11 Universal Gravitation - wittkamp's science site
11 Universal Gravitation Name
Reading: THE LAW OF UNIVERSAL GRAVITATION
Sir Isaac Newton, who probably contributed as much to science as any other person, once declared that if he had seen farther than others, it was because he was able to "stand on the shoulders of giants."
The story of the Law of Universal Gravitation starts with the ancient Greeks and ends with an apple. Newton is supposed to have thought of the final clue which led to his formulation of the Law of Universal Gravitation while watching an apple fall from a tree. Newton had been spending time at his mother's farm when the plague struck London. At this time, Newton saw an apple fall from a tree while the moon was visible in the sky, and began to formulate his ideas concerning gravitational attractions and planetary interactions.
The ancient Greeks were very interested in attempting to explain physical phenomena - both earthly and heavenly. Aristotle (384-322 B.C.) held that the fall of a body toward the earth was its natural motion because down was the natural place for such a body. Heavier objects sought down more than lighter objects and, therefore, heavier objects would fall faster than lighter ones. This was Aristotle's way of explaining the results of the phenomenon we call gravity. We know now that Aristotle was wrong, that heavier objects do not fall faster than lighter objects when air resistance is negligible, but his idea appeared quite reasonable at the time.
Aristotle and his fellow Greek philosophers also attempted to develop a blueprint for the universe. First, the earth was positioned motionless at the center of the universe (this is called the geocentric view). The planets, the moon, and the sun supposedly circled the earth. Second, the earth and the heavens were considered to be governed by different laws. The heavens, unlike the earth, were perfect. Because of their perfection, the heavens should be characterized by perfect motions. What could be more perfect than uniform circular motion? Thus, the planet's orbits were thought to be perfect circles traversed at uniform speeds.
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Nicolaus Copernicus 1
Figure 1. Nicolaus Copernicus
Figure 2. Nicolaus Copernicus
Nicolaus Copernicus 2
Nicolaus Copernicus 3
This view was to remain basically unchallenged until the time of Copernicus some 2000 years later. (It is interesting to note, however, that Aristarchus contended in 275 B.C. that the sun was the center of the universe and that the planets, including the earth, moved around the sun in large circular paths. His view was not widely accepted, however.) Nicolaus Copernicus (1473-1543; pictured at right), a Polish mathematician and astronomer, adopted a heliocentric, or sun-centered, view of the universe. According to Copernicus, the sun was at rest at the center of planetary motion; the earth, the moon, and the planets revolved around the sun. The earth was not motionless. In fact, it had two motions - it revolved around the sun and it rotated around its own axis. Aristotle and his fellow philosophers had hypothesized that the earth could not turn lest the objects on the earth be thrown off into space. Copernicus did not press his ideas and he did not experience much opposition from the religious leaders of the Catholic Church, who were firmly rooted in Aristotelian cosmology. He did not assert the physical reality of his heliocentric theory.
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One of history's most brilliant astronomical observers rose to prominence several years after Copernicus' death. Tycho Brahe (1546-1601, pictured at right), a Dane, made meticulous observations using equipment he himself either invented or improved. The Danish King, Frederick II, even provided Tycho with the funds to set up an extensive astronomy laboratory. Tycho made important observations concerning the comets, the planets, and the fixed stars. One of Tycho's best discoveries was a young German mathematician named Johannes Kepler. When Tycho died in 1601, Kepler obtained all of his mentor's records and observations of the motion of Mars.
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Kepler (1571-1630, pictured at left) was a Copernican and dedicated to perfecting the heliocentric theory. He used Tycho's observations of the motion of Mars as a starting point. He kept trying to squeeze circular orbits and geometric ideas from other astronomers into the data gathered by Tycho. Kepler worked out the best system he could (after seventy trials and a year and a half of work), but he discovered that his calculations diverged from Tycho's observations by eight minutes of arc. Eight minutes would be equivalent to about a fourth of the moon's diameter - which isn't much considering the equipment Tycho had to use.
Kepler had two choices at this point: he could throw out his system or consider Tycho's observations to be slightly erroneous. Kepler was so sure of Tycho's expertise that he decided to throw out his hard work and come up with an entirely new system not based on uniform circular motion, equants, epicycles, or any other previous ideas. The amazing results can be summarized in Kepler's three laws:
The Law of Elliptical Orbits - The paths of the planets are ellipses with the sun at one focus. The orbits are not circular, but elliptical!
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The Law of Equal Areas in Equal Times - The line from a planet to the sun sweeps out equal areas per unit of time. Simply put, this means that a planet will move from A to B in the same time it moves from C to D in the accompanying figure, with the areas of the pie wedges ABS and CDS being the same.
So much for uniform speed! This law indicates that a planet travels faster the closer it is to the sun and slower the farther away it is from the sun.
The Law of Periods - The ratio of the cube of the radius of the orbit to the square of the period for the planets is constant:
[pic] Where T is the period and r is the radius.
From this law we can determine the period of the orbit of any planet if we know the radius of its orbit. This is how we know it takes Pluto 248 of our years to orbit the sun even though it was only discovered in 1930.
At this point, you might wonder why we are using radii while working with elliptical motions. You should know that the orbits of most of the planets are very nearly circular. Pluto's orbit is the most elliptical, while the orbit of Venus is the closest to being circular. The orbit of the Earth is only a bit more elliptical than that of Venus. However, it would perhaps be more exacting to consider the average distance of the planet from the sun as being the value of the radius r.
Kepler had revolutionized astronomy. Through the use of Tycho's observations and his own mathematical genius, Kepler had abolished the uniform circular motion of the ancients and replaced it with elliptical, non-uniform motion.
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Galileo Galilei (1564-1642, pictured at right), the noted Italian scientist, corresponded with Kepler. Galileo was convinced of the validity of Copernicus' heliocentric view and spent much of his life trying to convince the religious leaders as well as the scientists of the time of its relevance. But Galileo's importance in our story deals with his work with terrestrial motion. Galileo determined that the weight of an object did not dictate the rate at which it would fall to earth. Galileo based his theories on empirical data and was more interested in the how's than the why's of free-fall.
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How can we work Newton and his Law of Universal Gravitation back into our story? Let's note that Newton (pictured at right) had an opportunity to spend two years pondering a multitude of problems while he was waiting for the plague to subside in London. One of his contemplations involved the fact that apples (and all other objects) fell perpendicularly to the ground - they never fell sideways or at an angle to the earth's center. But Newton did not stop there. He began to consider that the concept of gravity could be expanded to the motion of the planets as well as the motions of objects on earth. This was a revolutionary idea as the causes of physical phenomena on earth had long been considered separate from those of the heavens.
Newton wrote:
I began to think of gravity extending to the orb of the moon, and...from Kepler's rule (third law, law of periods)...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 to answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days (at age 21 or 22) I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since.
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Newton found that motion in an elliptical path could only occur when the central force was an inverse square force:
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The matter was settled when Newton showed that such a force law would also result in Kepler's Third Law. Let's look more closely at Newton's proof. We can rewrite the centripetal acceleration equation in terms of the radius of the revolution (or orbit, in the case of planets) and the period:
Kepler's Law of Periods stated a definite relationship between the orbital periods of planets and their average distance from the sun:
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We can substitute r3/k for T2 in our centripetal acceleration equation and obtain:
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Since 4π2k is a constant, we can write:
If we next consider Newton's Second Law of Motion, F=ma, we know that F ( a, and thus:
This means that the quantity of force exerted by the sun on a planet would be inversely proportional to the square of the radius of the planet's orbit.
Newton concluded that one general Law of Universal Gravitation applied to all bodies moving in the solar system - in fact, the law applied to any object moving in an orbit about a center of focus. Newton supposed that the planetary force was the same kind of force that caused objects near the earth's surface to fall. He first tested this idea on the earth's attraction for the moon.
According to the data available to Newton, the distance between the center of the earth and the center of the moon was about sixty times the radius of the earth (or 60 x 6.378x106 m). If the attractive force of the earth on the moon was proportional to the inverse of the square of the radius (see above proof) then the gravitational acceleration the earth exerts on the moon should be only 1/602 or 1/3600 of that exerted upon objects at the earth's surface. You know from your laboratory work that the acceleration due to gravity at the earth's surface is about 9.8 m/s2. Thus, the moon should fall toward the earth at 1/3600 of 9.8 m/s2, which is 2.72x10-3 m/s2. Let's see how close Newton was.
The orbital period of the moon is approximately 27 1/3 days or 2.3x106 seconds. The radius of the moon's orbit around the earth is about 3.9x108 meters. Using the equation for centripetal acceleration, we can calculate the acceleration of the moon toward the earth:
This is in very good agreement with the 2.72x10-3 m/s2 value predicted by Newton's idea that centripetal acceleration is directly proportional to the inverse of the radius, and thus so is gravity. Newton concluded that:
The force by which the moon is retained in its orbit becomes, at the very surface of the earth, equal to the force of gravity which we observe in heavy bodies there. And, therefore, the force by which the moon is retained in its orbit is that very force which we commonly call gravity...
Thus, the same gravity that brings apples down to the ground also holds the moon in its orbit. How simple! But, how revolutionary! (Pun intended.)
Newton's contemplations led him to believe that every object in the universe attracts every other object with a gravitational force. The sun attracts the earth, the earth attracts the moon, and you attract the person sitting next to you. (Cozy, isn't it!)
One last step needs to be taken before we can state Newton's Law of Universal Gravitation in mathematical terms. Your work with circular motion showed you that centripetal force was proportional to the mass of the orbiting object, or Fc ( m. Since, according to Newton's hypothesis, any two objects exert equal gravitational forces on each other, it follows that the gravitational attraction between two bodies is proportional to the product of their masses:
Combining this relationship with F ( 1/r2, we have:
We can write this as an equation by introducing a constant, G. G is the gravitational constant, and finally we get:
The value of the constant G presented a problem until Henry Cavendish (1731-1810), an English scientist, devised an instrument for determining the gravitational attraction between two pairs of lead spheres. Cavendish employed a sensitive instrument called a torsion balance. In this device, the gravitational attraction between two pairs of lead spheres twisted a wire holding up one of the pairs. The twist of the wire could be measured against the twist produced by known force. An experiment involving a 100 kg sphere and a 1 kg sphere at a center-to-center distance of .1 meter produced a force of 10-6 Newtons. From these data the value of G could be calculated:
The present accepted value of G is 6.67x10-11 Nm2/kg2 and is very close to this early calculated value.
The Law of Universal Gravitation was developed for planetary bodies where the use of "r" as a symbol for distance came from its being a "radius". Nevertheless, this law can be applied to any two objects in the universe, so it is often written with a "d" replacing the "r".
Thus the final form of the Law of Universal Gravitation is:
Whew! What a story! We have traveled from the third century B.C. to the middle of the eighteenth century A.D. What is the relevance of such a search? Newton's Law of Universal Gravitation was revolutionary in that it coordinated the physical laws of earth and heavens. It shows us that the same forces that cause objects to fall toward the center of the earth also hold planets in their orbits. Finally, it has reduced that mysterious phenomenon called gravity to a mathematically-described interaction between masses.
It is important to remember that Newton, like Galileo, was primarily interested in the how's and not the why's. The why's of gravitational attractions were addressed more fully in the twentieth century through Albert Einstein’s General Theory of Relativity. But Newton’s Law of Universal Gravitation remains the basis of our space missions, including interplanetary probes.
Watch the stars, and from them learn
To the Master’s honor all must turn,
each in its track, without a sound,
forever tracing Newton’s ground.
-Einstein
11 Universal Gravitation Name
READING CROSSWORD Inquiry Physics
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Across
3. Newton said the sun's gravity acted as this kind of force
4. Kepler showed orbital radius is proportional to this
5. Great Greek philosopher, but not a great physicist
8. Kepler showed orbits had this shape
9. Originator of earth-centered model with crystal spheres
12. Means "earth-centered"
13. Each planet's orbit has the sun at this point
15. Brahe's country
16. Gravity is proportional to the product of each object's...
18. Explained gravity as a warp in space-time
20. Kepler's country
22. Added epicycles, etc. to earth-centered model
23. Eudoxus' country
24. Copernicus' country
Down
1. Delayed publishing "On The Revolutions of Heavenly Bodies" until he died
2. Means "sun-centered"
6. First to measure universal gravitation constant
7. Copernicus thought orbits had this shape
10. Kepler showed that when farther from the sun, planets move...
11. Gravity is inversely proportional to the square of...
14. Published "Dialogue Concerning The Two Great World Systems" in 1632
17. Gravity prevents planets from flying off at a...
18. Newton's country
19. Galileo's country
21. Published "The New Astronomy" in 1609
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