Newton’s description of gravity is a crowning scientific achievement. In one insight, Newton united our physical understanding of the earth and the heavens. His description (theory) of gravity showed that not only could the heavens be understood, but that the rules they follow are the very same rules that familiar earthly objects follow. (How cool is that?!) This chapter describes how a very simple rule explains a tremendous number of phenomena.
pp. 161–162. The Universal Law of Gravity. This section outlines the very interesting story of how Newton developed the theory.
The main physics point to get from this section is in the middle of page 162: that the force of gravitational interaction is proportional (that’s what the book means by the “~” sign) to the product of the masses of the interacting objects divided by the square of the distance between them.
pp. 163–164. The Universal Gravitational Constant, G. G is the proportionality constant needed to upgrade the proportional relation of the previous section to an equation.
I need to point out that the statement in the book that “the magnitude of G is identical to the magnitude of the force…” is not strictly correct. For it to be true, the units need to be the same, which they aren’t. It is more correct to say that “the numerical value of G is identical…”.
The remainder of the section explains how G can be measured experimentally, and how knowing G allows us to calculate the mass of the earth.
pp. 165–166. Gravity and Distance: The Inverse-Square Law. This section conceptually justifies why the force of the gravitational interaction changes as 1/d2. It is as if objects with mass send out “gravity rays” that pull on other objects with mass.
Recommmended workbook exercise: p. 51, questions 1, 2, and 4.
pp, 166–172. Weight and Weightlessness; Ocean Tides. It breaks my heart that we won’t discuss this in class, but there isn’t time.
pp. 172–173. Gravitational Fields. Read this section. This is a good introduction to the concept of fields, which we will soon use extensively with electricity and magnetism.
In the representation of the Earth’s gravitational field by field lines, find the meaning of:
There is a typographic error in footnote 4 on page 173. The second line of the footnote should begin “ g = F/m” to be consistent with the words “Force F per unit of mass”. (It also must be “ g = F/m” to be correct.) The value g can be thought of as the acceleration of an object due to gravity or as the strength of the gravitational field. Both mean the same thing! Note that g depends on the mass of the earth and on the distance from the center of the earth. This does not change much between the places that are reasonably accessible to us, which are at or very near the surface of the earth.
We used the first part of this chapter earlier to learn how projectiles move in the approximately constant gravitational field near the surface of the Earth, where neither the direction nor the magnitude of the field varies much with position. Now we look at paths covering longer distances, for which the gravitational force cannot be treated as constant.
pp. 192–193. Fast-Moving Projectiles—Satellites. The discussion here explains how an object accelerating toward the Earth may never actually reach the Earth. You have already explored this idea: any object moving in uniform circular motion is constantly accelerating toward the center of its circular path, even though it never gets any closer to the center. (Weird? Perhaps. I recommend that you take some time to remind yourself why this is true and why it makes sense.)
pp. 194–195. Circular Satellite Orbits. You should be able to read this section quickly. The physics here should be entirely familiar to you: it is just the physics of uniform circular motion. The centripetal force accelerating the satellite toward the Earth is gravity.
Note that because the centripetal force is gravity, satellites far away from the Earth accelerate less than satellites closer in. This profoundly affects the timing of orbits.
Recommmended workbook exercise: p. 58, question 2.
pp. 196–197. Elliptical Orbits. Skim this section. The important idea to take away is that orbits are not necessarily circular. When a satellite travels in an elliptical orbit, the force of gravity is not always exactly perpendicular to its velocity, so it will speed up and slow down at different parts of the cycle. A circle is just one special instance of an infinity of ellipses. In our Solar system, orbits of planets and satellites are not circular, but most are fairly close.
Recommmended workbook exercise: p. 59.
p. 198. box “World Monitoring by Satellite.” Read this if you want.
pp. 199–200. Kepler’s Laws of Planetary Motion. Skip this section. These laws are important in astronomy, and they are direct consequences of Newton’s laws and Newton’s gravitational formula (but Kepler did not know that!). We will not delve into these in class because time is short. You should, however, understand the physics behind the laws.
pp. 200–201. Energy Conservation and Satellite Motion. There is no new physics here, but you should read this short section to see how conservation of energy is manifest in the orbits of satellites. Pay attention to the Figures, and try the Check Yourself questions. If you understand this section, you have mastered the concepts of orbital motion.
Recommmended workbook exercise: p. 60, questions 1–2.
p. 201–204. Escape Speed. Even though an object’s gravitational field exerts a force on all other masses, no matter how far away, that does not necessarily mean that gravity will eventually draw all objects together.
The key physics idea is on p. 202: the gravitational potential of an object infinitely far from the Earth is 62 × 106 J/kg, not infinity. (Gravitational potential at a point in space is the gravitational potential energy an object would have if it were located at that particular position, divided by the object’s mass.) It requires calculus to actually calculate this value, but you can get the idea of why it works that way by considering the discussion in the first full paragraph on p. 202.
The rest of this section discusses interstellar space flight. We will not address this in class.
We described the gravitational force acting on an object, such as the Moon in orbit around the earth, in terms of the Earth’s gravitational field. Recall that Newton’s third law requires that any force acting on one object is paired with an equal and opposite force acting on another object.
Copyright © 2008, Richard Barrans
Revised: 11 January 2010. Maintained by Richard Barrans.
URL: http://www.barransclass.com/astr1070/rguides/P1050F10rg_03-30.html