and in symbols
FΔt = kΔp,
or with units chosen so that the proportionality constant k is one,
FΔt = Δp.
If both sides of this equation is divided by Δt, and canceling, then
F = Δp / Δt,
where Δp / Δt is the time rate of change of momentum. Thus Newton’s Second Law may be restated: the rate of change of momentum is directly proportional to the net impressed force. This statement of the second law in terms of change of momentum allows for change of mass as well as change of velocity, that is,
F = Δ(mv) / Δt
F = (vΔm + mΔv) / Δt
F = vΔm / Δt + mΔv / Δt,
where Δm / Δt is the time rate change of mass and Δv / Δt is the time rate change of velocity. The change of mass occurs in a rocket, where some of its mass (the solid propellant) is burned and expelled, changing its mass, as the force lifts the rocket. But if the mass of the body involved remains constant (Δm = 0), the change of momentum will result from a change of velocity, and the time-rate of change of velocity is called “acceleration“, a. Thus Newton’s Second Law may again be restated: the acceleration of a body is proportional to the net impressed force and in the direction of that force. If the unit of force is chosen properly so that the proportionality constant is one, the second law may be expressed in symbols:
F = ma,
where F is the impressed net force, m is the mass of the body, and a is the acceleration of the body in the direction of the force, F. Mass has been defined as a measure of inertia, that is, the resistance to change of motion. Newton defined mass as the quantity of matter that a body possesses and is equal to the product of its density and its bulk (volume); volume is simply a measure of the physical space occupied by a body, measured in cubic inches, cubic centimeters, liters, etc.; it is “size” of the body, the amount of space that it occupies, and it is not the mass, nor the weight of the body. Density is a measure of how much mass is contained within a given volume, that is, “how tightly packed” mass is packed in space; specifically, it is the ratio of mass to volume:
density = mass / volume, or ρ = m / V.
Density is measured in units of grams per cubic centimeter, etc. If density is given terms of the density of water, it is called specific gravity, that is, the relative density of a substance compared to the density of water, which is 1 gm per cubic centimeter. For example, the specific gravity of gold is 19.3 or a density of 19.3 gm per cm3.
Mass is not weight, even though they are related: the weight of a body is directly proportional to the mass of the body, where the constant of proportionality is the acceleration due to the force of gravity; that is, weight equals mass times the acceleration due to gravity, or
w = mg,
where w is the weight of the body, m is the mass of the body and g is the acceleration of due to the force of gravity. This relation is a special case of Newton’s Second Law of Motion were the force is the force of gravity, called the weight of the body, and the acceleration is the acceleration due to the force of gravity. The weight of a body is the force of gravity exerted on the body. On earth the weight of a body is the force of the gravity of the earth exerted on the body; on the moon the weight of the body would be the force of the moon’s gravity exerted on the body. Weight is relative to where it is measured and not the same on every planet or satellite; but the mass of the body is the same every where.
In order to use Newton’s Second Law of motion, a system of units must be constructed, in which the units are interrelated such that from a set of primary units all other units are defined. Today there are three system of units that are currently in use: foot-pound-second system (FPS), centimeter-gram-second system (CGS), and meter-kilogram-second system (MKS). The FPS system is an English system of units in which the primary units are selected for length, weight, and time: foot for length, pound for weight, the second for time. The CGS system is the older metric system of units, in which the primary units are selected for length, mass, and time: centimeter for length, gram for mass, and the second for time. The MKS system of units is the newer and recommended metric system of units, in which the primary units are selected for length, mass and time: meter for length, kilogram for mass, and second for time. Other units are defined in terms of these primary units; for example, the unit of velocity in the FPS system is defined as the unit of length divided by the unit of time, that is, foot per second (the word “per” indicates division), based on the formula for velocity:
v = Δs / Δt,
where Δs is the change in space or distance through which a body travels in the elasped time Δt. The unit of force in the CGS system and in the MKS system is defined as the unit of mass times the unit of acceleration, that is, in the CGS system, the unit of force is the dyne, and is defined as gram·cm per sec2 and the unit of force in the MKS system the newton is defined as kilogram·meter per sec2;. These definitions of force units are based on Newton’s Second Law:
F = ma.
In the FPS system the unit of mass is the slug and is defined in terms of the unit of force divided by the unit of acceleration, that is, the slug is defined as pounds per feet per sec2 or lb sec2 / ft; this unit of mass is also based on Newton’s Second Law, solved for mass:
m = F / a.
Since w = mg, one slug weighs 32.2 pounds, one gram weighs 980 dynes, and one kilogram weighs 9.8 newtons. It may also be shown that 1 newton = 100,000 dynes = 0.22 pounds-force.
| FPS | CGS | MKS | |
|---|---|---|---|
| length | foot (ft) | centimeter (cm) | meter (m) |
| mass | slug | gram (gm) | kilogram (kg) |
| time | second (sec) | second (sec) | second (sec) |
| velocity | ft per sec (fps) | cm per sec (cm/sec) | m per sec (m/sec) |
| acceleration | ft per sec per sec (ft/sec²) | cm per sec per sec (cm/sec²) | m per sec per sec (m/sec²) |
| force | pound (lb) | dyne | newton (nt) |
Note also that Newton was careful to state that the direction of the forces in the same or opposite direction and are additive or subtractive respectively. In other words, the forces are vector quantities, whose direction must be taken into consideration when combining them with the use of vector addition. There are three special cases of the direction of the forces and of the motion:
(1) Force applied is in the same direction of the motion of the body; the speed increases and the body accelerates without change of direction,
(2) Force applied is in the opposite direction of the motion of the body; the speed decreases and the body decelerates without change of direction,
(3) Force applied is at right angles to the direction of the motion of the body; the direction changes, but not the speed.
As an example of case (3), a body at the end of string moving in a circle at a constant speed. The string exerts a force on the body at right angles to the direction of its motion and toward the center of the circle. This force is called a centripetal force (“walking toward the center”). The direction is constantly changing as the body moves about the circle, always pointing at right angles to the radial line from the body toward the center of the circle. This is accelerated motion because the velocity of the body is changing, that is, its direction, not its speed, is changing. This acceleration is called centripetal acceleration or radial acceleration; if the path of a moving body is a circle, then its magnitude may calculated by the following formula: centripetal acceleration = square of its constant speed / radius of circle, or
ac = v2 / r,
where ac = centripetal acceleration,
v = constant speed, and r = radius of circular path.
A thorough analysis of centripetal acceleration was first published in 1673 by Christian Huygens, and was probably known to Newton in all its essential details some years earlier. This formula may derived by the following considerations:
a. Suppose a body is moving at a constant speed v on the circumference of a circle whose center is at the point O. At some instant it is at position P and has a velocity of magnitude v and direction PE. By Newton’s first law, if there were no forces acting upon it, the body would continue to move in direction PE. In a short interval of time Δt the body will move to the position P´ along the circumference of the circle. The velocity of the body still has magnitude v, but its direction is now P´F. If the interval Δt is considered to be very small, the distance along the arc is nearly equal to straight-line distance from P to P´ equal to the product of its speed v and t.
b. As the body moves from P to P´, the direction of the velocity changes through an angle that is equal to the angle that the radial line from the body to the center of the circle sweeps through. These angles are equal because the direction of the velocity of the body at any point is always tangent to the circumference of the circle at that point and is always perpendicular to the radial line from the body to the center of the circle.
c. The change of the velocity of the body may be represented by a triangle OAB which has two of its sides, OA and OB, equal; these two equal sides represent the equal magnitude of the velocity v at points P and P´ and the included angle at O between the two equal sides represents the change of direction. The side AB opposite this angle at O represents the change of velocity Δv due to the change of direction.
d. Since this triangle OAB is similar to the triangle OPP’ formed by the motion of the body from P to P´ on the circumference of the circle whose center is at O. This triangle OPP´ has equal sides, OP and OP´, whose equal lengths are the radius r of the circle, and the angle at O in the triangle OPP´ is equal to the angle O in triangle OAB. Therefore, the following proportion holds:
AB : OA :: PP´ : OP
or
Δv is to v as (vΔt) is to r:
or
Δv / v = vΔt / r.
That is, ratio of Δv to Δt is equal to the ratio of v2 to r: that is,
Δv / Δt = v2 / r.
e. Since acceleration is the ratio of change of velocity to the time interval, that is,
a = Δv / Δt.
Therefore, a = v2 / r,
and this is the formula for centripetal acceleration or radial acceleration, if its path is a circle. Using the second law of motion, the formula for the magnitude of the centripetal force exerted on a body moving in a circle is
F = ma = mv2 / r,
this force is continually changing direction but is always directed to the center of the circular path and is thus sometimes called a central force.
“Law III. To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone; ….”
The first two laws were clearly drawn from Galileo’s analysis and statements; this third law is stated by Newton for the first time. Today this law is stated without the use of the terms “action” and “reaction” and in terms of forces: for every force there is a equal and opposite force. Combining these statements: for every force acting on a body there is equal and opposite reacting force exerted back upon the body exerting the acting force. Note that this law says three things about forces:
(1) forces always occurs in pairs, an acting force and reacting force.
(2) these two forces are not exerted on the same body, and, therefore, cannot cancel each other.
(3) these two forces are equal in magnitude but opposite in direction.
This third law may be expressed in symbols:
F12 = –F21,
where F12 is the force acting on a second body by a first body and F21 is the force reacting on the first body by the second body. For example, the earth is pulled on by the sun by a force of gravity there is an equal and opposite force of gravity pulling on the sun by the earth. Since the earth is in motion revolving about the sun, the force exerted by the sun on the earth is called a centripetal force (force acting toward the center of motion) and the equal and opposite force exerted back on the sun by the earth is called a centrifugal force (force acting away from the center of motion). As another example, the force exerted on a whirling body by the string is the centripetal force and the force exerted back on the hand holding the string is the centrifugal force and these forces are equal in magnitude and opposite in direction.
The Law of Universal Gravitation was expressed by Newton in two propositions:
“1. That there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they contain.
2. The force of gravity towards the several equal particles of any body is inversely as the square of the distance of places from the particles.”
A more recent statement of this law is the following:
“Between every pair of particles in the universe there exists a force of gravitational attraction. This force acts along the line joining the pair of particles and has a magnitude which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.”
In symbols this may be written:
F is proportional to m1m2 / d2,
or, as a mathematical formula,
F = Gm1m2 / d2,
where F is the magnitude of the force of attraction,
m1 and m2 are the masses of the two particles, d is the distance between them, and G is the constant of proportionality which is called the “Universal Gravitational Constant.”
Let F12 be the gravitational force exerted by mass one on mass two and F21 be the gravitational force exerted by mass two on mass one. These forces are equal in magnitude
( |F12| = |F21| )
but act in opposite directions
(F12 = –F21).
The magnitude is calculated by the mathematical formula above. This gravitational force acts at the center of the body if it is homogeneous (has same density throughout) and spherical. Since the law of universal gravitation, as stated above, apply to particles, not to such bodies, Newton invented the Integral Calculus in order to prove that the force of attraction exerted on or by a homogeneous spherical body is the same as that exerted on or by a particle having its mass (a particle is a mass-point, that is, a geometrical point having the property of inertia). Newton delayed publication of his law until he was able to prove mathematically that the law did apply to homogeneous, spherical bodies as well as to particles. This he was able to do by 1685.
The following summarizes the five steps given in the Principia by which Newton arrived at the law of gravitation:
- Step 1. In the first step, Newton established the direction of the force of gravitation by using Kepler’s Second Law of planetary motion. Any body moving in a closed path about a point, so that an imaginary line between the body and the point sweeps over or across equal areas in equal times (Kepler’s Second Law), has a force, called a centripetal force, exerted on it directed toward the point; it changes only the direction of the body but not its speed. Newton proved this theorem in Book I, Prop II, Theorem II.
- Step 2. The centripetal force acting on a body moving along a curved path is directly proportional to the product of the mass of the body and the square of its speed and inversely proportional to the distance between the body and the center of the path. This is called the Law of Centripetal Force and Newton proved it in Book I, Proposition IV, Theorem IV, Corollary I.
Newton’s proof is a combination of physical intuition and geometry. The following is a simplified form of that proof. For mathematical simplification, it will be assumed that the body revolves around the earth in perfect circle.
(1) As the body moves about the circular path, it may be considered as “falling” toward the earth at the center of the circle. The distance h that it falls in unit of time may be calculated by Galileo’s Law of Falling Bodies:
h = ½at2.
(2) At any instant of time at a point P the direction of the body is tangent to the circle at that point, that is, its direction is at right angles to the line between the body and the center of the circle, the radius of the circle.
(3) If at that instant the force on the body moving it in a circle was stopped, then the body would fly off on the tangent line and would travel a distance per unit time which may be calculated by the formula: d = vt; and it then would be at a point A whose distance from the center of the circle O would be equal to the radius of the circle plus the distance it would have “fallen,” that is, r + h, where r is the radius of the circle.
(4) Using the Pythagorean theorem on the right triangle OPA:
(r + h)2 = r2 + d2, or
r2 + 2rh + h2 = r2 + d2, and,
canceling r2,
2rh + h2 = d2.
(5) Since h is very small with respect with d and r, h2 is very small compared to 2rh. Also as the time interval t becomes very small, h2 becomes very small with respect to 2rh. Therefore, a good approximation is found by neglecting h2: thus,
2rh = d2, and solving for h:
h = d2 / 2r.
(6) From step (3) substitute d = vt for d in formula (5), and from step (1) substitute h = ½at2 for h in formula (5), it becomes
½at2 = [(vt)2] / 2r
and solving for a we get
a = v2 / r,
which is the formula for centripetal acceleration.
(7) Using the Second Law of Motion, F = ma, the formula for the magnitude of the centripetal force exerted on a body moving in a circle is
Fc = mv2 / r;
this force is continually changing direction but is always directed to the center of the circular path and is thus sometimes called a central force.
- Step 3. In the third step Newton assumed the existence of a force of attraction between the moving body m2 and a stationary body m1 at a focus of its orbit, then he proved that the body m2 moving according to Kepler’s Third Law of Planetary Motion is acted upon a centripetal force which is directly proportional to the mass of the moving body m2 and inversely proportional to the square of the distance between the moving body m2 and the attracting body m1. The proof for the case of motion in a circle is as follows:
- Step 3. In the third step Newton assumed the existence of a force of attraction between the moving body m2 and a stationary body m1 at a focus of its orbit, then he proved that the body m2 moving according to Kepler’s Third Law of Planetary Motion is acted upon a centripetal force which is directly proportional to the mass of the moving body m2 and inversely proportional to the square of the distance between the moving body m2 and the attracting body m1. The proof for the case of motion in a circle is as follows:
The circumference of the circle C is
C = 2πr,
where r is the radius of the circle.
The speed of the moving body m around the circle is
v = Δs / Δt = C / T = 2πr / T,
where T is the period of the moving body m.
The centripetal force exerted on the moving body is
Fc = mv2 / r = m[(v)2] / r = m[(2πr / T)2] / r = m[4π2r2 / T2] / r = 4π2mr / T2.
Since T2 = kr3,
which is Kepler’s Third Law of Planetary Motion, then
Fc = 4π2mr / kr3 = 4π2m / kr2 = (4π2 / k)(m / r2),
where 4π2 / k is a constant.
Since Fc = F12, then
F12 is proportional to m2 / r2.
That is, the force of gravity varies inversely as the square of distance between the attracting body and the attracted body.
- Step 4. By Newton’s Third Law there is a reaction force of attraction exerted on body one by body two, F21, which is equal in magnitude but opposite indirection to the action force of attraction exerted by body one on body two, F12. So this force F12 is therefore directly proportional to the product of both the attracted mass m2 and the attracting mass m1: F12 is proportional to m1 times m2, or F is proportional to m1m2.
- Step 4. By Newton’s Third Law there is a reaction force of attraction exerted on body one by body two, F21, which is equal in magnitude but opposite indirection to the action force of attraction exerted by body one on body two, F12. So this force F12 is therefore directly proportional to the product of both the attracted mass m2 and the attracting mass m1: F12 is proportional to m1 times m2, or F is proportional to m1m2.
- Step 5. Combining equations from steps 3 and 4, Newton arrives at the statement that the force of attraction exerted by body one on body two, F12, is directly proportional to the product of their masses and inversely proportional to the square of the distance between them:
- Step 5. Combining equations from steps 3 and 4, Newton arrives at the statement that the force of attraction exerted by body one on body two, F12, is directly proportional to the product of their masses and inversely proportional to the square of the distance between them:
F12 is proportional to m1m2 / r2,
where G = 4π2 / km1 is the constant of proportionality called the universal gravitational constant.
And since by Newton’s Third Law,
|F12| = |F21| = F,
therefore,
F = Gm1m2 / r2.
This is Newton’s Law of Universal Gravitation.
Observations:
(1) Newton’s Law of Gravitation is based on the Kepler’s Second and Third Laws of Planetary Motion and Newton’s Second and Third Laws of Motion.
(2) What kind of law is it? What is the character of this law of gravitation? As a law it is describes how the force of gravity behaves. But it does more than just describe the force of gravity; it also explains why bodies fall as they do and why the planets and the moon move as they do as described by Kepler’s first and second laws of planetary motion. It is a theory which explains why these laws of planetary motion are as they are. But it is not an explanation of gravity; it does not explain what gravity is. And Newton refused to form such a hypothesis to explain the nature of gravity. At the end of Book III, he says,
“Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power… But hitherto I have not been able to discover the cause of those properties of gravity from phenomena [observations and experimentation], and I frame no hypotheses… And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all motions of the celestial bodies and of our sea.”
One of Newton’s first test of the law of gravitation involved observations of the moon. Having used Kepler’s second and third laws of planetary motion, in the formulation of the law of gravitation was not a sufficient confirmation of it. What was needed was a direct quantitative confirmation of the law of gravity. Newton proposed to test the law of gravitation with factual observations of the moon. This was the moon problem: does the motion of the moon about the earth confirm the law of gravitation? In the view of Newton, it is gravitational attraction of the earth for the moon that accelerates the moon toward the earth, hence keeps it in its orbit about the earth. Also the earth’s attraction for a body on or near the surface of the earth is the cause of the acceleration of free fall. If this view is correct, the moon’s constant “fall” toward the earth and the fall of stone dropped from a cliff may be ascribed to the same cause. Of course the acceleration of the moon, being at a much greater distance from the earth, will be much smaller than the acceleration of the stone near surface of the earth. But how much smaller? But before this can be determined a common point of reference must be chosen from which to reckon the two distances. If the center of the earth is chosen for this purpose, then this implied that that point is center of the action of the earth’s force of gravity, even though the mass of earth is distributed throughout a large volume. How can it be proven that the force of attraction exerted on or by a homogeneous spherical body is the same as that exerted on or by a particle having its mass, on or by a mass-point? The search for the solution of this problem kept Newton from publishing his law of gravitation until he was able to solve it, and it required the invention of “inverse fluxion” or the integral calculus, which he did by 1685 and before the publication of the Principia in 1687. Without repeating Newton’s proof, we will assume that the theorem is proven. Gravitational force, like other forces, must satisfy the second law of motion. Thus,
F = ma = GmM / r2,
where a is the acceleration of the falling body of mass m (here either the moon or a stone) at a distance r from the other body of mass M (here the earth). Solving for a, canceling m, we get
a = GM / r2.
The distance from the earth center to the moon is known roughly 240,000 miles, and the distance from the earth center to the stone on or near the surface of the earth is about 4000 miles. Using these values, the acceleration a of the moon in its orbit toward the earth should be
a = GM / (240000)2,
and the acceleration g of a stone at the surface of the earth should be
g = GM / (4000)2.
At this time Newton did not know either G, gravitational constant, nor M, the mass of the earth; so he divided the equation for the acceleration of the moon, a, by the equation of the acceleration of the stone, g, both G and M cancel out, he obtained
a / g = (4000 /240000)2 = 1 / 3600,
and solving for a, a = g / 3600.
This is a prediction of the law of gravitation, and if the distance to the center of the earth from the moon and from the stone on the surface of the earth, then the acceleration of the moon toward the earth is 1/3600 of the acceleration g. Since the measured value of g = 32.2 ft per sec per sec, the moon accelerate toward the earth 32.2 / 3600 ft per sec per sec or 0.00894 ft per sec per sec.
To verify this value, the actual value of acceleration of the moon must be found. To obtain the actual value the formula for centripetal acceleration will be used, assuming that the moon’s orbit is circular and that its centripetal acceleration is the result of the force of gravity, a = v2 / r. If the moon traverses a circle of radius r in time T, called its period, then the speed v of the moon in its orbit is the circumference (2πr) divided by the time period T, that is, v = 2πr / T. The centripetal acceleration of the moon is
ac = 4π2r / T2.
Since the period T of the moon is 27.3 days = 27.3 * 24 * 3600 = 2,358,720 seconds and r = 240000 miles * 5280 ft/mi = 1,267,200,000 ft, then a = 0.00896 ft per sec per sec. This is very close to the predicted value of 0.00894 ft per sec per sec. The difference is due to the assumption that the orbit of the moon is circular instead of slightly elliptical and to the approximate distance from the moon to the center of the earth and from the stone to the center of the earth. But as Newton put it, he had “computed the force requisite to keep the moon in her orb with the force of gravity at the surface of the earth, and found them to answer pretty nearly.” This was the first direct independent confirmation of the inverse-square gravitation law. Some have suggested that Newton withheld for so long the publication of his law of gravitation because that when Newton first made his calculation, the ratio of the radius of moon’s orbit to the radius of the earth was not very precisely known, and his calculation showed only approximate agreement. It used to be supposed that this decrepency was reason for him laying the matter aside, but this view is not generally accepted now.
The solution of the moon problem only confirmed that law of gravitation held for the moon-earth system. What was needed was a test to solve the larger problem involving the sun and the planets: does the planets movement about the sun according to the law of gravitation? To solve this larger problem Newton proposed to derive Kepler’s three Laws of Planetary Motion from the law of gravitation, showing that Kepler’s three laws as empirical laws are consistent with the law of gravitation and that the law of gravitation accounts for the observational data that they contained.
Newton’s invention of the calculus made the calculation of the elliptical motions of the planets and many other difficult computations relatively simple. With the use of the calculus he performed these calculation but we will not repeat them here. Without using the calculus and making use of the simplifying assumption that the orbits of the planets are practically circular, we will derive Kepler’s third law. If the centripetal force Fc on any particular planet with a period of T and a mass of m, by Newton’s second law of motion,
Fc = mac,
where ac is the centripetal acceleration due to the force Fc.
Fc = mac = mv2 / R
where R is the radius of the orbit of the planet.
Since v = 2πR / T, then
Fc = mv2 / R = 4π2mR / T2.
Assuming the centripetal force Fc is due to the force of gravity Fg of the sun with a mass of M exerted on the planet with a mass m. That is,
Fg = Fc = 4π2mR / T2,
and by the law of gravitation,
Fg = GmM / R2,
where G is assumed to be a universal constant, then
GmM / R2 = 4π2mR / T2.
Solving this equation for T2, we get
T2 = [4π2 / GM]R3.
Since the quantity in brackets contains the gravitational constant G, the mass of the sun M, and numerical factors, it is a constant k; that is,
k = 4π2 / GM.
Thus T2 = kR3.
This is Kepler’s Third Law of Planetary Motion.
The most serious technical problems in directly verifying the law of gravitation were solved by Henry Cavendish (1731-1810 A.D.) in 1798, over 100 years after the publication of the Principia. He used a torsion balance, an instrument operating on the torsion principle similar to that of a spring balance, in which a wire is twisted instead of stretched, that had been developed earlier for this purpose by his friend Rev. John Mitchell. The instrument consisted of two small metal balls of equal masses, m, attached to each other at the ends of a horizontal metal bar which is suspended at its middle by a vertical wire. After the two small balls comes to a position of equilibrium, any sideward force applied on either or both small balls will cause the suspension wire to twist. The amount of twist produced, as determined from the new position of the two small balls, is proportional to the amount sideward force applied. When two large metal balls of equal masses, M, are brought near the two small metal balls, one large ball near one small ball, the law of gravitation predicts that the large balls will attract the small balls and cause the two small balls to move turning the horizontal bar and the vertical wire to twist. If the magnitude of the force is determined by the amount of twist, and if the distance between the masses M and m are measured, the gravitation constant G can be found, using the formula for universal gravitation. The actual forces observed by Cavendish amounted to only about one five-millionth of the weight of the small masses m; extreme care had to be taken to shield the apparatus from outside disturbances such as air currents and temperature variations. The most modern measurements, using improved versions of essentially the same instruments, gives a value for G = 6.67×10-8 dynes · cm2 per gm2 = 6.673×10-11 newton · m2 per kg2, where mass is measured in grams, distance is measured in centimeters, and force is dynes. In other words, if two one gram masses are one centimeter apart the force of gravitational attraction between them will be less than one ten-millionth of a dyne. It is almost miraculous that gravitational forces so small were measured at all.
In addition to the direct verification of the law of gravitation there were a number of indirect verifications of the law of gravity.