1 Background Information The Second Law of Motion and The Law of Gravitation Student Activities 1. Round and Round They Go! 2. onic Sections - Movement in Newton s Gravitational orce Notes to Teachers
2 Teacher Information Teachers: This material explains Newton s Law of Gravitation in a way that will help you teach the law to your students. The photocopy-ready Student Activity pages will give students the opportunity to learn aspects of the Law of Gravitation in a way that they will find interesting and fun. Notes about each activity appear in the Notes to Teachers section. Each activity can be tailored for the level of your students, and can be completed individually or in groups. In addition, students will create a logbook, called Newton s Lawbook, in which they can take notes and track their findings from the scientific experiments offered in the Student Activity pages. Newton s Law of Gravitation Two objects with masses m 1 and m 2 that have a distance r between their centers attract each other with a force equal to Gm 1 m2, where G is the Gravitational onstant equal to x Nm 2 /kg 2 r 2 The Second Law of Motion and The Law of Gravitation Sir Isaac Newton ( ) established the scientific laws that govern 99% or more of our everyday experiences. He also explained our relationship to the Universe through his Laws of Motion and his universal theory of gravitation - which are considered by many to be the most important laws of all physical science. Newton was the first to see that such apparently diverse phenomena as a satellite moving near the Earth s surface and the planets orbiting the Sun operate by the same principle: orce equals mass multiplied by acceleration, or =ma. Our everyday lives are influenced by different forces. As you know, the Earth exerts a force on us that we call gravity. We feel the force required to lift an object from the floor to a table. We can see and feel a magnet s pull on a pile of metal paper clips. But exactly how does Newton s Second Law of Motion relate to gravity? To understand Newton s theories, you must first know about the nature of force and acceleration when applied to circular motion, rather than motion in a straight line. Newton s irst Law of Motion tells us that, without the interaction of some sort of force, everything travels in a straight line forever. This means that an object traveling in a circular path must be influenced by a net (outside) force. The circulating object has a velocity that is constantly changing, not because its speed is changing but because its direction is changing. A change in velocity is called an acceleration. Newton s Second Law explains it this way: A net force changes the velocity of an object by changing either its speed or its direction. Therefore, an object moving in a circle is undergoing an acceleration. The direction of the acceleration is toward the center of the circle. The magnitude of the acceleration is equal to, v r 2 where v is the
3 constant speed along the circular path and r is the radius of the circular path. This acceleration is called centripetal (literally, center-seeking ) acceleration. The force needed to produce the centripetal acceleration is called the centripetal force, which is equal to ma according to Newton s Second Law (=ma). But since a is v 2, the centripetal force is equal to mv 2. r r Majestic examples of circular motion can be found throughout our Universe: Planets orbit around the Sun in nearly circular paths; moons orbit around their planets in nearly circular paths; and satellites orbit the Earth in nearly circular paths. Newton was smart enough to realize that his Second Law (=ma) must apply to the force of gravity. He theorized that the Sun must be providing a centripetal acceleration to the Earth (and all the other planets in our solar system) in order for it to maintain its roughly circular orbit. Newton figured out that the force of gravity is the source for the acceleration. He calculated that, where m p is the mass of the planet and R is the distance from the planet to the Sun. = Newton learned from his predecessor, Johannes Kepler ( ), that the square of the orbital period for a planet is proportional to the cube of its orbital radius, or T 2 ~ R 3 (known as Kepler s Third Law). Since T is the time taken to complete one orbit, by definition: T = 2πR v So v ~ R T Putting these equations together, we see that v 2 ~ 1. Thus, the force that the Sun exerts on a planet must take the form of: R m p v = 2 ~ R R 2 Newton s Third Law of Motion says: or every force, there is an equal and opposite force. rom this, Newton calculated that a planet must exert an equal (but oppositely directed) force on the Sun that the Sun exerts on the planet. Due to this symmetry, he concluded that the two forces must depend on the masses of both objects in the same way. This means they take the form of: ~ In his final equation, Newton added the Universal Gravitational onstant, or G, which accounts for all of the constants of proportionality. His final equation reads: = m p v 2 R The gravitational constant cannot be derived or predicted by theory. It must be determined by experimental measurement. The value of G was first measured by Henry avendish in The currently accepted value of G is x Nm 2 /kg 2. Newton s model of gravity is one of the most important scientific models in history. It applies to everything in the Universe, from apples falling from trees to baseballs soaring into the outfield; from m p m p M s R 2 GmM R 2
4 the Earth orbiting the Sun to a moon orbiting a planet. It applies to the motion of all of the objects in our solar system, as well as to the distant stars and galaxies. Interesting Side Note If we put the above equations together, we learn something very interesting! We know that: = ma = In this case, a is the centripetal acceleration of an object in circular motion. rom our discussion, we know that, a = v 2 R ombining these equations, we see that: GmM R 2 v = sqrt ( GM ) R In other words, the mass of an object does not influence the orbit of the object. At a given distance, a baseball or a moon will orbit in exactly the same way at exactly the same speed!
5 Student Activities Students: These activities will help you to learn all about Newton s Law of Gravitation. Use the notebook, which you have designated as your Newton s Lawbook, to take notes, track your progress, and evaluate findings from the experiments you will conduct. Start by writing down Newton s Law of Gravitation: Newton s Law of Gravitation Two objects with masses m 1 and m 2 that have a distance r between their centers attract each other with a force equal to Gm 1 m2, where G is the Gravitational onstant equal to x Nm 2 /kg 2 r 2 Activity #1: Round and Round They Go! As you learn about the orbits of the planets around our Sun, it is important to learn more than just the order of the planets based on their distances from the Sun. Think about the amount of time it takes to complete one orbit. Why do the outer planets take so much longer to complete an orbit than the inner planets? Is it just because they have so much farther to go? Or is some other factor involved? This lesson investigates these questions. Perhaps you ve seen a scale model of the solar system with pieces of string representing the orbital paths of each of the nine planets. Stretching the strings out from a common point will show you that the outer planets indeed have much farther to go to complete one revolution. However, this by itself does not explain the orbital periods. There is something more to it. The following activity will show you what else is involved. Materials You will need the following items for this experiment: one sturdy plastic drinking straw (cut in half) or one piece of 1/2 plastic tubing about 5 long string, long two rubber washers or stoppers Procedure In the following exercise you will build a model of the solar system. 1. Tie one end of the string to one of the washers. Next, run the string through the straw and tie it to the other washer (as shown here). 2. Before you begin, make sure you have lots of room around you.
6 3. Hold the straw and washer assembly upright and pull the string upward until the straw rests against the bottom washer. 4. Hold the straw with one hand and fully extend your arm out in front of you. 5. Rapidly rotate your wrist, spinning the string washer assembly until the string is fully extended. ontinue spinning it at a steady rate. 6. Use your other hand to slowly pull down on the bottom washer. As the orbit of the washer gets shorter, what do you notice about the speed at which the spinning washer is traveling? Use your Newton s Lawbook to make notes about your discovery. Think About It With your class, discuss the following topics. Write your conclusions and answers to the questions in your Newton s Lawbook. 1. What does this exercise have to do with planets orbiting the Sun? 2. Using the Solar System Data hart below, calculate the orbital speed of Mercury to see how long it would take Pluto to make its orbit if it traveled at Mercury s speed. Then compare the time to how long it actually takes Pluto to complete an orbit. Write down your conclusion. Make the same calculations and comparisons for several other planets as well. Remember, you can calculate the average speed of a planet s orbit by dividing the length of the orbital path by the time it takes to make one complete orbit. The orbital path (for this exercise, assume all orbits are circular) is simply 2πr, where r is the distance from the planet to the Sun. 3. With your class, make a graph that compares the distance of each planet to the Sun with the planet s average speed in its orbit. Look at the graph. As you can see, the planets farthest from the Sun move more slowly than the others. This is very similar to the behavior of a pendulum. The longer the pendulum s arm, the longer it takes to complete one swing. Investigate this concept by swinging your string apparatus like as a pendulum. You will see that its the length of the swing arm - and not the weight or mass of the washer - that affects the period of time it takes your pendulum to complete one swing. Isaac Newton was the first person to realize that there must be a connection between the period of a pendulum s swing and the period of a planet s orbit. He realized that the pendulum bob was attracted to the Earth, which made it swing like it did. He also realized and that the Moon was similarly attracted to the Earth, with the result that it orbited like it did.
7 Go back to your straw and string apparatus. Hold onto one end of the string (not the straw) and swing the washer over your head. Do you feel it pull on your hand? The washer would fly off if your hand wasn t pulling back on the string with a force equal to your swing. Newton s Model of Gravity Experimenting just as you did, Sir Isaac Newton was led to consider the relationship between the Earth and Moon, which he knew weren t connected by a string. He imagined an invisible string called gravity that provides a pull just like a real string. He came to realize that the amount of gravitational pull was dependent on the amount of matter or mass in each object (in this case, the masses of the Earth and Moon). rom your experiment, you can imagine that swinging a larger washer from a string would require more pull. Newton s law says that the amount of pull between two objects depends on the distance between the objects - just as you discovered in your experiment. Swing your string/washer apparatus one more time. This time, decrease the distance between your hand and the washer. You will find that doing so makes it much easier to sustain the orbit of the washer. Newton worked out the mathematics for this and found that the pull (which we now call the gravitational pull) can be expressed as: g = Gm 1 m 2 r 2 In other words, the pull exerted on one body by another equals the Gravitational onstant (G in the equation above) times the mass of the first body times the mass of the second body divided by the square of the distance between the centers of the two bodies. Notice that in this equation no distinction is made between which body is pulling and which is being pulled. Think about what this means: The Earth s pull on your body is exactly equal to your body s pull on the Earth! Newton s model of gravity is one of the most important scientific models in history. It applies to apples falling from trees, baseballs soaring into the outfield, and the milk being spilt in your school cafeteria. The exact same model applies to the motion of all of the objects in our solar system, as well as to the distant stars and galaxies. Solar System Data hart: Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Mean Distance rom Sun (km) 57,910, ,200, ,600, ,940, ,300,000 1,429,400,000 2,870,990,000 4,504,300,000 5,913,520,000 Sidereal Orbital Period (days) , , , , ,800.00
8 Activity #2: onic Sections - Movement in Newton s Gravitational orce When a right circular cone is intersected by a plane, a conic section is formed. Any plane that s perpendicular to the axis of the cone cuts a section that is a circle. Incline the plane a bit and the section forms an ellipse. Tilt the plane still more until it is parallel to an outside edge of the cone and a parabola is formed. ontinue tilting until the plane is parallel to the axis of the cone and the section is a hyperbola. So what do conic sections have to do with Newton s Laws? onic sections play a fundamental role in space science. Any object under the influence of an inverse square law force (such as gravity) must have a trajectory (curved path) that is one of the conic sections. or example, when talking about orbits, a celestial body (such as a planet, comet, or artificial satellite) moves according to its gravitational attraction to a primary celestial body. The primary body s center of mass is one focus of the conic section along which the satellite moves. The conic section you follow is determined by your energy (or velocity). Starting with a lowvelocity, you follow an ellipse (oval). As your velocity (speed) increases, you eventually follow a circle. As your velocity increases some more, your path becomes an ellipse again. As your velocity continues to increase, your path becomes a parabola, and finally, with an even higher velocity, a hyperbola. The specific values of the velocities that transition you between the conic sections are determined by the gravitational pull you are experiencing 1 your kinetic energy mv 2 Note that all closed orbits are either circles or ellipses. Part 1 You can model conic sections by simple paper folding. Try to predict which conic sections you will make from each of the following set-ups. Materials 2 You will need the following items for this exercise: g = Gm 1 m 2 r 2 and by wax paper or tracing paper a pencil Procedure 1. ut out or trace a circle on wax paper or tracing paper. Make the circle at least 7 cm in radius. 2. Locate and mark the center of your paper circle. In the illustration below, the center of the circle lies at Point. 3. hoose and mark a point within the interior of your circle (Point in the example). The location of this point is not important. or the most dramatic results, however, pick a point that is not too close to the circle s center. crease
9 4. As shown above, fold the circle so that any point on the circle s circumference (edge) intersects Point. Make a sharp crease so that you have evidence of your fold. When you open your fold, you will have a straight line on the circle where you made the crease. 5. Make several more creases so that different points on the circle s circumference are folded through Point. 6. You should start seeing a curve appear on your paper circle. Think About It A. The creases on your circle form the outline of what appears to be a/an. Where is/are the focus/foci? B. If you moved Point closer to the edge of the circle and folded another curve, describe how you think the curve s shape would change.. If you moved Point closer to the center of the circle and folded another curve, describe how you think the curve s shape would change. D. How does this paper folding construction work? Part 2 In this exercise, you will create another conic section. The paper-folding activity will show you the mathematical characteristics found with the focus and directrix. Materials You will need the following items for this exercise: wax paper or tracing paper a pencil Procedure 1. ut out a rectangle from the wax paper or tracing paper. You need at least a 7 cm by 7 cm area. 2. Mark a point near the bottom edge of the rectangle, roughly in the center (Point in the illustration).
10 3. old the paper so that a point on the bottom edge of the paper lands on Point. Make a sharp crease so that you have evidence of your fold. When you open your rectangle, you will have a straight line where you made the crease. 4. ontinue to make creases from the bottom edge of the paper through Point. After you ve made several creases, predict what kind of curve these creases are forming. bottom edge crease Think About It A. Describe any symmetries you see in your curve. B. How do you think the appearance of your curve would have been different if Point had been closer (or farther away from) the bottom edge?. How does this paper-folding construction work? Use mathematical vocabulary such as directrix and focus in your explanation. Part 3 In this exercise, you will fold yet another piece of paper in order to discover a third type of conic section. Materials You will need the following items for this exercise: a blank sheet of paper a pencil a compass
11 Procedure 1. Use a compass to draw a circle on your piece of hole paper. Make the circle as dark as you can. The location and size of your circle does not matter, but try to keep the radius no larger than 3 cm. Mark the center of your circle as Point. 2. Mark a Point outside the circle s circumference anywhere on the paper. crease 3. Use the sharp end of your compass to punch a small hole through Point. The hole should be large enough so that you can see through it. 4. old the paper so that the hole at Point lands somewhere on the circle s edge. You should be able to see the circle s edge through the hole. Hold the paper up to the light to see it even better. Make a sharp crease so that you have evidence of your fold. When you open the paper, there will be a straight line where you made the crease. 5. old more creases, making certain that Point falls somewhere on the circumference of the circle. After you ve made several creases, predict what kind of curve these creases are forming. Think About It A. What similarities and differences are there between this construction and the previous two? B. The curve that you made looks like a(n). Where is/are the focus/foci?. Describe how you think the shape of the curve would change if you varied the location of Point.
12 Notes to Teachers Activity #1: Round and Round They Go! Have students spin the washer at a quick but steady rate before starting to pull the string. Tell them to keep spinning as they pull the washer, until the point that the spinning becomes self-sustaining. Students will wonder why the washer orbits faster as its orbit radius gets smaller. Shorter string lengths produce a more manageable assembly for younger or smaller students. They can even spin their planet in a vertical circle if they can t maintain the horizontal circle. While the science of this experiment is not as technically correct with a vertical circle, this may be the only way younger students can enjoy the experience independently. Make each student describe what they feel in their hand that is holding the straw and also what they see as the orbital circle gets smaller. This will help them to focus on their orbiting washer. Remind students to pull the string slowly enough for the washer to make complete orbits at each radius before pulling more on the string. This is hard to do at larger orbits, but easy to do as the orbits get smaller. Be forewarned that students will inevitably hit themselves with their orbiting washers. This is why rubber washers are suggested for this exercise, something soft and not hard and heavy. Also, washers have enough mass to be effective orbiters. Discussion Point: As you know, planets farthest from the Sun have a longer orbital path AND move more slowly around that path than the planets closer to the Sun. It is important to understand that there are two reasons the outer planets have longer orbital periods than the inner planets. Activity #2: onic Sections - Movement in Newton s Gravitational orce The elliptical shape of planetary orbits was first asserted by German astronomer Johannes Kepler ( ). His declaration was based on the painstaking observations he made in conjunction with those of his predecessor, Danish astronomer Tycho Brahe. However, it took Isaac Newton s great genius years later (between ) to establish mathematically that the inverse square law of gravi- e=1.2 tation must produce a trajectory that is one of the conic sections. (The proof required the use e=0.25 of calculus and will not be shown here.) Newton concluded that the nature of a trajectory depends on the total energy, E, such that: 4 5 D e=0.5 If E=0, the trajectory is a parabola If E<0, the trajectory is an ellipse If E>0, the trajectory is a hyperbola e=0.75 e=1
13 Recall that E, the total energy, is defined as: 1 E= mv 2 - GMm r 2 Note that 1 mv 2 is the kinetic energy of the system and GMm is the gravitational potential r energy of the system. 2 At a meeting of the Royal Society in 1684, some of the great thinkers of the time took up the question of the relationship between the inverse square attraction of the Sun for a planet and the nature of the planet s elliptical orbit. The connection between the two had to be mathematical, which was a profoundly different way of thinking for the time. Until then, most understanding of the physical world came from human experience: We could feel it, hear it, or see it. To complicate things, the force of gravity acting on a planet is not something a human can directly experience. It is even hard for the brain to comprehend: It acts at a distance. It acts immediately. And it acts everywhere! Scientists searched for mathematical proof of why the properties of a force led to the orbit of a planet. Looking for answers, in 1684 Edmund Halley went to visit Isaac Newton in ambridge, England, and posed the problem to him. A famous account of this meeting has been provided by mathematician Abraham De Moivre. In it, De Moivre wrote that after Halley posed the problem, Newton immediately responded that the planet must move in an ellipse if it moves under the inverse square force of gravitational attraction to the Sun. Halley asked how he knew this to be so, and Newton purportedly replied, I have calculated it. Halley asked Newton to write down his proof. Later that year Newton delivered his treatise, On the Motion of Bodies in an Orbit. In the space of nine pages, Newton calculated that if the planets are moving in elliptical orbits, they must be under the control of an inverse square force directed toward one focus of the ellipses in which they move. He went on to show that under the influence of an inverse square force, the orbits of all objects (whether moving slowly or swiftly) must describe a conic section. The relationship between physical science and mathematics was changed forever. There was now no doubt that mathematical methods can provide a comprehensive description of the world around us. Answers to Part 1 Think About It or the sake of understanding the answers, it should be assumed that a conic section is made from a locus of points whose distance from a fixed point to a fixed line is in a constant ratio. The fixed point is the focus of the conic and the fixed line is the directrix. A. The creases on your circle form the outline of what appears to be a/an ellipse. Where is/are the focus/foci? The center (point ) and point. B. If you moved Point closer to the edge of the circle and folded another curve, describe how you think the curve s shape would change. The ellipse would be stretched and become less circular.
14 . If you moved Point closer to the center of the circle and folded another curve, describe how you think the curve s shape would change. The ellipse would become more circular, until both and coincide, where the ellipse would have one focus and actually be a circle. D. How does this paper folding construction work? Explain. Each crease that is created is a line from which the distance to the focus AND the directrix are equal. In this case, when an ellipse is made, the focus is point and the directrix is the circumference of the original circle. Answers Part 2 Think About It A. Describe any symmetries you see in your curve. There is a line of symmetry through the vertex of the parabola that extends vertically with the left and right branch of the parabola on either side of it. B. How do you think the appearance of your curve would have been different if Point had been closer (or farther away from) the bottom edge? The farther point is from point, the more elongated or the more eccentric the curve becomes.. How does this paper-folding construction work? Use mathematical vocabulary such as directrix and focus in your explanation. Like the ellipse made in Part 1, each crease that is created is a line from which the distance to the focus AND the directrix are equal. In this case, when a parabola is made, the focus is point and the directrix is the bottom edge of the rectangle. Answers to Part 3 Think About It A. What similarities and differences are there between this construction and the previous two? Answers may vary. or example, in all of the constructions, point is a focus or foci and all curves are created by folding the directrix onto the foci or focus. However, if the shape of the directrix changes, the shape of the conic changes as well. B. The curve that you made looks like a(n) hyperbola. Where is/are the focus/foci? The hole at point.. Describe how you think the shape of the curve would change if you varied the location of Point. As point moves farther and farther away from point, the two branches of the hyperbola become wider and the bend in the hyperbola becomes less sharp.