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Transcript
Chapter 14 THE LAW OF GRAVITY INTRODUCTION In this chapter we study the law of gravity. Emphasis is placed on describing the motion of the planets, because astronomical data provide an important test of the validity of the law of gravity. We show that the laws of planetary motion developed by Johannes Kepler follow from the law of gravity and the concept of the conservation of angular momentum. A general expression for the gravitational potential energy is derived, and the energetics of planetary and satellite motion are treated. The law of gravity is also used to determine the force between a partide and an extended body. EQUATIONS AND CONCEPTS The universal law of gravity states that any two particles attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of their separation. The constant G is called the universal gravitational constant. The acceleration due to gravity, g*, decreases with increasing altitude, h, measured from the Earth*s surface. A gravitational field g exists at some point in space if a particle of mass m experiences a gravitational force Fg = mg at that point. That is, the gravitational field represents the ratio of the gravitational force experienced by the mass divided by that mass. The gravitational field at a distance r from the center of the Earth points radially inward toward the center of the Earth. Over a small region near the Earth*s surface, g is an approximately uniform downward field. G = 6.673 x 10-11 N m2/kg2 Since the gravitational force is conservative, we can define a gravitational energy function corresponding to that force. As a mass m moves from one position to another in the presence of the Earth*s gravity, its potential energy changes by an amount given by Equation to the right, where r1 and are the initial and final distances of the mass from the center of the Earth. The gravitational potential energy associated with any pair of particles of masses m1 and m2 separated by a distance r is given by Equation. The negative sign in this expression corresponds to the attractive nature of the gravitational force. An external agent must do positive work to increase the separation of the particles. In this expression it is assumed that Ui = 0 at ri = 4, and the equation is valid for an Earth-particle system when r > RE. As a body of mass m moves in an orbit around a very massive body of mass M (where M>> m), the total energy of the system is the sum of the kinetic energy of m (taking the massive body to be at rest) and the potential energy of the system. When the two contributions are evaluated for a circular orbit, one finds that the total energy E is negative, and given by equation. This arises from the fact that the (positive) kinetic energy is equal to one half of the magnitude of the (negative) potential energy. The escape velocity is defined as the minimum velocity a body must have, when projected from the Earth whose mass is ME and radius is RE, in order to escape the Earth*s gravitational field (that is, to just reach r = 4 with zero speed). Note that Vesc does not depend on the mass of the projected body. The potential energy associated with a particle of mass m and an extended body of mass M can be evaluated using equation, where the extended body is divided into segments of mass dM, and r is the distance from dM to the particle. If a particle of mass m is outside a uniform solid sphere or spherical shell of radius R, the sphere attracts the particle as though the mass of the sphere were concentrated at its center. If a particle is located inside a uniform spherical shell, the force acting on the partide is zero. If a particle of mass m is inside a homogeneous solid sphere of mass M and radius R, the gravitational force acting on the particle acts toward the center of the sphere and is proportional to the distance r from the center to the partide. This force is due only to that portion of the mass contained within the portion of the sphere of radius r < R. SUGGESTIONS, SKILLS, AND STRATEGIES In this chapter we made use of the definite integral in evaluating the potential energy function associated with the conservative gravitational force. You should be familiar with the following type of definite integral: For example we used the above expression as follows: REVIEW CHECKLIST • State Kepler*s three laws of planetary motion and recognize that the laws are empirical in nature; that is, they are based on astronomical data. Describe the nature of Newton*s universal law of gravity, and the method of deriving Kepler*s third law (T2 % r3) from this law for circular orbits. Recognize that Kepler*s second law is a consequence of conservation of angular momentum and the central nature of the gravitational force. • Understand the concepts of the gravitational field and the gravitational potential energy, and know how to derive the expression for the potential energy for a pair of partides separated by a distance r. • Describe the total energy of a planet or Earth satellite moving in a circular orbit about a large body located at the center of motion. Note that the total energy is negative, as it must be for any dosed orbit. • Understand the meaning of escape velocity, and know how to obtain the expression for vesc using the principle of conservation of energy. • Learn the method for calculating the gravitational force between a particle and an extended object. In particular, you should be familiar with the force between a particle and a spherical body when the partide is located outside and inside the spherical body.