1443-501 Spring 2002Lecture #21Dr. Jaehoon Yu1. Kepler’s Laws2. The Law of Gravity & The Motion of Planets3. The Gravitational Field4. Gravitational Potential Energy5. Energy in Planetary and Satellite MotionsToday’s Homework Assignment would have been #10 but I will assign next Monday.Apr. 17, 20021443-501 Spring 2002Dr. J. Yu, Lecture #212Example 14.3Using the fact that g=9.80m/s2at the Earth’s surface, find the average density of the Earth.21121067.6EEEERMRMGg−×==Since the gravitational acceleration is So the mass of the Earth is GgRMEE2=Therefore the density of the Earth is 3361132/1050.51037.61067.6480.93434mkgGRgRGgRVMEEEEE×=×××××==3==−πππρApr. 17, 20021443-501 Spring 2002Dr. J. Yu, Lecture #213Kepler’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 the direct consequence of law of gravitation being inverse square law.•All planets move in elliptical orbits with the Sun at one focal point.•The radius vector drawn from the Sun to a planet sweeps out equal area in equal time intervals. (Angular momentum conservation)•The square of the orbital period of any planet is proportional to the cube of the semi-major axis of the elliptical orbit.F1F2bcaEllipses 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 axisApr. 17, 20021443-501 Spring 2002Dr. J. Yu, Lecture #214The Law of Gravity and the Motion 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 aMto the apple’s acceleration g would be ( )( )42862221075.21084.31037.6/1/1−×=××===MEEMMrRRrgaREMoonApplegaMv234/1070.280.91075.2 smaM−−×=××=Newton also calculated the Moon’s orbital acceleration aMfrom the knowledge of its distance from the Earth and its orbital period, T=27.32 days=2.36x106s()( )22368226080.9/1072.21036.21084.344/2≈×=×××====−22smTrrTrrvaMMMMMπππThis means that the Moon’s distance is about 60 times that of the Earth’s radius, its acceleration is reduced by the square of the ratio. This proves that the inverse square law is valid.Apr. 17, 20021443-501 Spring 2002Dr. J. Yu, Lecture #215Kepler’s Third LawIt is crucial to show that Keper’s third law can be predicted from the inverse square law for circular orbits.Since the orbital speed, v, of the planet with period T isSince the gravitational force exerted by the Sun is radiallydirected toward the Sun to keep the planet circle, we can apply Newton’s second lawrvMrMGMpPs22=Trvπ2=The above can be writtenThis is Keper’s third law. It’s also valid for ellipse for r being the length of the semi-major axis. The constant Ksis independent of mass of the planet. Msssvr()rTrMrMGMpPs22/2π=Solving for T one can obtain 3324rKrGMTss==π32192/1097.24msGMKss−×==πandApr. 17, 20021443-501 Spring 2002Dr. J. Yu, Lecture #216Example 14.4Calculate the mass of the Sun using the fact that the period of the Earth’s orbit around the Sun is 3.16x107s and its distance from the Sun is 1.496x1011m.Using Kepler’s third law.The mass of the Sun, Ms, is3324rKrGMTss==π( )kgrGTMs303117112321099.110496.11016.31067.644×=×××××==−ππApr. 17, 20021443-501 Spring 2002Dr. J. Yu, Lecture #217Kepler’s Second Law and Angular Momentum ConservationSince the gravitational force acting on the planet is always toward radial direction, it is a central forceConsider a planet of mass Mpmoving around the Sun in an elliptical orbit.0ˆ=×=×= rFrFrτBecause the gravitational force exerted on a planet by the Sun results in no torque, the angular momentum L of the planet is constant. This is Keper’s second law which states that the radius vector from the Sun to a planet sweeps our equal areas in equal time intervals. dtMLdtvrrdrdAp221=×=×=Therefore the torque acting on the planet by this force is always 0.Since torque is the time rate change of angular momentum L, the angular momentum is constant.constLdtLd=== ;0τconstvrMvMrprLpp=×=×=×=SBADCrdrSince the area swept by the motion of the planet is constMLdtdAp==2Apr. 17, 20021443-501 Spring 2002Dr. J. Yu, Lecture #218The Gravitational FieldThe force exists every point in the space.The gravitational force is a field force.If one were to place a test object of mass m at a any point in the space in the existence of another object of mass M, the test object will fill the gravitational force, , exerted by MgmFg=In other words, the gravitational field at a point in space is the gravitational force experienced by a test particle placed at the point divided by the mass of the test particle.Therefore the gravitational field g is defined as mFgg≡So how does the Earth’s gravitational field look like?rRGMmFgEEgˆ2−==Where is the unit vector pointing outward from the center of the EarthrˆEFar away from the Earth’s surfaceClose to the Earth’s surfaceApr. 17, 20021443-501 Spring 2002Dr. J. Yu, Lecture #219The Gravitational Potential EnergyWhat is the potential energy of an object at the height y from the surface of the Earth?No, it would not. Because gravitational force is a central force and a central force is a conservative force, the work done by the gravitational force is independent of the path.The path can be looked at as consisting of many tangential and radial motions. mgyU=Do you think this would work in general cases?Why not?Because this formula is only valid for the case where the gravitational force is constant, near the surface of the Earth and the generalized gravitational force is inversely proportional to the square of the distance.OK. Then how would we generalize the potential energy in the gravitational field?REmmriFgrfFgApr. 17, 20021443-501 Spring 2002Dr. J. Yu, Lecture #2110More on The Gravitational Potential EnergySince the gravitational force is a radial force, it only performed work while the path was radial direction only.
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