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The Law of Universal Gravitation Physics Montwood High School R. Casao Newton’s Universal Law of Gravity Legend has it that Newton was struck on the head by a falling apple while napping under a tree. This prompted Newton to imagine that all bodies in the universe are attracted to each other in the same way that the apple was attracted to the Earth. Newton analyzed astronomical data on the motion of the Moon around the Earth and stated that the law of force governing the motion of the planets has the same mathematical form as the force law that attracts the falling apple to the Earth. Newton’s Universal Law of Gravity Newton’s law of gravitation: every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. If the particles have masses m1 and m2 and are separated by a distance r, the magnitude of the gravitational force is: Fg G m1 m 2 r 2 Newton’s Universal Law of Gravity G is the universal gravitational constant, which has been measured experimentally as 6.672 x 10-11 . N m2 kg2 The distance r between m1 and m2 is measured from the center of m1 to the center of m2. Newton’s Universal Law of Gravity By Newton’s third law, the magnitude of the force exerted by m1 on m2 is equal to the force exerted by m2 on m1, but opposite in direction. These gravitational forces form an actionreaction pair. Properties of the Gravitational Force: The gravitational force acts as an action-ata-distance force, which also exists between two particles, regardless of the medium that separates them. The force varies as the inverse square of the distance between the particles and therefore decreases rapidly with increasing distance between the particles. The gravitational force is proportional to the mass of each particle. Properties of the Gravitational Force: The force on a particle of mass m at the Earth’s surface has the magnitude: G ME m Fg 2 RE ME is the Earth’s mass (5.98 x 1024 kg); RE is the radius of the Earth (6.37 x 106 m). The net force is directed toward the center of the Earth; both masses accelerate, but the Earth’s acceleration is not noticeable due to its extremely large mass. The smaller mass accelerates towards the Earth. Weight and Gravitational Force Weight was previously defined as FW = m·g, where g is the magnitude of the acceleration due to gravity. With the new perspective related to the attractive forces existing between any two objects in the universe, G ME m mg R E2 The mass m cancels out, giving us (also called surface gravity): G ME g 2 RE Bodies Above the Surface of a Mass Consider a body of mass m at a distance h above the Earth’s surface, or a distance r from the Earth’s center, where r = Re + h. The magnitude of the gravitational force acting on the mass is given by: G ME m Fg 2 RE h Gravity/Radius Ratio If the body is in free fall, then the acceleration of gravity at the altitude h is given by: g G ME R E h 2 Thus, it follows that g decreases with increasing altitude. Gravity/Radius Ratio The value of g at any given location can be determined using the following proportional relationship: 2 g1 r2 2 g 2 r1 This proportional relationship can also be applied to the weight of an object: Fw1 Fw 2 r2 2 2 r1 Kepler’s Laws Kepler formulated three kinematic laws to describe the motion of planets about the Sun: Kepler’s First Law: All planets move in elliptical orbits with the sun at one of the focal points. Kepler’s Second Law: The radius vector drawn from the sun to any planet sweeps out equal areas in equal time intervals. Kepler’s Third Law: The square of the orbital period of any planet is proportional to the cube of the semi-major axis of the elliptical orbit. Kepler’s Laws First law: Second law: Kepler’s Laws Third law equation: k r3 T2 where k is a constant 3.35 x 1018 m3/s2 r is the radius of rotation T is the period of rotation (the time necessary to complete one revolution) Kepler’s laws apply to any body that orbits the Sun, manmade spaceship as well as planets, comets, and other natural objects. The mass of the orbiting body does not enter into the calculation. Kepler’s Laws The ratio of the squares of the periods (T) of any two planets revolving about the Sun is equal to the ratio of the cubes of their average distances r from the Sun: 2 T1 2 T2 3 r1 3 r2 Period of a Satellite The period of a satellite or planet orbiting about a central body is given by: 2 3 4 r 2 T G Mbody Mbody is the mass of the central body being orbited. The Gravitational Field The gravitational field concept revolves around the general idea that an object modifies the space surrounding it by establishing a gravitational field which extends outward in all directions, falling to zero at infinity. Any other mass located within this field experiences a force because of its location. So, it is the strength of the gravitational field at that location that produces the force not the distant object. The situation is symmetrical - each object experiences a gravitational force because of the field set up by any other object. The Gravitational Field The gravitational field is a vector quantity equal to the gravitational force acting on a particle divided by the mass of the particle: g Fg on m m The gravitational field equation can be used to determine the value of g at any location by: GM g r 2 Escape Velocity Suppose you want to launch a rocket vertically upward and give it just enough kinetic energy (energy of motion) to escape the Earth’s gravitational pull. The minimum initial velocity of an object at the Earth’s surface that would allow the object to escape the Earth, never to return, is the escape velocity. Escape velocity from Earth: v escape 2 G ME rE Satellite Orbits A satellite is held in a circular orbit because the force of gravity supplies the necessary centripetal force to keep the object moving in a circular path about the central body. In order for a satellite to orbit around a central body, such as the Earth, there must be a net force on the object directed toward the center of the circular orbit, a centripetal force. For a satellite in orbit around Earth, the centripetal force is equal to the gravitational force exerted by the Earth on the satellite. Satellite Orbits A satellite does not fall because it is moving, being given a tangential velocity by the rocket that launched it. It does not travel off in a straight line because Earth’s gravity pulls it toward the Earth. The tangential speed of an object in a circular orbit is given by: v G ME r If the period of the orbit is known, the velocity may be determined using: 2 r v T Satellite Orbits The period of a satellite can be determined by: 2 r T v Satellite Orbits The Goldilocks principle can be used to explain the relationship between the speed of a satellite and its orbit. The velocity of the satellite is critical, and the velocity described by the equation: v r g min describes the minimum velocity necessary for the satellite Vmin to maintain its proper circular orbit (JUST RIGHT). If the satellite velocity is TOO HOT (greater than the vmin), it will not maintain the proper circular orbit and fly into space. If the satellite velocity is TOO COLD (less than vmin), it will be pulled into the Earth’s atmosphere by the Earth’s gravitational force, where it will either burn up in the atmosphere or slam into the Earth’s surface. Gravitational Potential Energy Revisited Gravitational potential energy near the surface of the Earth is given by the equation Ug = m·g·h, where h is the height of the object above or below a reference level. This equation is only valid for an object near the Earth’s surface. For objects high above the Earth’s surface, the equation for potential energy is: G ME m U r Gravitational Potential Energy The negative sign comes from the work done against the gravity force in bringing a mass in from infinity where the potential energy is assigned the value zero, towards the Earth. This work is stored in the mass as potential energy. As r gets larger, the potential energy gets smaller; the gravitational force approaches zero as r approaches infinity. Gravitational Potential Energy Only changes in gravitational potential energy are important. For an object that moves from point B to point A, the expression for the change in potential energy is: G ME m G ME m U U A UB rA rB Web Sites Kepler's Laws (with animations) Kepler's Three Laws Kepler's Three Laws of Planetary Motion Kepler's Laws of Planetary Motion