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Gravity Gravity is one of the four fundamental forces in the Universe which include: Tycho Brahe and Johannes Kepler For centuries, it was thought that the sun, planets, and even the stars rotated around the earth. It wasn't until 1543 that Copernicus suggested that it would be easier to explain the motion of the planets if they orbited the sun rather than the earth. (He actually was not the first, but the most famous). This of course was not readily accepted: it was even considered insulting. However, the Danish astronomer Tycho Brahe, who was born right after Copernicus died, accepted the view that the earth was the center of the Universe and took great pains to prove it, rather than use philosophical arguements. Taking actual measurements and using data to support an arguement was actually a novel idea back then. This is exactly what Tycho did. This story is remarkable. You have to know that telescopes were invented around the 1600's (Galileo did not actually invent the telescope. He only improved it). Brahe made his measurements starting in 1575! He did this without the aid of a telescope. He built massive brass protracter-like instruments he called quadrants. (Don't ask me how they worked). It was with these primitive instruments that he painstakingly recorded the data of the planet's positions. He spent 20 years of his life recording the data. You might think that these measurements were approximations and not all that accurate. It turns out that these measurements were accurate to 1/60 of a degree! They are so good, we still use them today! Before he died, he gave the data to Johannes Kepler, a young collegue of his. Kepler studied over the data and discovered three relationships that were common for all of the planets. Remarkably, Kepler had to adjust the data in order to find the relationships. Brahe's data was relative to the earth's position. Kepler had to re-orientate the data so that it would be from the perspective of looking down on top of the Solar System. That took some ingenuity. Believing in the Copernican view, Kepler used the data to prove that the planets revolved around the sun. He also set out to prove that the orbits were perfect circles. What he actually found was a bit of a surprise. Kepler's Three Laws From the data, he discovered three things about the motion of the planets around the sun. Law One: Each planet has an elliptical orbit with the sun at one focus. An ellipse is an elongated circle which has two points, each of which is called the focus.  The foci are important to the ellipse because they hold a special property. If you attached one end of a string to one focus and the other end to the other focus and placed your pencil at the middle of the string and made it tight, when you traced out the curve, the result would be an ellipse with foci at each end of the string.  So, the sun is located at one of the foci, and the planet follows the path of an ellipse.  This is a bit exaggerated because the orbits are really close to circular, but they are still ellipses nonetheless. Law Two: Each planet moves so that it sweeps out equal areas in equal time intervals. It is best to explain this one using a picture. consider the picture below.  This is not drawn to scale, but hopefully you see the connection. The time interval on the left is 10 days, and in those ten days the planet moved around the ellipse. The shaded region shows you the area, called area one, swept during that time period. On the right side, the time period is the same, ten days. Therefore, the area swept, area two, is the same as area one. This is true for any period of time and at any position on the elliptical orbit. Law Three: The ratio between the square of a planet's period of revolution and the cube of its average distance from the sun has the same value for all planets. This relationship took Kepler the longest to discover as you can imagine. In fact, it took him an additional ten years to discover this! In equation form this law looks like,  Where T is the period, r the radius, and K is the constant, which is true for all planets. Let's evaluate two planets to see if this is true. We can use the earth and Jupiter. The earth's period is 365.3 days and average orbital radius of 1.49(10^11) meters. (That number is 1.49 followed by 11 zero's). We have to use average orbital radius because remember the orbit is an ellipse so the radius is not a constant. Plugging in the numbers we have, Now let's do Jupiter. Jupiter's period is 11.86 years which is approximately 4332.5 days. Its average orbital radius is 7.78(10^11) meters. So, plugging in the numbers gives, Those numbers are very close, and we should note that the data used was not the most accurate numbers available. Sir Isaac Newton Kepler finished his work in 1619. Newton began working on the question of gravity about 40 years later. Everyone is familiar with the apple falling off of the tree story. Like so many neat stories, it is not for certain whether that is exactly what happened. However, Newton did make the connection that falling objects on the earth and the moon's motion are attributed to the same exact thing: the gravitational force from the earth. Now he just needed to discover an equation that would describe that force. This is a very condensed version of how that happened. He knew that mass had no affect on objects as they fell because of Galileo's work 20 years earlier. Galileo also determined that objects fall at a rate of 32 ft per second every second, which translates to the familiar 9.8 m/s^2, which is g. Newton had already developed his Three Laws which gave him the use of F = ma. Therefore, if you are talking about the gravitational force from the earth you would substitute g in for a since that is the accelertation due to the earth's force, and you get the familiar equation for weight: F = mg. The F is the gravitational force of the earth. The trick now is to find an equation that describes that force. If you set mg equal to a gravitational force equation, the m would have to cancel since the mass of the falling object does not matter. So, he knew that the gravitational force equation would have to probably be proportional to the product of the masses, which means you multiply the two masses in question. This would allow the object's mass to cancel and still keep the mass of the earth. Now all he needed was to find the relationship between the force and the distance separating the masses. This is where Brahe's and Kepler's work comes in. Newton had access to this data and was able to determine that the force keeping the planets in orbit around the sun was inversely proportional to the distance between them squared. Putting the two relationships together in equation form looks like this,  The inverse relationsip means the distance, denoted by r, goes on the bottom of the equation. Therefore, if r goes up, F goes down and vice versa. The square means when r goes up, F goes down in a hurry. For example, if you double your distance, F drops by one forth. If you triple your distance, F drops by one ninth. It turns out that due to Kepler's three laws, the only possibility for this relationship is this inverse square relationship. There is a very detailed and highly complicated geometric proof that shows because the force is inversely proportional to the distance between the masses squared, the orbits have to be ellipses and cannot be perfect circles. Kepler's three laws are a result of this relationship! Now Newton needed a constant to make this an equation and not a proportionality. He used the letter G for this constant. So, the equation becomes,  This is the Universal Gravitational Equation. The cool thing about this is that it is all inclusive. Every object with a mass has a gravitational pull. Of course, it takes a massive mass (sorry for the play on words) to generate any noticeable force. It is a good thing, because it would be hard to walk down the street. Henry Cavendish Now the problem becomes finding the value of G. Although Newton had a value for G, a truly accurate value was not found until over a hundred years after Newton published his theories. Henry Cavendish set up an experiment that agaisnt all odds actually found G. He used what was called a torsion balance. (See picture below).  The black circles are massive spheres that are rigidly fixed. The blue circles are smaller spheres suspended together by the yellow cord. Since all objects attract all other objects, the larger spheres will pull on the smaller spheres making them move ever so slightly. We are talking very slightly. The result of putting the larger spheres on opposite sides is the the twisting of the balance.  Since the movement is very slight, Cavendish put a mirror on the cord. He then shined a very fine point of light on the mirror which reflected it back to an observable screen. When the balance moved, he could see the fine point of light move and thus measure the distance. As you can imagine, this was a very sensitive and tedious experiment. This could not be duplicated near traffic, because the vibrations of the cars would mess it up. Incredibly, Cavendish pulled it off and found a value of G (Of course he did not have to contend with cars back then). It was so accurate, it would not be improved upon for almost 100 years! The value is,  If you were to write that number out the long way it would be, That is pretty small, but it has to be. Remember, if it were a large number, smaller objects would have a more noticeable force. With an accurate value of G, it is now possible to calculate a more precise value for the mass of the earth. That is why Cavendish called this experiment "weighing the earth." If we use the gravitational equation and set it equal to the weight of an object (because that's what weight really is, the gravitational force on an object), and use little m for the object's weight and Me for the mass of the earth and re for the radius of the earth we have,  The little m cancels, which is what has to happen remember. You are left with g, which they new from Galileo, the radius of the earth, which they also knew, and the newly measured G, all that is left is the mass of the earth to solve. Solving for Me we have,  Which equals 6.02(10^24) kg! The long way would be, I should point out that when find the gravitational force from a planet to an object on its surface, you have to use the radius of that planet for the disance and not zero even though the object is on the surface. It is as if all of the mass is concentrated at the very center of the object. So, when dealing with big planets, you have to consider the distance between their centers, not their surfaces. Let's now use the equation to determine the force the earth exerts on the moon. Here are the given information: Distance from earth to moon = 3.8(10^8) m (From the earth's center to the moon's center) Mass of moon = 7.36(10^22) kg Mass of earth = 6.02(10^24) kg The gravitation equation becomes, Which equals 2.0(10^20) Newtons which is 4.6(10^19) pounds! Gravity and Weight Let's now turn our attention to the consequences of the inverse square relationship. If the force of gravity was inversely proportional to just the distance, then if you were twice the distance from an object, the force would be half. If you were three times the distance, the force would be 1/3 as much. If this was reality, we probably would not be here because that would have major consequences in the early Universe as it was forming. Anyway, the fact is, gravity depends on the inverse square relationship. So, now if you are twice the distance away, the force is not one half as much, it is one fourth as much. That is a big difference. It gets even bigger the farther you move from the object. Consider the picture below.  As you can see, the force drastically drops off the farther you move out. There is a major difference between one fourth and a sixteenth. At the surface of the earth, g is...well g. So, a scale would read 100 pounds for a 100 pound person. When a person moves a distance equal to two earth radii, g is now one fourth the value on earth which is 9.8/4 = 2.45. Since it is one fourth the value, the scale would now read 25 pounds for a person that weighs 100 pounds on the surface of the earth. Remember, the mass of the person stays a constant. It is only the weight that changes. In fact, weight is really a relative measurement. The interesting thing here is when you apply this to free fall. Let's suppose a person is located two earth radii away from the earth's center, and he is not moving. Since g out there is 2.45 m/s^2, he would actually free fall toward the earth with an acceleration of 2.45. It is no different from falling at the surface of the earth with an acceleration of 9.8. Of course, the closer he got to the earth, the more he would accelerate because g would be increasing until it reached 9.8. You would have an acceleration of your acceleration, because it would be changing as you got closer to the earth's surface. (FYI: acceleration of acceleration is actually called a jerk). You might be wondering why astronauts seem to be "weightless" when they are in orbit. Weightlessness Remember, I said the person in our last example was not moving. Astronauts in orbit are moving. They do not, I REPEAT, DO NOT appear "weightless" because there is no gravity in space! In reality, gravity is very much alive in their orbits. In fact, the gravity that exists in their orbits is not much less then the gravity we feel on earth. Here is why. Their distance from the surface is not very much in the grand scheme of things. They are not even close to being two earth radii away. They are so close, g for them is still pretty close to 9.8! So, why do they seem weightless? Let me answer this by using an analogy of a person in an elavator.  Obviously, if the elevator is at rest the scale will read the person's weight. (Say it is 100 lb). When the elevator starts to move down with a certain acceleration, the person will "feel" lighter, and the scale will read a smaller weight. For example, if the acceleration of the elevator was 4.9 m/s^2, you would have to take 9.8 and subtract 4.9 to find the new "g" that the person is experiencing. 9.8 minus 4.9 equals 4.9. So, it is just like the person is on a new planet that has a g of 4.9. Since that is half of the earth's g, his weight will be half, so the scale would read 50 pounds. If the acceleration is 7.35 m/s^2, the new g would be 9.8 minus 7.35 which equals 2.45 m/s^2. 2.45 is one fourth of the earth's g, so the weight will be one fourth and the scale will read 25 pounds. What happens if the cable of the elevator snaps and the elevator is in free fall?  If the elevator is in free fall, what will be its acceleration? It would be the acceleration due to gravity, 9.8. Since the elevator's acceleration is 9.8, 9.8 minus 9.8 is equal to zero, which means the scale will read zero pounds! The person feels "weightless" because both he and the elevator are falling with the same acceleration. This means the elevator floor will no longer push up on the person, so he is now free to float around the room. The point is, whenever you are in free fall, you will feel weightless. When you jump off of a roof, or high dive at the swimming pool, for that brief amount of time you are experiencing weighltessness. Why? What is weight? It is the sensation we "feel" when the floor or chair is pushing up on our body. There is nothing to push up on you when falling, so you no longer feel that sensation. Now, while you are free falling, if you happen to be inside a room or elevator that is also free falling at the same rate, you can see how that does not change the fact that you are still falling. So, the reason you float is because you are actually falling.  What does that have to do with astronauts in orbit? The astronauts are in free fall! But, you might be wondering if they are in free fall, why don't they hit the earth? The answer: they are falling around the earth. They are traveling at just the right speed, according to their orbital distance from the earth, so that they will not fall toward the earth. Remember circular motion? The earth's gravity is the centripetal force. That force is pulling them toward the earth, but their tangential speed causes them to want to go straight. The result: they fall around the earth. Since both the shuttle and the astronaut are "falling" at the same rate, they float around inside the shuttle. If you have ever seen the movie Apollo 13, you saw the three actors float inside their cabin. Since they didn't actually go into orbit, how did they make it look so real? This is very cool. They actually built the cabin inside a Jumbo Jet. The airplane went up as high as it could and dove with an acceleration of 9.8 m/s^2. Just like in the elevator example, the actors were in free fall right along with the jet, so they floated. Of course, they did not have much time for each take. I saw a show on this once, and I think they had around 5 seconds for each shoot! (I am not completely sure so don't quote me on that). Calculating g's For Different Planets There is an easy way to determine the gravitational force for different planets that have different sizes and masses compared to the earth. We will do this by using the equation for the earth's g. I will use ge to represent the earth's g to avoid confusion.  If you plug in the numbers in that equation, it will equal 9.8. Let's now consider a new planet that has the same mass as the earth but twice the size. Do you think this new planet will have a stronger force or weaker force compared to the earth? Most people would think the force would be strong because it is twice the size. Here is the easy way to figure it out. We will use the g equation, and since the mass of this new planet is the same as the earth, we will us Me for the mass. Since it is twice the size, it has twice the radius of the earth, so we will use 2 times re for the distance. This means we have to square both the 2 and the re. This will result in,  Mathematically, we have to square both the 2 and the re,  Which becomes,  We can bring the 1/4 out and view the equation this way,  Notice the GMe/re^2. That is exactly the value for the earth's g. So the new planet's g is equal to,  This means that the new planet's gravitational force is one fourth the earth's force. If you weigh 100 pounds on the earth, you would weigh 25 pounds on this planet. It is interesting that a planet that is twice the size of the earth actually has a smaller gravitational force. We will get to the reason in a second. Let's consider a new planet with the same mass as the earth but half the radius. The equation becomes,  When you square the 1/2 and re, you end up with,  Whenever you have a fraction in the denominator, the bottom number of the fraction moves up to the numerator. So, after substituting ge like before the equation becomes,  This means a planet with half the size of the earth but same mass, will have four times the gravitational force. If you weigh 100 pounds on the earth, you will weigh 400 pounds on this planet! |