Physics For Beginners page 08
Main Physics Table of Contents
Table of Contents for Page 08
- Gravity (continued)
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6. Gravity (continued)
- Gravity In Detail
Well, we've just spent an entire section talking about the effects of gravity near the earth's surface. We found out that, near the surface of the earth, gravity causes all objects to accelerate downward at a constant rate of g = 9.81 (m/sec)/sec. In addition, we also saw that, near the surface of the earth, your weight is just the force of gravity pulling you downward. Finally, there was the startling, tabloid-esque revelation that the acceleration due to gravity wasn't really constant. It just changed little enough near the surface of the earth that we could assume the acceleration was constant. We will discuss this last point in more detail later.
There was something, however, that was quite unsatisfying about the previous discussion. Though we were able to calculate the force of gravity and predict its effects on objects, we never really got to the source of it. What is the source of gravity and what types of objects does it affect? In addition, how does it affect objects in general, not just near the surface of the earth?
Simply put, gravity is the attraction between any two objects that have mass. For instance, the earth and the moon are attracted toward each other because they both have mass. Likewise, the sun and the earth are attracted toward each other.
Now, here's a thought question involving gravity. Let's say there are two objects: A and B. They are attracted toward each other by the force of gravity between them. Furthermore, let's say object A pulls object B towards it with a force of 10 Newtons. What is the force with which object B pulls object A? Refer to the picture below.
(Hint: If you get stuck, think about Newton's Laws of Motion. I won't tell you which one, so you'll have to read through all three if you are unsure of the answer.)So, it turns out that object B pulls object A towards it with the same amount of force (10 Newtons), just in the opposite direction. Hmmmm.... I wonder which of Newton's Laws of Motion would have told us the answer to this question.
This is a very important point to understand. The force of gravity is a mutual attraction that two objects of mass have toward one another. In addition, the force of gravity between the two objects is the same except in opposite directions because of Newton's 3rd Law of Motion. In the figure above, Object A and Object B have the same amount of gravitational attraction toward one another. Object A is attracted to Object B with the same amount of force as Object B is attracted to Object A, just in opposite directions. As an aside, I will sometimes refer to the force of gravity as gravitational attraction. I will use these terms interchangeably.
Before I introduce the general formula for the force of gravity between two objects with mass, I want to give some qualitative observations about gravity in the hopes that, when I introduce the formula, it won't seem like I am just pulling it out of thin air. I want to make the formula seem at least plausible. (As an aside, what I'm about to do is officially called "hand waving".)
- Gravity's Dependence on Mass
As we noted above, gravity is the mutual attraction between two objects that have mass, so the formula should obviously depend on the masses of the two objects under consideration. For instance, in the figure above, the force of gravity between Object A and Object B should depend on the mass of Object A and the mass of Object B. But how does gravity depend on the masses?
To answer this question, I will propose another question. (I know it is impolite to answer a question with another question, but then again, no one has ever accused me of being polite.)
- Thought Question: If gravity is the mutual attraction between any two objects that have mass, then how come two coins don't seem to be attracted toward one another if they are placed next to each other on a table?
No matter how long you wait, those coins won't come together. For those of you too busy to stare aimlessly for hours at two coins on a table, I've gone ahead and done that for you, and I will attest that they, indeed, do not come any closer after several hours of careful mind-numbing observation.
Does this mean we are wrong about gravity's dependence on mass? Well, I'm going to be stubborn again and say that gravity does indeed depend on mass. So, if we are correct about gravity's dependence on mass, what could account for why the two coins don't move toward one another on the table?
To help answer this question, let's take a step back and think about just one coin and the earth. Is the coin attracted to the earth? Obviously, the answer is "yes" because we can throw the coin up in the air and see that it falls back down to the earth, so there is an attraction between the coin and the earth. Yet, why don't we see this behavior with the two coins sitting on a table? How come the two coins don't seem to be attracted to each other while a coin is obviously attracted to the earth? What could account for this apparent difference in behavior?
Well, the answer lies in the fact that the earth has a very large mass. The gravitational attraction between the earth and the coin is apparent because the mass of the earth is so large. On the other hand, the gravitational attraction between the two coins is not very apparent because they have relatively small masses (when compared to the earth). A gravitational attraction does indeed exist between the two coins, but it is so tiny that it is not apparent or perceptible. In fact, the gravity between the two coins is insufficient to overcome the friction between the coins and the table.
In conclusion, the force of gravity between two objects of mass depends on the masses of both objects and increases with the masses of those objects. Well, this was probably obvious to you, so let's see what else gravity depends on.
- Thought Question: If gravity is the mutual attraction between any two objects that have mass, then how come two coins don't seem to be attracted toward one another if they are placed next to each other on a table?
- Gravity's Dependence on Distance
We haven't touched on this point, but nevertheless it is true. The force of gravity between two objects depends on the distance between the two objects. In fact, as the distance between the two objects gets farther apart, the force of gravity between them decreases rapidly.
- General Gravity Formula Revealed
Now that we've seen what the force of gravity depends on, let me introduce the formula. Please refer to the picture below when looking at the formula.
First of all, let me point out that this is the formula for the magnitude of the force of gravity between two objects. If you will recall, whenever we talk about a force, we have to give its value (in Newtons) and a direction in which the force is acting. The value of the force is called its magnitude. In other words, the magnitude of a force is just the amount of force (in Newtons) without its direction. This is precisely what the above formula gives us.
As for the direction of the force of gravity, please refer to the picture above. You should notice that the force lies in a direction along the line connecting the centers of objects A and B. In addition, the gravitational attraction of object A on object B is towards object A. This makes sense because we know gravity is an attractive force. In this sense, object A is pulling object B towards it. In a similar fashion, object B pulls object A towards it.
In addition, as we discussed above, we know that the magnitude of the force with which object A pulls object B towards it is the same as the magnitude of the force with which object B pulls object A towards it. The only difference is that they are in opposite directions.
As for the distance, if you look carefully at the picture above, you will notice that the distance is measured from the center of one object to the center of the other object. In the figure above, the distance is measured from the center of object A to the center of object B. This is an important point to keep in mind. The distance, r, is not measured from the surface of object A to the surface of object B.
Finally, let me say something briefly about, G, the gravitational constant. We will have more to say about it later, but for now just think of it as a number. As the name implies, the value of G won't change because it is a constant. The main thing to keep in mind is that it is different from g, the acceleration caused by gravity near the surface of the earth.
Next, let's take a look at this formula in a little more detail. In fact, we will come to see that this formula accurately reflects what we already qualitatively know about gravity.
- Dependence on Mass
First, let's take a look at its dependence on mass. Looking at the formula above, we see that the mass of object A and the mass of object B are multiplied together on the right hand side of the formula. In other words, the magnitude of the force of gravity depends on the product of the mass of object A and the mass of object B. By product, I mean the force of gravity doesn't just depend on the mass of object A or the mass of object B, but it depends on the two multiplied together. Well, so far so good. This is what we suspected from our observation above that the force of gravity should depend on the masses of both objects, not just on one.In addition, this is what we expected from the discussion following the thought question above. While the force of gravity between two objects (like the two coins above) with relatively small masses is very small and nearly unnoticeable, the force of gravity between an object with a small mass (like a coin) and an object with a very large mass (like the earth) can be noticeable and apparent. You can think of the two coins as multiplying two small positive numbers together. The result is another small positive number. (In other words, the force of gravity between two objects of small mass is small. The reason we are using positive numbers is because mass is positive.) However, when you multiply a small positive number by a very large positive number, the result is something noticeable. (In other words, the force of gravity between an object of small mass and an object of very large mass is not small and can be noticeable. Once again, we are using positive numbers because mass is positive.) This is what I mean when I say the force of gravity depends on the product of the masses of the two objects. It depends on the mass of one object multiplied by the mass of the other object. If this point is a bit confusing, take some time with it and look at the formula above. Just imagine the force of gravity between two objects both with very large masses.
Often, when we are looking at a formula's dependence on certain things (like mass) it is helpful to look at just the terms involving that quantity. To do this, let's isolate the term in the formula above that involves mass. Looking at the formula above, we see that its dependence on mass is just the mass of object A multiplied by the mass of object B. I have written that term below by itself.
By doing this, we are seeing how mass affects the force of gravity without looking at any other factors. In particular, we are ignoring gravity's dependence on distance for now. This is not to say that gravity doesn't depend on distance, but rather we are just looking at the aspect of gravity that is affected by mass for now. We will do the same thing for gravity's dependence on distance in the next section.
As we know, gravity depends on mass. Specifically, it depends on the product of the mass of object A multiplied by the mass of object B. This is exactly what the term above tells us.
Next, let's try some numbers for the masses to see how gravity depends on mass. Because we know masses are positive, the numbers we use will be positive as well. Normally, we would include units, but for now, we will ignore them out of convenience. We can ignore units for now because we are not actually going to calculate anything real and because we just want to see gravity's rough dependence on mass.
- Example 1:
Let's say the mass of object A is 0.1 and the mass of object B is 0.2. What is the product of these two masses?Well, 0.1 multiplied by 0.2 gives us a result of 0.02. When two small positive numbers are multiplied together, the result is something small. This is like the force of gravity between two objects both with small masses. It reflects the fact that the force of gravity depends on both masses and is small when both objects have small masses. This is like the case of the two coins in the thought question above. Because both coins have a very small mass, the force of gravity between them is very small and fairly unnoticeable.
- Example 2:
Assume the mass of object A is 0.1 and the mass of object B is 1000000. What is the product of these two masses?0.1 multiplied by 1000000 gives us a result of 100000. When a small positive number is multiplied by a very large positive number, the result is still fairly large. This is like the force of gravity between an object with a small mass and an object with a very large mass. While the force of gravity between two small masses is small, the force of gravity between a small mass and a very large mass is noticeable. This is like the case of a coin and the earth in the thought question above. Because the mass of the earth is so large, the force of gravity between the earth and the coin is still noticeable even though the mass of the coin is small. This example really shows us that the force of gravity doesn't depend on one mass alone, rather it depends on the product of the two masses multiplied together. While the mass of object A is small, the product of the mass of object A multiplied by the mass of object B is still fairly large because the mass of object B is very large.
In other words, we may say gravity's dependence on mass is 
- Example 1:
- Dependence on Distance
Next, let's take a look at gravity's dependence on the distance between the two objects that are attracted to each other from the standpoint of the formula above. As we briefly discussed above, the force of gravity should depend on the distance between the two objects that are attracted to each other. In addition, this force should drop as we increase the distance between the two objects. Let's see if the formula above reflects this observation.Looking at the formula above, we see that the distance, r, is on the bottom of the formula on the right hand side. In fact, it appears twice on the bottom because it is squared. What this means is that whatever is on top (on the right hand side of the formula) is divided by the distance, r, twice. In other words, gravity's dependence on the distance between two objects is basically like dividing by the distance twice. So, if we know what it is like to divide things, then we kind of have an idea of how gravity depends on distance. Well, I'm sure everyone knows what it's like to divide things, but I'll go ahead and point out some obvious facts. If you divide by a larger positive number, the result is something smaller. For example, consider 1/2 which equals 0.5 and 1/10 which is 0.10. You should notice that the result of dividing 1 by 10 (which is larger than 2) is smaller than dividing 1 by 2. In a sense, this is like the dependence of gravity on distance. Since the distance, r, is on the bottom of the formula, you are in essence dividing by it. In fact, you are dividing by it twice. As an aside, you should notice we are using positive numbers for the distance, r, because distances are positive.
Therefore, the larger the distance, r, between the two objects, the smaller the force of gravity between the two objects. This is precisely what we expected. The closer two things are to one another, the larger the force of gravity between them. And, the farther two things are from one another, the smaller the force of gravity between them. I hope this last discussion was clear. If you are having difficulty with this last point, take a calculator and try dividing a lot of numbers. Pay particular attention to what happens when you divide by larger and larger positive numbers.
Because this last point is fairly important, let's take a look at it in a little more detail. In fact, let's take a look at just the dependence on the distance, r, between the two objects that are attracted to each other. So, if we just isolate the formula's dependence on the distance between the two objects, it should look like the term below.
In other words, we may say that gravity's dependence on distance is 
If you will notice, I have explicitly written out what r2 means in the bottom of the fraction. For those unfamiliar with r2 , it is just a fancy way of writing "r x r". In particular, this is an example of what we call "squaring". In words, the term above is called "1 over r squared" because we are dividing by "r squared".
Okay, now that we've isolated gravity's dependence on distance, let's try some numbers just to see how it depends on distance. Remember, by isolating the term above, we are just looking at gravity's dependence on distance and nothing else. In particular, we are ignoring gravity's dependence on mass for now. In addition, we will be using positive numbers for r because distances are positive.
Specifically, let's take a look at two distances. For the first one, let's make r = 10. Once again, I have left off the units for now. Since we are looking at just the rough dependence of gravity on distance, we can do this, but, in general, units are very important and must not be left out. Ok, let's go ahead and put r = 10 into the term above and see what we get.
Second, let's make r = 100 and then compare the results.
Let's take a look at what we just did and see if we can learn anything interesting from it. By going from r= 10 to r = 100, we have increased the distance because that is what r is. So, what happened to the term when we increased the distance from r= 10 to r = 100? Well, looking at the calculations above, it decreased from 1/100 to 1/10000. In fact, it decreased dramatically.
If you will recall, the term above is gravity's dependence on distance. Therefore, as we increase r (the distance between the two objects that are attracted to one another), the force of gravity decreases rapidly.
In particular, we may say that gravity's dependence on distance is This tells us specifically how gravity depends on distance and also tells us that the force of gravity decreases rapidly as the distance between the two objects increases.
Once again, we see the beauty of mathematics. While it took me several paragraphs to describe gravity's dependence on mass and distance, the same information is stored in the formula above. All we had to do was interpret what the formula meant. While this might seem a bit hard or confusing at first, you will get better at it with practice and experience.
Since we said a lot about the force of gravity, let's summarize all the important points in one place.
- Summary of Important Points about the Gravity Formula
- The force of gravity is an attractive force between two objects that have mass.
- The magnitude of the force of gravity is the same for the two objects that are attracted to one another. (In accordance with Newton's 3rd Law of Motion.)
- The gravitational attraction of object A pulling on object B is in a direction opposite to the gravitational attraction of object B pulling on object A. (Once again, in accordance with Newton's 3rd Law of Motion.) In addition, the direction is along the line connecting the centers of the objects. Please refer to the figure above.
- The force of gravity between two objects depends on both masses of the two objects that are attracted to one another. Specifically, it depends on the product of the mass of object A multiplied by the mass of object B. In addition, the larger the product of these two masses, the larger the force of gravity between the two objects.
- The force of gravity depends on the distance between the two objects that are attracted to one another. Specifically, the greater the distance between the two objects, the smaller the force of gravity between them. In addition, the distance between the two objects is measured from the center of one to the center of the other.
- The force of gravity is a relatively weak force. Its effects only become readily apparent when objects (like the earth) of very large mass are involved.
- Dependence on Mass
- Some Interesting Observations about Gravity
Now that we've discussed the force of gravity and what it depends on in great, yawn-inducing detail, let's see if there is anything interesting about gravity.
- Thought Question: If your weight at the surface of the earth is 800 N, what is the equal and opposite reaction to this force? Obviously, you know the force must be 800 N and in the opposite direction. However, what is the source of the reaction force? (Hint: Think of what you learned about gravity on this page and the previous page and Newton's 3rd Law of Motion.)
This question is a little tough because it requires you to combine several of the concepts you have learned so far. In addition, it requires you to put together several details to get the answer. As always, you should try to figure this question out on your own first. If you run into problems, make sure you really understand everything we said about gravity and Newton's 3rd Law of Motion before tackling the problem. If you don't get the answer, try again using a different approach to your thinking. Trying and failing is a part of learning, and even if you don't get the answer, the effort you put into solving a hard problem is not wasted. With that in mind, give the problem a go before reading further.
Well, to answer this question, we must first understand what weight is. Weight is just the force of gravity pulling you down at the earth's surface. Specifically, weight is the force of gravity of the earth pulling you down. Therefore, in our example, the earth is pulling you downward (at the earth's surface) with a gravitational force of 800 N.
Next, according to Newton's 3rd Law of Motion and what we learned about gravity on this page, there must be an equal and opposite reaction force. Before we continue, recall that the force of gravity is the attraction that occurs between two objects with mass. In this case, what are the two objects that are attracted to one another? If you can answer this question, then you should be able to solve this problem with a little more thought.
In this example, you and the earth are the two objects that are attracted to one another by the force of gravity. Because both you and the earth have mass, you are both attracted to each other by gravity. Specifically, the earth is pulling you downward with a force of 800 N because of this gravitational attraction.
Next, since we know there must be an equal and opposite reaction force, you must also be exerting a force on the earth because of this gravitational attraction. In fact, since it must be equal and opposite, you are pulling the earth upward with a gravitational force of 800 N. Once again, this is because of the nature of gravity and Newton's 3rd Law of Motion. The force of gravity between two objects that have mass are equal and in opposite directions. If the earth is pulling you downward with a gravitational force of 800 N, then you must be pulling the earth upward with a gravitational force of 800 N. This is precisely the answer to the question. The equal and opposite reaction force to your weight of 800 N (at the surface of the earth) is just you pulling upward on the earth with a gravitational force of 800 N.
If this was a bit confusing, take some time with it. There is a lot to keep track of. Remember, the force of gravity exists between any two objects that have a mass. You are no exception to the rule, unless you are massless, though I don't advocate dieting to such an extreme extent.
If you were able to answer the above thought question, give yourself a well deserved pat on the back. The idea of an equal and opposite force is often very confusing to someone new to physics. However, there is another reason that might have made the above thought question hard. This reason has to do with your common sense and intuition about how the world works. By the time you read this page, you will already have spent several years on this planet. You will already have formed ideas and conceptions about how the world works. Some of these conceptions will be correct, however, some of them will be incorrect. The above thought question just happens to be one of those cases which might seem counter-intuitive to what you've already formulated about how the world works. However, there must be some basis for this apparent confusion because something about the answer to the above thought question seems wrong. Uh oh, I feel another thought question coming.
- Thought Question: Once again assume your weight is 800N. If the earth is pulling you downward with a gravitational force of 800 N and you are pulling upward on the earth with a equal and opposite gravitational force of 800 N, then how come you fall down to the earth when you jump up in the air while the earth does not move up toward you?
While it is easy to believe the earth is pulling you downward with a gravitational force of 800 N, it is hard to believe that you are also pulling upward on the earth with the same amount of force. This is what seems counter-intuitive in the answer to the thought question above. We can believe the earth is pulling you downward because we see evidence of that everyday. Whenever you jump in the air, you are immediately pulled downward by the gravitational force of the earth pulling you down. However, we don't see evidence of you pulling upward on the earth. Because whenever you jump into the air, you do not see the earth moving upward. You only see yourself falling back down toward the earth. Could the answer in the previous thought question be wrong? It certainly seems possible because every intuition seems to tell us that it is wrong.
Well, once again, I will be stubborn and say that, if the earth is pulling you down with a gravitational force of 800 N, you are also pulling upward on the earth with the same amount of force. So, what could account for this seemingly counter-intuitive fact? Try to solve this thought question on your own first before reading further. (Hint: Think about Newton's 2nd Law of Motion. In addition, think about whether the mass of the earth is large or small.)
The solution to this thought question does indeed lie in Newton's 2nd Law of Motion. Therefore, before we continue, let's briefly review it. Newton's 2nd Law of Motion tells us that acceleration is the response an object experiences when an external force is applied to it. In addition, it tells us that the acceleration an object experiences depends both on the amount of force applied and on the mass of the object which is experiencing the force. Specifically, if the same amount of force is applied to two objects of different mass, the object with more mass will experience a smaller acceleration (or response) to the force applied to it.
This is precisely what we need to solve this thought question. Both the earth and you feel the same amount of force (800 N). Therefore, whichever one has a smaller mass will experience a larger acceleration and whichever one has a larger mass will experience a smaller acceleration.
Well, compared to the earth, your mass is very small. In other words, the earth has much more mass than you. So, while 800 N of gravitational force (from the earth pulling you down) is enough to cause you to accelerate downward at g = 9.81 (m/sec)/sec, the 800 N of gravitational force (from you pulling the earth upward) is not enough to cause the earth to accelerate upward.
The 800 N of gravitational force from you pulling upward on the earth is so small compared to the mass of the earth that the earth's response to it is unnoticeable. This accounts for why the earth does not move upward when you jump into the air, while you are pulled down toward the earth.
Therefore, while the answer to the previous thought question might have seemed counter-intuitive, it is nevertheless correct. Newton's 3rd Law of Motion is still valid, and the laws of physics are still successful at explaining what we see and don't see in the real world. After all, that is the goal of physics. Applying what we know about gravity and Newton's Laws of Motion, we successfully explained why we fall toward the earth when we jump into the air, while the earth does not move upward when we jump into the air. Finally, we saw how the study of physics builds upon itself. We haven't talked too much about Newton's Laws of Motion since the first page, but here we are using them to help us solve a problem.
Next, we will discuss the main reason for studying gravity. However, before we do that, let me take a slight detour. You will see the reason for this later.
- A Slight Detour and a Trip to the Park
Imagine you are at a park on a nice sunny day. You reach into your pocket and find a string attached to a ball on one end. All of a sudden, you feel an uncontrollable urge to take that string and swing the ball around your head in a circle. I'm sure everyone instinctively knows how to make the ball swing in a circle. However, I want you to really think about the direction in which you have to pull on the string to make the ball move in a circle. In addition, draw a top-down picture of the situation. When you are done, compare it to the picture below.In the figure above, you are the yellow circle in the middle. In order to keep the ball moving in a circle, you need to pull the string toward the center of the yellow circle. If you are having trouble believing this, try this experiment on your own and pay close attention to the direction in which you have to pull the string in order to keep the ball moving in a circle. In addition, you should notice that the harder you pull on the string, the faster the ball swings around the circle. It is important that you convince yourself this is true before moving on. There are far worse things to do than spending an entire afternoon swinging a ball around.
Ok, so why this weird detour? Not enough sunshine and too much typing? Both answers are correct, but there was a more legitimate reason for this detour, and it is directly related to the main reason for studying gravity. Can you think of the reason?
Well, instead of thinking about swinging a ball attached to a string, let's pretend the yellow circle in the picture above is the sun and the blue circle in the picture above is our planet, earth. Hmmm.......
As we all know, our planet, earth, orbits around the sun. Therefore, like the ball in the picture above, there must be some force pulling the earth toward the center of the sun as it orbits around the sun. What could this force be?
The answer should be fairly obvious. It is the gravitational force of the sun pulling on the earth which is keeping the earth in orbit. As the earth moves in orbit around the sun, the direction of the force changes, but it is always pointing from the center of the earth to the center of the sun. While the ball on a string required us to directly pull on the string to keep the ball moving in a circle, there is no string required to hold the earth in orbit around the sun. However, in both cases, there is a force pulling on the orbiting object which keeps it in orbit. This is what both cases have in common.
Therefore, we know that it is the gravitational attraction between the earth and the sun which keeps the earth in orbit around the sun. Without this gravitational attraction, the earth would not remain in orbit, much like what happens when you cut the string as you swing the ball in a circle over your head.
If you've been paying attention, you'll notice I haven't said anything about the gravitational force of the earth pulling on the sun. If you caught this, you might be wondering about the effect of the gravitational force of the earth pulling on the sun. Well, first of all, we know that it is equal and opposite to the gravitational force of the sun pulling on the earth. If this is true, then why is it that the earth orbits around the sun instead of the other way around?
The answer to this lies in the previous thought question. The mass of the sun is very large compared to the mass of the earth. Even though both the sun and the earth experience the same amount of force (but in opposite directions), their responses are different because they have very different masses. Because the sun's mass is very large compared to the earth's mass, the acceleration it experiences is much smaller than the acceleration the earth experiences. It is because of this that the earth orbits around the sun instead of the sun orbiting around the earth. For our purposes here, we may assume the acceleration the sun experiences is so small that it is nearly motionless as the earth orbits around it.
- Thought Question: If your weight at the surface of the earth is 800 N, what is the equal and opposite reaction to this force? Obviously, you know the force must be 800 N and in the opposite direction. However, what is the source of the reaction force? (Hint: Think of what you learned about gravity on this page and the previous page and Newton's 3rd Law of Motion.)
Well, we've covered a lot of ground in this section. The interesting thing to note is that, while we looked at the general formula for the force of gravity, we really didn't do any calculations. We were able to figure out some general properties of gravity by looking at the formula without doing explicit calculations. Finally, we were able to see some interesting observations about gravity. In particular, we saw that gravity is the force responsible for keeping the planets in orbit around the sun. If you have a clear understanding of gravity at this point, then you are doing well. It is far more important to understand the concepts instead of blindly using a formula. If you were able to sit through this in one sitting, then you really deserve a break right about now.
Next time, we will spend more time talking about the mysterious gravitational constant, G, and we will do some explicit calculations with the gravity formula, unless another accidental detour pops up.
- Gravity's Dependence on Mass
