chapter thirteen notes: universal gravitation. in the early 1600's, german mathematician and...
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Chapter Thirteen Notes:Universal Gravitation
In the early 1600's, German mathematician and astronomer Johannes Kepler mathematically analyzed known astronomical data in order to develop three laws to describe the motion of planets about the sun. Kepler's three laws emerged from the analysis of data carefully collected over a span of several years by his Danish predecessor and teacher, Tycho Brahe. Kepler's three laws of planetary motion can be briefly described as follows:The path of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)
While Kepler's laws provided a suitable framework for describing the motion and paths of planets about the sun, there was no accepted explanation for why such paths existed. The cause for how the planets moved as they did was never stated. Kepler could only suggest that there was some sort of interaction between the sun and the planets which provided the driving force for the planet's motion. To Kepler, the planets were somehow "magnetically" driven by the sun to orbit in their elliptical trajectories. There was however no interaction between the planets themselves. Newton was troubled by the lack of explanation for the planet's orbits. To Newton, there must be some cause for such elliptical motion. Even more troubling was the circular motion of the moon about the earth. Newton knew that there must be some sort of force which governed the heavens; for the motion of the moon in a circular path and of the planets in an elliptical path required that there be an inward component of force. Circular and elliptical motion were
clearly departures from the inertial paths (straight-line) of objects. And as such, these celestial motions required a cause in the form of an unbalanced force. As learned in Lesson 1, circular motion (as well as elliptical motion) requires a centripetal force. The nature of such a force - its cause and its origin - bothered Newton for some time and was the fuel for much mental pondering. And according to legend, a breakthrough came at age 24 in an apple orchard in England. Newton never wrote of such an event, yet it is often claimed that the notion of gravity as the cause of all heavenly motion was instigated when he was struck in the head by an apple while lying under a tree in an orchard in England. Whether it is a myth or a reality, the fact is certain that it was Newton's ability to relate the cause for heavenly motion (the orbit of the moon about the earth) to the cause for Earthly motion (the falling of an apple to the Earth) which led him to his notion of universal gravitation.
Newton realized that the moons circular path around the earth could be caused in this way by the same gravitational force that would hold such a cannonball in low orbit, in other words, the same force that causes bodies to fall. To think about this idea, let us consider the moons motion, beginning at some particular instant, as deviating downwardsfallingfrom some initial horizontal line, just as for the cannonball shot horizontally from a high mountain. The first obvious question is: does the moon fall five meters below the horizontal line, that is, towards the earth, in the first second? This was not difficult for Newton to check, because the path of the moon was precisely known by this time. The moons orbit is approximately a circle of radius about 384,000 kilometers (240,000 miles), which it goes around in a month (to be precise, in 27.3 days), so the
distance covered in one second is, conveniently, very close to one kilometer. It is then a matter of geometry to figure out how far the curved path falls below a horizontal line in one second of flight, and the answer turns out to be not five meters, but only a little over one millimeter! (Actually around 1.37 millimeters.)Its completely impossible to draw a diagram showing how far it falls in one second, but the geometry is the same if we look how far it falls in one day, so here it is:
triangle ABC is really thin, but we can still use Pythagoras theorem!Thus the natural acceleration of the moon towards the earth, measured by how far it falls below straight line motion in one second, is less than that of an apple here on earth by the ratio of five meters to 1.37 millimeters, which works out to be about 3,600. What can be the significance of this much smaller rate of fall? Newtons answer was that the natural acceleration of the moon was much smaller than that of the cannonball because they were both caused by a forcea gravitational attraction towards the earth, and that the gravitational force became weaker on going away from the earth.
In fact, the figures we have given about the moons orbit enable us to compute how fast the gravitational attraction dies away with distance. The distance from the center of the earth to the earths surface is about 6,350 kilometers (4,000 miles), so the moon is about 60 times further from the center of the earth than we and the cannonball are. From our discussion of how fast the moon falls below a straight line in one second in its orbit, we found that the gravitational acceleration for the moon is down by a factor of 3,600 from the cannonballs (or the apples). Putting these two facts together, and noting that 3,600 = 60 x 60, led Newton to his famous inverse square law: the force of gravitational attraction between two bodies decreases with increasing distance between them as the inverse of the square of that distance, so if the distance is doubled, the force is down by a factor of four.
A survey of Newton's writings reveals an illustration similar to the one shown at the right. The illustration was accompanied by an extensive discussion of the motion of the moon as a projectile. Newton's reasoning proceeded as follows. Suppose a cannonball is fired horizontally from a very high mountainIn a region devoid of air resistance. In the Absence of gravity, the cannonball would travel in a straight-line, tangential path. Yet in the presence of gravity, the cannonball would drop below this straight-line path and eventually fall to Earth (as in path A). Now suppose that the cannonball is fired horizontally again, yet with a greater speed. In this case, the cannonball would still fall below its straight-line tangential path and eventually drop to earth. Only this time, the cannonball would travel further before striking the ground (as in path B).
Now suppose that there is a speed at which the cannonball could be fired such that the trajectory of the falling cannonball matched the curvature of the earth. If such a speed could be obtained, then the cannonball would fall around the earth instead of into it. The cannonball would fall towards the Earth without ever colliding into it and subsequently become a satellite orbiting in circular motion (as in path C). And then at even greater launch speeds, a cannonball would once more orbit the earth, but in an elliptical path (as in path D). The motion of the cannonball orbiting to the earth under the influence of gravity is analogous to the motion of the moon orbiting the Earth. And if the orbiting moon can be compared to the falling cannonball, it can even be compared to a falling apple. The same force which causes objects on Earth to fall to the earth also causes objects in the heavens to move along their circular and elliptical paths. Quite amazingly, the laws of mechanics which govern the motions of objects on Earth also govern the movement of objects in the heavens.
Newtons theory of gravity confirmed the Copernican theory of the solar system. No longer was earth considered to be the center of the universe. The earth and other planets orbit the sun, the same way the moon orbits earth. The planets continually fall around the sun in closed paths.
Newton discovered that gravity is universal. Everything pulls on everything else in the universe in a way that involves only mass and distance.Isaac Newton compared the acceleration of the moon to the acceleration of objects on earth. Believing that gravitational forces were responsible for each, Newton was able to draw an important conclusion about the dependence of gravity upon distance. This comparison led him to conclude that the
force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center. But distance is not the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equationFnet = m aNewton knew that the force which caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third
law), that force must also depend upon the mass of the earth. So for Newton, the force of gravity acting between the earth and any other object is directly proportional to the mass of the earth, directly proportional to the mass of the object, and inversely proportional to the square of the distance which separates the centers of the earth and the object.But Newton's law of universal gravitation extends gravity beyond earth. Newton's law of universal gravitation is about the universality of gravity. Newton's place in the Gravity Hall of Fame is not due to his discovery of gravity, but rather due to his
discovery that gra