PHYS 1443 – Section 501Lecture #11AnnouncementsResistive Force Proportional to SpeedNewton’s Law of Universal GravitationMore on Law of Universal GravitationFree Fall Acceleration & Gravitational ForceExample for Gravitational ForceExample for GravitationKepler’s Laws & EllipseThe Law of Gravity and the Motion of PlanetsKepler’s Third LawExample of Kepler’s Third LawKepler’s Second Law and Angular Momentum ConservationPHYS 1443 – Section 501Lecture #11Monday Mar 1, 2004Dr. Andrew Brandt•Newton’s Law of Gravitation•Kepler’s LawsMonday, Feb. 23, 2004 1PHYS 1443-501, Spring 2004Dr. Andrew BrandtAnnouncements• HW#5 on Ch. 6 will be assigned tomorrow due Mon. March 8 at midnightMonday, Feb. 23, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt2Resistive Force Proportional to Speed Since the resistive force is proportional to speed, we can write R=bvmLet’s consider that a ball of mass m is falling through a liquid.RmgvRFFg+=∑This equation also tells you that0 when , ==−= vgvmbgdtdvThe above equation also tells us that as time goes on the speed increases and the acceleration decreases, eventually reaching 0.An object moving in a viscous medium will obtain speed to a certain speed (terminal speed) and then maintain the same speed without any more acceleration.What does this mean?What is the terminal speed in above case?bmgvvmbgdtdvt==−= ;0How do the speed and acceleration depend on time?;0 when 0 ;1 ==⎟⎠⎞⎜⎝⎛−=−tvebmgvmbtThe time needed to reach 63.2% of the terminal speed is defined as the time constant, τ=m/b.0=∑xFvmbgdtdv−=dtdvmmabvmgFy==−=∑;0 when ; =====−−tgageembbmgdtdvatmbtτvmbgembbmgembbmgdtdvtt−=⎟⎠⎞⎜⎝⎛+−==−−ττ11Monday, Feb. 23, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt3Newton’s Law of Universal GravitationPeople have been very curious about the stars in the sky, makingobservations for a long time. But the data people collected have not been explained until Newton 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 principle mathematically?21221rmmFg∝With G21221rmmGFg=G is the universal gravitational constant, and its value isUnit?22/ kgmN ⋅1110673.6−×=GThis constant is not given by the theory but must be measured by experiment.This form of force is known as an inverse-square law, because the magnitude of the force is inversely proportional to the square of the distance between the objects.Monday, Feb. 23, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt4More on Law of Universal GravitationConsider two particles exerting gravitational forces on each other.Monday, Feb. 23, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt5It means that the force exerted on the particle 2 by particle 1 is attractive force, pulling #2 toward #1.Gravitational force is a field force: Forces act on object without physical contact between the objects independent of medium between them.1222112ˆrrmmGF −=The gravitational force exerted by a finite size, spherically symmetric mass distribution on a particle outside the distribution is the same as if the entire mass of the distribution was concentrated at the center.m1m2rF21F1212ˆrTwo objects exert gravitational force on each other following Newton’s 3rdlaw.Taking as the unit vector, we can write the force m2experiences as12ˆrWhat do you think the negative sign mean?gFHow would you describe the gravitational force on the surface of the Earth?2EERmMG=Free 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 getmg2EERmMG=g2EERMG=Monday, March 1, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt6•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?gF'mg=2rmMGE=()2hRmMGEE+='g()2hRMGEE+=What does this tell us about the gravitational acceleration?Example 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 isMEEEmg=2EERmMG=N61022.4 ×=GEFSince the orbit is at 350km above the surface of the Earth, the gravitational force at that height isTherefore the weight in orbit isOF'mg=()2hRmMGEE+=()GEEEFhRR22+=()GEEEFhRR22+=()()N66256261080.31022.41050.31037.61037.6×=×××+××=OFMonday, Feb. 23, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt7Example for GravitationUsing the fact that g=9.80m/s2at the Earth’s surface, find the average density of the Earth.Since the gravitational acceleration is 2EERMG=2111067.6EERM−×=gSo the mass of the Earth is GgRMEE2=Therefore the density of the Earth is EEVM=324EERGgR3=πEGRgπ43=ρ33611/1050.51037.61067.6480.93mkg×=×××××=−πMonday, Feb. 23, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt8Kepler’s Laws & EllipseF1F2bcaEllipses have two different axis, major (long) and minor (short) axis, and two focal points, F1& F2a is the length of a semi-major axisb is the length of a semi-minor axisKepler lived in Germany and discovered the law’s governing planet’s movement some 70 years before Newton, by analyzing data.1. 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, Feb. 23, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt9Kepler’s laws can be derived from Newton’s laws. Kepler’s third law is the direct consequence of the inverse square nature of the law of gravitation.The Law of Gravity and the Motion of Planets•Newton assumed that the law of gravitation applies the same whether it is acting on the Moon or an apple on the surface of the Earth. The interacting
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