16.3.1 Gravity.rtf

# GRAVITY Program 16 Lesson 3.1

## 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.

## 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.

## 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.

### 3.2. .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

### 3.3. 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.

### 3.4. 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

## 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.

### 4.2. 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.

#### 4.2.4. inverse square ==> conic section orbit

Note how this is different from 4.2.2.

### 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.

### 4.4. 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?

## 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.

## 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.

### 6.2. 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?

The second law!

### 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.

### 6.5. 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.

## 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.