Slide 1Newton’s Law of Universal GravitationMore on Law of Universal GravitationExample for GravitationFree Fall Acceleration & Gravitational ForceExample for Gravitational ForceKepler’s Laws & EllipseThe Law of Gravity and Motions of PlanetsKepler’s Third LawExample of Kepler’s Third LawKepler’s Second Law and Angular Momentum ConservationMotion in Accelerated FramesExample of Motion in Accelerated FramesMonday, Oct. 4, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu11. Newton’s Law of Universal Gravitation2. Kepler’s Laws3. Motion in Accelerated FramesPHYS 1443 – Section 003Lecture #11Monday, Oct. 4, 2004Dr. Jaehoon YuMonday, Oct. 4, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu2Newton’s Law of Universal GravitationPeople have been very curious about stars in the sky, making observations for a long time. But the data people collected have not been explained until Newton has discovered the 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.How would you write this law mathematically?gF1110673.6GG is the universal gravitational constant, and its value isThis constant is not given by the theory but must be measured by experiments.With GgF =Unit?22/ kgmN This form of forces is known as the inverse-square law, because the magnitude of the force is inversely proportional to the square of the distances between the objects.�1m1 2212m mr2m 212rGMonday, Oct. 4, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu3It means that the force exerted on the particle 2 by particle 1 is an attractive force, pulling #2 toward #1.More on Law of Universal GravitationConsider two particles exerting gravitational forces to each other.Gravitational force is a field force: Forces act on object without physical contact between the objects at all times, independent of medium between them.1222112ˆrrmmGF The gravitational force exerted by a finite size, spherically symmetric mass distribution on an object outside of it is the same as when the entire mass of the distributions is concentrated at the center of the object.m1m2rF21F1212ˆrTwo objects exert gravitational force on each other following Newton’s 3rd law.Taking as the unit vector, we can write the force m2 experiences as12ˆrWhat do you think the negative sign mean?gFWhat do you think the gravitational force on the surface of the earth look?2EERmMGMonday, Oct. 4, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu4Example for GravitationUsing the fact that g=9.80m/s2 at the Earth’s surface, find the average density of the Earth.gSince the gravitational acceleration is GgRMEE2Therefore the density of the Earth is 2EERMG2111067.6EERMEEVM324EERGgREGRg4333611/1050.51037.61067.6480.93mkgSolving for MEMonday, Oct. 4, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu5Free Fall Acceleration & Gravitational ForceWeight of an object with mass m is mg. Using the force exerting on a particle of mass m on the surface of the Earth, one can obtain•The gravitational acceleration is independent of the mass of the object•The gravitational acceleration decreases as the altitude increases•If the distance from the surface of the Earth gets infinitely large, the weight of the object approaches 0.What would the gravitational acceleration be if the object is at an altitude h above the surface of the Earth?mgWhat do these tell us about the gravitational acceleration?gF2EERmMGg2EERMG'mg2rmMGE 2hRmMGEE'g 2hRMGEEDistance from the center of the Earth to the object at the altitude h.Monday, Oct. 4, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu6Example for Gravitational ForceThe international space station is designed to operate at an altitude of 350km. When completed, it will have a weight (measured on the surface of the Earth) of 4.22x106N. What is its weight when in its orbit?The total weight of the station on the surface of the Earth isTherefore the weight in the orbit isGEFOFSince the orbit is at 350km above the surface of the Earth, the gravitational force at that height isMEEOFmg2EERmMGN61022.4 'mg 2hRmMGEE GEEEFhRR22 GEEEFhRR22 N66256261080.31022.41050.31037.61037.6Monday, Oct. 4, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu7Kepler’s Laws & EllipseKepler lived in Germany and discovered the law’s governing planets’ movement some 70 years before Newton, by analyzing data.Newton’s laws explain the cause of the above laws. Kepler’s third law is a direct consequence of law of gravitation being inverse square law.F1F2bcaEllipses have two different axis, major (long) and minor (short) axis, and two focal points, F1 & F2 a is the length of a semi-major axisb is the length of a semi-minor axis1. All planets move in elliptical orbits with the Sun at one focal point.2. The radius vector drawn from the Sun to a planet sweeps out equal area in equal time intervals. (Angular momentum conservation)3. The square of the orbital period of any planet is proportional to the cube of the semi-major axis of the elliptical orbit.Monday, Oct. 4, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu8The Law of Gravity and Motions of Planets•Newton assumed that the law of gravitation applies the same whether it is on the Moon or the apple on the surface of the Earth.•The interacting bodies are assumed to be point like particles.Therefore the centripetal acceleration of the Moon, aM, isNewton predicted that the ratio of the Moon’s acceleration aM to the apple’s acceleration g would be gaMREMoonApplegaMv234/1070.280.91075.2 smaMNewton also calculated the Moon’s orbital acceleration aM from the knowledge of its distance from the Earth and its orbital period, T=27.32 days=2.36x106sMaThis means that the Moon’s distance is about 60 times that of the Earth’s radius, and its acceleration is reduced by the square of the ratio. This proves that the inverse square law is valid. 22/1/1EMRr2MErR42861075.21084.31037.6Mrv2 MMrTr2/224TrM2 23268/1072.21036.21084.34sm 26080.9Monday, Oct. 4, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon
View Full Document
Unlocking...