11 Newton s Law of Universal Gravitation

Size: px

Start display at page:

Download "11 Newton s Law of Universal Gravitation"

Alaina Williamson
5 years ago
Views:

1 Physics 1A, Fall 2003 E. Abers 11 Newton s Law of Universal Gravitation 11.1 The Inverse Square Law The Moon and Kepler s Third Law Things fall down, not in some other direction, because that s where the earth is. The earth is in some sense the cause of gravitational acceleration. Perhaps the power of the earth that causes objects to accelerate downward weakens as you go out. The moon does, after all, orbit the earth, approximately in a circle. Let s compute its acceleration, and compare to g. 1 Actually, I did this computation earlier, but it is worth repeating here. As stated, the radius of the moon s orbit is about meters. Its period is a month. Note: The relevant period is the sidereal period, its period relative to a fixed direction in space, about 27.3 days, or T =2, 360, 000 seconds. One month is the synodic period, the period between two new moons. The acceleration of the moon is a = 4π2 r T 2 = 4π ( ) 2 = meters per sec per sec The acceleration of the moon is much smaller than g. The ratio is g 9.78 = 3626 a whereas the ratio of the distances, in the two problems, is r Moon r Earth = Very nearly, the acceleration due to the earth s gravity seems to decrease inversely in proportion to the square of the distance from the earth s center. Is this the general rule? If one could observe the accelerations of several objects, at a variety of distances from the center of the earth, it could be checked, but all we have is things near the surface and the moon. But for the sun, there is more data. Each of the planets moves in an approximately circular orbit, so we can compute their accelerations and compare, using again the formula a = 4π2 R T 2 1 Read Chapter 6, sections 6-1 through 6-5. The remaining sections in Chapter 6 are not required, but you should find them interesting.

2 Physics 1A, Fall 2003 Universal Gravitation 2 where now r is the radius of a planet s orbit, and T its period of revolution around the sun. We could make a table of R and T for each planet, and compute a, but Kepler s third law saves us the trouble. R and T are not independent, but rather T 2 = kr 3 for all the planets, for some number k which is a constant. Therefore a = 4π2 R kr 3 = 4π2 kr 2 The accelerations are indeed inversely proportional to the squares of the distances. Gravity is not a constant, as we assumed earlier, studying motion near the earth s surface; it diminishes inversely as the square of the distance from the center A Difficult Mathematical Problem This much was known in the early 1680 s to those few who followed natural philosophy. Robert Hooke, whose discoveries you will learn more about later, posed the following question: Suppose the acceleration of the planets towards the sun obeys a 1/ law, as Kepler s Third Law for circular orbits requires. But the orbits are not exactly circles. What is the path in general of a planet obeying the inverse square law? Hooke bet that he could solve the problem in a few months; The architect Sir Christopher Wren and the astronomer Edmund Halley, took him on. Hooke managed a numerical or graphical solution, but was never able to solve the problem in closed form. It is clear today what Hooke s difficulty was. The mathematics of his day was insufficient to deal with continuously changing rates of change. For that, one needs the methods of the calculus. Realizing that the problem was more difficult that had been thought, Halley traveled to Cambridge to discuss it with the reclusive Isaac Newton. Newton was already known for his explanation of the nature of color, for the invention of the reflecting telescope, and as a learned mathematician. Newton answered that he had solved, then mislaid, the solution to the central scientific problem of his age! Halley urged Newton to reconstruct his proof and publish it; the result was the Principia Mathematica Philosophiae Naturalis, in which the correct general methods for solving all problems concerning the motion of objects was presented. The solution of the problem of the planets essentially proving Kepler s first law is one of the more difficult results in the Principia Universal Gravitation Form of the Law of Gravity Newton s first law tells us how things move when there are no forces: forces are needed to explain accelerations. The second law tells us quantitatively how much acceleration is produced by forces. The third law states a property all forces must have. Finally, Newton provider one example of a force the force of gravity. The sun exerts a force on the earth proportional to the inverse square of the distance. But Kepler s third law says that it is the accelerations of the planets, not the forces on them, that obey the inverse square law.

3 Physics 1A, Fall 2003 Universal Gravitation 3 Therefore the force of the sun on the earth or any planet must also be proportional to the planets mass, so that the mass can cancel out in the acceleration: Force on earth = F = constant m earth 1 The earth also exerts a force on the sun, which similarly is Force on sun = F = constant m sun 1 These must be the same numbers, by the third law! It follows that the force of gravity of the sun on the earth, or the earth on the sun, is proportional to the mass of the sun and the mass of the earth. The astonishing discovery is that the force of the sun s gravitational force on the earth is proportional not only to the mass of the earth, but also to the mass of the sun. The mass of the sun or the earth, introduced in the second law as the inertia, the resistance to acceleration, is also the strength of the gravitational force. Gravitation is universal everything attracts everything. The moon attracts the earth, Jupiter the sun, and you the earth. The rule for the magnitude of the force between two bodies of masses m 1 and m 2 is F = G m 1m 2 The acceleration of m 1 is F/m 1 = Gm 1 /. It is independent on m 1, as it is supposed to be. Same for m 2. The constant gravitational acceleration we experience here near the earth s surface is but a special case. It was of course one of the original observed facts which suggested the universal law Gravitational Mass and Inertial Mass There is no logical reason why the same property of matter, the mass, appears in both the law of gravitation and Newton s second law. The strength of other forces do not depend on an object s mass. For example you probably have heard of Coulomb s law, that tells us the force between two charged objects. The force is F = constant Q 1Q 2 where Q 1 and Q 2 are the objects electric charges. You will study this electrical force next term. Suppose we called the m s that occur in the law of gravitation an object s gravitational mass, and for the moment used for it the letter N instead of m. That is the gravitational force between two objects is F = GN 1N 2 and call the number m that occurs is the second law an objects inertial mass. Then if an object of gravitational mass N and inertial mass m is acted on by the gravitation attraction of the earth, the force on it is F = G N earthn

4 Physics 1A, Fall 2003 Universal Gravitation 4 so its acceleration is a = F m = N m GN earth Now it might have been that the ratio N/m is different for different bodies. But it seems that nature did not make that choice. We know ever since Galileo dropped a heavy ball and a light ball off the tower of Pisa that the acceleration of two objects under the force of gravity is independent of the objects mass. Later experiments have checked this identity to very high accuracy. The upshot is that the ratio N/m is the same for everything. N/m is a universal constant. Since it multiplies G, we can never measure it independently, so it s value is just absorbed into the definition of Newton s constant G, and we say that gravitational mass and inertial mass is the same for all objects. This is a profound and powerful principle, called the principle of equivalence, and is at the foundation of the modern theory of gravity, called the general theory of relativity Vector Form of the Law The rule above gives the magnitude of the force of gravitational attraction. But force is a vector. The vector way of writing it is: If is the position of the object of mass m 2 in a coordinate system where the mass m 1 is at the origin, then the force m 1 exerts on m 2 is F = G m 1m 2 r r 3 Check that it has the right magnitude and direction. In a general coordinate system, where r 1 and are the positions of the masses, the force of 1 on 2 is F 12 = G m 1m 2 ( r 1 ) r Kepler s Laws So Kepler s third law is a consequence of the inverse square law of universal gravitation. But what exactly is the r that goes into it? I have worked it out in the approximation that the orbits are circles, but you already know they are ellipses. We were just lucky that the planets orbits have small eccentricity. For a general elliptical orbit, the answer is that T 2 is proportional to a 3, where a is the major, or long axis of the ellipse. We will not prove that here, nor derive Kepler s first law from the force law, but it is not too hard and could easily be done at the end of the term if there were a bit more time. You will learn it if you take the next level course in mechanics. The second law is easier, it is a consequence of the conservation of angular momentum. We ll study that subject in a few weeks. If enough of you are interested I will post some notes on Kepler s laws after you study energy and angular momentum.

5 Physics 1A, Fall 2003 Universal Gravitation Galileo s Constant The force of the earth s gravitational attraction on an object of mass m near the earth s surface is F = GmM earth earth so its acceleration is g = GM earth earth This provides the relation between g and G Effect of the Size of the Attracting Mass But now you might worry that it is not correct to say that the earth s gravity here on the surface is the same as if all the earth s mass were located at the center. Newton worried about this problem too. It delayed his writing the Principia for fifteen years, since Newton, essentially, had to invent the integral calculus to solve the problem. Eventually he proved the following: For any spherically symmetric mass distribution, the result of adding up the gravitational forces due to each little piece of the sphere on an object outside is that the force is given by GMm/, where M is the total mass and r the distance to the center. It isn t very hard to prove, but I won t do it in this course. The upshot is that the magnitude of the earth s gravitational force on an object near its surface is whence one can indeed identify F = g = GMm R Earth 2 GM R Earth 2 Objects in free fall near the earth s surface do indeed obey Kepler s laws with respect to the center of the earth, at least until the strike the surface and other forces act on them. How is that possible? If an ellipse is very eccentric, then near the end far away from the focus, the ellipse looks very nearly like a parabola. These are the parabolas of projectile motion, which replace Kepler s ellipses as long as the earth s curvature can be ignored The Value of Newton s Constant G is a universal constant. How can it be measured? Obviously not by comparing Galileo s g to the acceleration of the moon, or the planets to each other using Kepler s third law. Suppose we compare the attraction of the earth on something, to the attraction of the sun on things The Mass of the Sun Let a be the acceleration of the earth as it goes around the sun, and g the acceleration of objects near the surface of the earth. Let r e be the radius of the earth, and R e the radius

6 Physics 1A, Fall 2003 Universal Gravitation 6 of its orbit. Then a = GM S 2 R e and g = GM e r 2 e whence one can compute the ratio of the masses (exercise!) but not G. So now you can know the mass of the sun, relative to the mass of the earth. But this measurement does not give either mass independent of G Weighing the Earth By now it should be clear that to measure G you need to know the mass of both objects. So to determine G from g you need you know the mass of the earth. Newton could guess the earth s mass roughly, since he know its volume and the density of dirt and water, but no one really knew what the earth is made of inside. Since everything attracts everything, there should be a force between two terrestrial objects. It was finally measured by Henry Cavendish in 1798, over a century after Newton proposed the inverse square law. is value is G = newtons per mete per kg 2 Cavendish He used a torsion pendulum, with two known masses. There is a good description ion the book. He called the experiment weighing the earth Other Simple Effects Perturbations All the examples treat one object under the attraction of a much larger object. Only in that case is there the simplicity of Kepler s laws. If we lived in a solar system with a double star, like many are, the planets motions would be so complicated that nobody would ever have figured out the inverse square law. There are small effects, since everything attracts everything. Learn for instance about the Cavendish torsion balance experiment he called weighing the earth that let to the measurement of G. Even in our solar system, there are small corrections to Kepler s laws. The attraction on the earth by the other planets, principally Jupiter and Venus, cause the earth s orbit, instead of being exactly a closed curve, to precess. The perihelion, or aphelion, point in a slightly different direction every year. Every century the axis of the orbit precesses nine or ten minutes or arc because of the gravitation of the other planets. These effects are calculated so precisely that they once led to the discovery of a new planet. In the early nineteenth century astronomers were unable to account precisely for the motions of the known planets. Uranus especially seemed not to obey Newton s law exactly. It was even suggested that perhaps the inverse square law is not exactly correct so far from the sun. But Leverrier in France and others explained the discrepancy as due to another, as yet unknown planet, and very elaborate computations let to the successful prediction, in 1846, of the planet Neptune.

7 Physics 1A, Fall 2003 Universal Gravitation The Rotation of the Earth The weight of an object varies slightly as you move around the earth, because of the earth s rotation. Suppose you Suspend a mass m from a spring scale at the north pole. A spring scale is any device that measures the force W t. When it comes to rest, mg W = 0, so the scale reads W = mg. Now do the same experiment at the equator. The vertical acceleration is not zero, because of the earth s rotation. Supposing g does not change, the scale reads W, where The weight is now mg W = m v2 r earth ) W = m (g v2 r earth What is the acceleration of a falling object at the equator? Well, if the force on it is the weight W above, its acceleration, by Newton s second law, is Geosynchronous Orbits g = W m = g v2 r earth A television satellite, or a spy satellite, has to stay in the same place in the sky, as viewed from the earth. How is this possible? We have worked out the period of a low orbit satellite (about 87 minutes) and the moon (about 27 days). What is needed for a geosynchronous satellite is that has a period of exactly one day (and it ought to be suspended over the equator why?). If the radius of its orbit is r, then, if T is the period, a = v2 r r =4π2 T = GM earth 2 or r 3 = GM earth 4π 2 T 2 = gr2 earth 4π 2 T 2 Plug in the numbers: You get r 43, 300 km This is the height (measured from the center of the earth) of a geosynchronous satellite This is the right answer, but the derivation is a little glib. After all, why is it allowed to use a coordinate system that is rotating. Careful analysis, which can get a bit complicated, shows that this is the right answer.

Physics 12. Unit 5 Circular Motion and Gravitation Part 2

Physics 12. Unit 5 Circular Motion and Gravitation Part 2 Physics 12 Unit 5 Circular Motion and Gravitation Part 2 1. Newton s law of gravitation We have seen in Physics 11 that the force acting on an object due to gravity is given by a well known formula: F