16.3.1 Gravity.rtf

©1998 RCBrill. All rights reserved


GRAVITY
Program 16
Lesson 3.1


Text References

Spielberg & Anderson 96-101
Booth & Bloom 122-125; 127-131

Coming Up

Before we finish this lesson we will have learned how Sir Isaac Newton formulated his famous Theory of Universal Gravitation. We will look at the components of the Newtonian Synthesis, Newton's use of inductive and deductive reasoning, and his creative problems solving methods. Then we will see how Newton put it all together to write the famous equation. Finally, we will see how Newton designed a test of the theory using the orbit of the moon, thinking of it like a ball on a string in a circular orbit.

Objectives

1. Be able to state the meaning and terms of Newton's law of universal gravitation
2. Describe how each of the elements of the Newtonian synthesis was used in formulating the law of gravity.
3. Describe Newton's use of creative problem solving in determining the direction, magnitude, physical nature and continuity of the gravitational force
4. Describe the methods used by Newton to test his theory of gravity using the moon's orbit.

Questions

1. What other physical quantities besides gravity show an inverse square relationship?
2. Discuss Newtons's use of logic in regards to the direction, magnitude, physical nature and continuity of the gravitational force.
3. What is the "distance" problem and what solution did Newton offer?
4. How would be the acceleration of gravity at a distance of 10 Earth radii from Earth's center compare with gravity at Earth's surface? At 60 Earth radii?
5. How did Newton use the moon to test his inverse square hypothesis?
6. Discuss the role of Newton's laws of motion in the development of the Law of Universal Gravitation.
7. Discuss the statement, "Projectile motion is a specialized case of orbital motion." Relate both to the conic sections.
8. What is the significance of Newton's use of the word, Universal, in his description of gravity.
9. In Newton's gravitational equation, define the terms and state the meaning of the equation in words.
10. According to the third law, how strong is the moon's gravitational pull on Earth compared with Earth's pull on the moon?
11. Discuss the statement, "Newton dealt the final blow to Aristotle's universe."

1. Introduction

Newton's synthesis of universal gravitation stands as one of the most significant intellectual achievements in the history of thought. He used a variety of methods including inductive and deductive reasoning and various forms of mathematical analysis, many of which he invented.

In this lesson we will examine the ways in which Newton synthesized information from various sources to piece together the theory of gravity and test it using the orbit of the moon.

2. Universal Law of Gravitation

Newton's law was universal because it applied to all objects regardless of their location, sublunar or cosmic. It is really a simple law and easily stated. That such a simple law can have such far reaching implications and uses attests to its validity and its generality.

Be sure you understand the meaning of the law in verbal and mathematical forms. You should know what each of the terms in the mathematical relationship stands for. Pay special attention to "G", the universal gravitational constant. It is a constant of proportion. If you have forgotten what a constant of proportion is, then go back to lesson 12.2.4 and review.

2.1. F = (GMm)/r^2 : All objects in the universe attract each other with a force which is proportional to their masses and inversely proportional to the square of the distance between their centers

2.2. m1, m2 are masses
2.3. r is distance between centers
2.4. G is gravitational constant

3. Newtonian Synthesis

Newton drew upon the work of many who went before him. He brought together Euclidian geometry, Galileo's kinematics, Kepler's laws, central force, inverse square, the laws of motion, circular motion and data about the orbit of the moon. Combining inductive and deductive reasoning like no one before him, he added his own tools of analysis including new forms of mathematics.

3.1. Laws of Motion

If the laws of motion applied to rocks and stones, then they should also apply to planet earth, which is, after all, composed of rocks and stones.

If that was the case, then there was no need of an explanation for why planets kept moving (recall Aristotle's Prime Mover). The real question was, "what keeps the planets from moving in straight lines?" (Recall the first two laws of motion).

Furthermore, if a force is exerted on a planet to keep it in orbit, then the planet must exert an equal but opposite force.

3.1.1. planets also obey
3.1.2. no force required to sustain motion
3.1.3. force required to change motion

3.1.3.1. in proportion to mass and acceleration
3.1.3.2. speed or direction

3.1.4. forces exerted in pairs, equal and opposite

3.2. .Central force

The idea that something within the orbit of a planet acts to hold it in orbit is a modern concept. Nothing in Aristotle's cosmology suggested that. In his view, all the planetary motions were controlled from outside, as with the Prime Mover.

Kepler recognized the need to keep the planets moving in closed paths, but incorrectly credited magnetism as the cause. It was obvious to most

3.2.1. modern concept
3.2.2. Kepler recognized need to keep planets moving in closed paths

3.2.2.1. incorrectly theorized magnetism

3.3. Logical continuity

The argument that different laws apply in different parts of the universe was challenged by Newton when considering the relationship between freefall motion and orbital motion. This will be discussed in more detail in section 4.4 below.

3.3.1. force pulls apple to Earth, where does it stop
3.3.2. if Moon is also pulled to Earth why doesn't it fall?

3.4. Centripetal acceleration

Newton worked out the relationship between the radius and velocity of circular motion and centripetal acceleration, unaware that Huygens in Holland had already done a similar analysis.

The Booth and Bloom text has a good explanation and analysis of circular motion beginning on page 122. You may also wish t review our coverage of this material in lesson 15.2.7

3.4.1. force is required to hold object in circular motion

3.4.2. amount of force is proportional to acceleration

3.4.2.1. F = ma

3.4.3. acceleration can be calculated

3.4.3.1. equals velocity squared divided by radius
3.4.3.2. ac = v2/r

3.5. Inverse square

3.5.1. light and sound intensity decreases with square of distance
3.5.2. gravity might emanate from massive objects in a similar way

4. Creative Problem solving

Newton was able to solve the gravity problem by solving a series of smaller problems. This he did by analyzing planetary motion using his laws of motion and vector geometry. Newton was one of the first to systematically tackle a large problem by breaking it into smaller problems. We might say that Newton combined analysis and synthesis. To solve the following problems required the mind of a genius who could see to the root of the problem and who could concentrate on the problem until the solution presented itself.

When looking at the outline and viewing the program do not be concerned if you cannot follow every detail. More importantly, look at the big picture to see the way in which the problem was solved rather than the details of the solution.

4.1. Direction of Planetary Force

Newton used a simple geometric argument to show that an object under the influence of a central force will always describe equal areas in equal times. He used an intermittent force directed towards a point at regular intervals combined with the second law of motion and his vector algebra. The video program shows how this argument works.

4.1.1. Derivation of Kepler's Second Law

4.1.1.1. object moving under influence of continuously applied centrally directed force must sweep out equal areas in equal times
4.1.1.2. proof by geometric induction

4.1.2. object which sweeps out equal areas in equal times must be moving under the influence of a central force

4.1.2.1. deduction from first and second mechanical laws

4.1.3. Inductive/deductive combination leads to necessary/sufficient connection:
4.1.4. equal areas in equal times <==> influence of central force

4.2. Magnitude of Planetary Force

By analyzing the motion of a planet and the direction and magnitude of forces acting on it a different portions of its orbit, Newton generated a differential equation. When he solved the equation in general terms he found that the solution was of the form of a conic section. From this he was able to show that an object under the influence of an INVERSE SQUARE force will describe an orbit that is normally elliptical.

You may wish to review the focus on conic sections in lesson 10.2.2.

4.2.1. How does it vary during orbit of planet in elliptical orbit?
4.2.2. conic section orbit ==> inverse square

4.2.2.1. force is inversely proportional to the square of the body's distance from the focus of the conic section
4.2.2.2. inverse square

4.2.3. conic sections

4.2.4. inverse square ==> conic section orbit

Note how this is different from 4.2.2.

4.2.4.1. geometric analysis of forces on planets generated equation
4.2.4.2. solutions to equations are conic sections

4.2.5. Inverse square relationship alone does not allow for the calculation of the magnitude of the force

4.3. Physical Nature of the Force

Newton simply assumed that there was some as yet described force which attracted the planets to the sun as it also attracted objects to earth and caused them to accelerate in free fall.

For the first time gravity was viewed as a mutual property of two objects. In Aristotle's view, and also to Galileo, gravity was something possessed by the apple or the rock. Newton viewed the acceleration of the apple as due to the interaction of the apple and the earth, Certainly the earth was much larger than the apple, but the force exerted on the apple depended equally as much on the apple as on the earth.

The mechanism for this force, and how it could operate over millions of miles of empty space remained problems even in Newton's own mind. The concept of "action at a distance" bothered him, but he did not let the lack of a solution to WHAT gravity is interfere with describing HOW gravity affects objects and how it varies with distance.

4.3.1. Kepler speculated magnetism to be the cause

4.3.1.1. not fruitful, but called attention to the sun as a relevant factor

4.3.2. Descartes postulated a space filled with invisible magnetic fluid

4.3.2.1. planets caught in swirling vortex of fluid around the sun

4.3.3. Newton proposed a dramatic solution

4.3.3.1. All objects attract one another with a gravitational force like that existing between a falling stone and Earth
4.3.3.1.1. central force on a planet is gravitational attraction of the sun
4.3.3.1.2. central force on moon is gravitational attraction of Earth
4.3.3.1.3. Rule 3

4.3.4. THE SAME MECHANICS APPLY TO BOTH TERRESTRIAL AND CELESTIAL MOTION

4.3.4.1. projectiles and planets obey the same laws
4.3.4.2. planetary motion can be described the same way as projectile motion
4.3.4.3. Kepler's laws became a consequence of the same all-pervading mechanics

4.4. Continuity of the Force

The idea that the force which caused the apple to fall to the ground would suddenly stop at some arbitrary level above earth simply did not make sense. You will recall that Aristotle and the Scholastics had maintained that a different set of laws were in effect outside the moon's orbit.

Newton imagined what would happen if you released the apple from successively higher and higher altitudes. Would there be a boundary above which it would no longer fall? If so, what would happen if you released it while part of it was below the boundary and part was above?

4.4.1. Newton to William Stukeley during tea in a garden under apple trees:

4.4.1.1. "I was just in the same situation, as when formerly, the notion of gravitation came to my mind. It was occasion'd by the fall of an apple, as I sat in a contemplative mood."
4.4.1.2. 'contemplative mood' is the key element, rather than the apple.

4.4.2. Apple falls downward under the influence of Earth's gravity

4.4.2.1. by second law, downward force is proportional to mass and acceleration
4.4.2.2. Guessed that force is proportional to mass of both objects
4.4.2.2.1. based on Gilbert's incorrect assertion that strength of a magnet depends on the mass of the magnet
4.4.2.2.2. by third law, it is a mutual effect

4.4.3. Could the force extend above the treetops all the way to the moon ?

4.4.3.1. more reasonable to think that it might than to think that it does not

5. Putting It Together

To put the theory together was simple after the pieces were formulated.

It is a simple step to turn a series of proportions into a mathematical equation.

5.1. Force is inversely proportional to square of distance of separation

5.1.1. force decreases with the second power of distance
5.1.2. Kepler's laws are consistent with central inverse square force
5.1.3. inverse square central force generates elliptical and other conic orbits.

5.2. Force is proportional to both masses

5.2.1. third law suggests a mutual effect
5.2.2. A exerts force on B, B exerts equal and opposite force on A
5.2.3. changing the size of either mass will affect the forces on both masses

5.3. Force is proportional to product of masses, inversely proportional to square of distance between centers

6. Testing the Theory

Although the relationships seemed to be true according t the methods of analysis which Newton had so cleverly invented, it was not a proof. To prove the theory it would be necessary to make a prediction deductively from the law and then compare it with data.

How could the theory be tested? Gravity is far too weak to be tested in the laboratory. Newton's synthesis was the result of a series of inductions and deductions, but it was not clear whether it corresponded to fact. To avoid the mistakes of the Greek philosophers, it needed to be be tested to see if it corresponded to reality.

Even so, the gravitational equation does not allow the actual magnitude of the planetary force to be calculated. To do that would require that we know the masses of the earth and the sun, and the value of the gravitational constant. None of these were known in Newton's time. In fact, these physical quantities can be calculated form the Gravitational equation. Even today, our knowledge of the masses of the planets comes from the gravitational equation.

We will explore this in more detail in lesson 17.3.2.

6.1. Is the gravitational equation correct?

6.1.1. It is the result of a series of deductions and inductions
6.1.2. Dose it correspond to fact?
6.1.3. It does not answer the question of the magnitude of the planetary force

6.2. Acceleration vs. force

The gravitational force on the earth or the moon could not be measured or calculated because the masses and the gravitational constant are not known. But the moon is in a nearly circular orbit around earth and circular motion requires a centripetal acceleration. Is is possible that earth's gravity provides exactly the correct amount of centripetal acceleration to keep the moon in orbit?

On one hand Newton could calculate the amount of centripetal acceleration required to keep the moon in orbit. This can be done once the radius of the moon's orbit and its speed are known.

On the other hand, the acceleration of earth's gravity at the moon's distance should be inversely proportional to the gravitational acceleration at earth's surface.

But how do you figure a proportion if one of the terms is zero? After all we live at zero distance from earth's surface, don't we?

6.2.1. Calculation of Gravitational Force is not possible
6.2.2. Gravitational constant is not known!
6.2.3. Mass of Earth, moon, sun are not known!
6.2.4. acceleration is proportional to force

The second law!

6.2.5. if gravitational force decreases with the square of distance acceleration should also show inverse square relationship

6.3. How is the distance between Earth and the apple or between Earth and the moon to be determined?

The problem is that the earth and moon are both extended objects, so there is a great difference in the distance between earth and moon depending on where the distance is measured from.

6.3.1. some parts of Earth are very close to the apple and some are very far away
6.3.2. Some parts of the moon are quite a bit further from any point on Earth than others.

6.4. Distance must be measured from the center of mass

Newton used the gravitational proportions and a symmetry argument to show that the distance r should be measured from the center of mass and is directed towards the center of the planet.

To do this required the use of vector algebra and a method which he invented for summing large numbers of infinitely small parts. Today we call this method integral calculus.

The crux of Newton's proof is that for every small piece of mass in the earth there is an identical piece of mass which is exactly opposite it so that the vector sum of the forces due to the two pieces points half way between them towards the center.

6.4.1. if objects are spherical and homogeneous or made of spherical, homogeneous shells
6.4.2. consider a spherical object to be made of many small particles
6.4.3. each particle attracts each other particle with a force related by gravitational equation
6.4.4. mutual gravitational forces is the sum of individual gravitational forces
6.4.5. analysis required the methods now called integral calculus

6.5. The moon problem

To test the inverse square relationship Newton compared the centripetal acceleration of the moon with the acceleration of earth's gravity reduced by the inverse square of the moon's distance and found them to "agree pretty nearly".

In doing so he also defined for us the limits on expectations of perfect accuracy in science based upon the degree of approximations made and the accuracy of the data on which we base our calculations.

 

6.5.1. considered moon's orbit as circular
6.5.2. compare inverse square gravitational acceleration with centripetal acceleration

6.5.2.1. moon's orbit is very nearly circular
6.5.2.2. gravity provides the necessary centripetal force as moon "falls" towards Earth

6.5.2.3. rate of acceleration is less than at Earth's surface by inverse square factor
6.5.2.4. Earth-moon distance is about 60 Earth radius
6.5.2.5. Freefall acceleration of moon should be about 1/3600 that of surface value

6.5.3. Newton found the calculation to "answer pretty nearly"

6.5.3.1. 0.0023 vs. 0.0027 m/s2 in modern units

6.5.4. assumption of circular orbit and use of rough values for Earth radius and Earth-moon distance made it clear that "no perfect agreement" could be expected

6.5.4.1. "perfect agreement" should be defined as correspondence of the values within the expected margin of error

6.5.5. "perfect agreement" should be defined as correspondence of the values within the expected margin of error
6.5.6. note the modified use of the word "perfect"both in context and in meaning

7. Summary

The gravitational equation stands even today as the single most significant relationship in the history of science. It allows the orbits of the planets to be calculated with whatever precision is desired. It is the first universal relationship. It unifies celestial and terrestrial motion, it provides explanations for many phenomena which were formerly not imagined to be related.

We will study the implications of gravitation in the next lesson.

7.1. Newton combined several models and creative problem solving techniques to show conclusively:

7.1.1. F = (GMm)/r2 : All objects in the universe attract each other with a force which is proportional to their masses and inversely proportional to the square of the distance between their centers

7.1.2. central force <==> areas proportional to time
7.1.3. inverse square force <==> conic section orbits
7.1.4. gravity of extended homogeneous object such as a planet "emanates" from the center of mass diminishing with the square of the distance between centers
7.1.5. objects exert mutual attractive forces in proportion to both of their masses
7.1.6. the centripetal acceleration of the moon in its orbit is consistent with the above

7.1.6.1. to within the accuracy of existing measurements of Earth's radius and Earth-moon distance
7.1.6.2. assuming a circular orbit for the moon (which it is very nearly)