Tipo y diseño de investigación

Gravity

Even before taking an astronomy class, most people have a sense of how gravity works. No mathematics is needed to understand the idea that every mass attracts every other mass and that gravity is the force that causes apples to fall from trees. But what if you want to know how much you’d weigh on Saturn’s moon Titan, or why the Moon doesn’t come crashing down onto the Earth, or how it can possibly be true that you’re tugging on the Earth exactly as hard as the Earth is tugging on you? The best way to answer questions like that is to gain a practical understanding of Newton’s Law of Gravity and related principles.

This chapter is designed to help you achieve that understanding. It begins with an overview of Newton’s Law of Gravity, in which you’ll find a detailed explanation of the meaning of each term. You’ll also find plenty of examples showing how to use this law – with or without a calculator. Later sections of this chapter deal with Newton’s Laws of Motion as well as Kepler’s Laws.

And like every chapter in this book, this one is modular. So, if you’re solid on gravity but would like a review of Newton’s Third Law, you can skip to that section and dive right in.

2.1 Newton’s Law of Gravity

The equation for Newton’s Law of Gravity may look a bit daunting at first but, like most equations, it becomes far less imposing when you take it apart and examine each term. To help with that process, we’ll write “expanded” versions of some of the important equations in this book, of which you can see an exam-ple in Figure 2.1. As you can see, in an expanded equation, the meaning and units of each term are readily available in a text block with an arrow pointing to the relevant term. After the figure, you’ll find additional explanations of the

41

42 Gravity

F

= G m

m

R

mass 2 (kg) mass 1

(kg)

Distance from the center of mass 1 to the center of mass 2 (m) The universal

gravitational constant (N m²/kg²) Force of gravity

between mass 1 and mass 2 (N)

Figure 2.1 Newton’s Law of Gravity.

terms as well as examples of how to apply the equation using both the absolute method and the ratio method.

2.1.1 Description of terms in the gravity equation

Whenever you encounter an equation such as that shown in Figure 2.1, it’s a good idea to make sure you understand not only the meaning (and units) of each term, but also what the placement and powers of those terms are telling you.

The force of gravity, Fg, appears on the left side of this equation in units of newtons (N). The force occurs between two objects such as those shown in Figure 2.2; each object produces the gravitational force Fgon the other. Here are detailed descriptions of each of the terms on the right side of the gravity equation:

G The first term is G, the universal gravitational constant.¹To the best of sci-entists’ knowledge, this constant has the same value throughout the known Universe, and that value in SI units is 6.67 × 10⁻¹¹N m²/kg².

m1, m2 The variables in the numerator of the fraction, m1and m2, represent the amount of mass (in units of kilograms) in each of the two objects for which the force of gravity is being calculated. Most astronomy texts use lowercase “m,” as we have here, as a variable (a placeholder for an unspeci-fied quantity) to represent mass. Be wary that this invites confusion with the

1 This constant is always written with an uppercase G – do not confuse this with lowercase g, which is usually used to denote the acceleration produced by gravity at a specific location. G and g have different units and different meanings.

2.1 Newton’s Law of Gravity 43

m₁

m₂ F_g

F_g R

Figure 2.2 Two masses tugging on each other.

abbreviation for the distance unit meters, which is also abbreviated with a lowercase “m.” Take care never to confuse the two. Some texts, like this one, italicize variables but not units – so “m” would represent a variable for mass, and “m” would be an abbreviation for the unit meters – but convention varies between texts, so you may have to scrutinize your text to ascertain whether or not it uses this convention. Moreover, if there’s a subscript after the m, that’s a strong hint that m represents a variable for the quantity mass, not the units meters – but again, convention varies between texts. You may have to judge from context which meaning “m” has in different circumstances.

You should note that the mass of an object is a measure of the total amount of material that makes up the object and is not the same as the weight of the object. As you will see in the first example below, the weight of an object is simply the force of gravity (usually expressed in pounds in everyday life, rather than newtons), and that force depends on exactly where the object is located. So, your weight on the surface of the Earth is greater than your weight on the surface of the Moon because the Moon produces a smaller force of gravity at its surface. But your mass is the same no matter where you are.

R It’s quite common for students to assume that the R in the denominator of the gravity equation means the “radius” of an object, but in fact it represents the distance (in units of meters) between the center²of mass 1 and the center of mass 2. There are certainly some cases in which the distance R turns out to be approximately equal to the radius of a sphere (such as a planet), but you should not fall into the habit of thinking of R as always being a radius.

Once you’re comfortable with the meaning and units of each term, it’s time to step back and consider what the placement and power of those terms is telling you about the force of gravity. The fact that both masses appear in the

2 The word “center” in this context means center of gravity, but for simplicity most astronomy texts assume spherically symmetric objects, for which the geometric center and the center of gravity are the same thing.

44 Gravity

numerator on the right side of the equation tells you that the force of gravity is directly proportional to each of the masses. So, if you double one of the masses while keeping everything else the same, the force of gravity between the masses will also double. Notice also that it doesn’t matter which mass you call m1and which you call m2; since multiplication is commutative, they are interchangeable. And since you calculate only one force Fgbetween the two masses, the force of gravity of m1on m2is exactly the same as the force of gravity of m2on m1. This is an example of Newton’s Third Law (which you can read about later in this chapter) and it means that right now you’re pulling on the Earth exactly as hard as the Earth is pulling on you.

Now consider the placement and power of the R term in the gravity equation.

Since the distance between the objects appears in the denominator, that means force and distance squared are inversely proportional, so greater distance will result in smaller gravitational force (exactly as you would expect, since com-mon sense tells you that nearby objects exert a greater gravitational pull than far-away objects). Since R is squared, that means that the force of gravity drops off rapidly with distance. So doubling the distance while keeping everything else the same does not cause the force to decrease to one-half its original value (as it would if distance appeared to the first power in the denominator). Instead, doubling the distance reduces the force to one-quarter its original value (since

1

2² = ¹₄). This is called the “inverse-square” law relating force and distance – inverse because of the inverse proportionality, and square because the R term is raised to the second power.

With an understanding of the meaning of the gravity equation, you’re ready to use this equation to solve astronomy problems. As described in Section 1.2, there are two ways to use an equation like this to solve problems. The absolute method can be used to find the value of the force of gravity (in newtons) by plugging numerical values into the gravity equation. The ratio method is useful if you wish to compare the force of gravity between objects under two different sets of circumstances. You can see an example of the use of that approach a bit later in this section.

2.1.2 Calculating the force of gravity

Using the absolute method, you enter the values of all variables (in this case, the masses of m1and m2in kilograms and the distance in meters) and constants (here, only G) in appropriate units. If you’re given the values of any variables in different units, you’ll need to convert to the required units. Then perform the necessary mathematical operations to arrive at an “absolute” answer – that is, an answer that represents a value with appropriate units rather than a relative

2.1 Newton’s Law of Gravity 45

answer. This is the approach to use if you’re trying to find the force of gravity between two objects of known mass at a known distance. Here’s an example:

Example: Calculate the force of gravity between the Sun and the planet Uranus.

A good way to begin any problem is to write down exactly what you’re given, what you’re trying to find, and what relationship connects what you’re given to what you’re trying to find.

In this case, you’re given the names of two objects (the Sun and Uranus), and you’re asked to find the force of gravity between them. You know that Newton’s Law of Gravity can be used to find the force of gravity between any two objects, as long as you know the mass of each object and the distance between them. And although the problem statement doesn’t give you the mass of either the Sun or Uranus or the distance between them, you can find that information in most comprehensive astronomy texts or on-line.

Using those resources, you should be able to find that the mass of the Sun is about 2× 10³⁰ kg, the mass of the planet Uranus is about 8.7 × 10²⁵ kg, and Uranus’s distance from the Sun varies from about 2.74×10⁹to 3.01×10⁹km.

Since the problem doesn’t specify the point in Uranus’s orbit at which you should find the force of gravity, you’re free to use either of those values or something in between. If you take the middle of that range (2.87 × 10⁹km) as the distance, you have all the quantities needed to find the force of gravity.

But before you can start plugging values into Newton’s Law of Gravity, it’s essential that you remember to convert the distance into the required units of meters:

Now you can plug in the values for the masses and distance, like this:

Fg= Gm1m2

Exercise 2.1. Calculate the force of gravity between two people, each with