Simple Nature - Light and Matter

trical energy is the Lennard-Jones formula,[ (a ) 12 ( a) ] 6U(r) = k − 2 ,r rwhere r is the center-to-center distance between the atoms and k isa positive constant. Show that (a) there is an equilibrium point atr = a,(b) the equilibrium is stable, and(c) the energy required to bring the atoms from their equilibriumseparation to infinity is k. ⊲ Hint, p. 91821 The International Space Station orbits at an altitude of about360 to 400 km. What is the gravitational field of the earth at this√altitude?22 (a) A geosynchronous orbit is one in which the satellite orbitsabove the equator, and has an orbital period of 24 hours, so that itis always above the same point on the spinning earth. Calculate the√altitude of such a satellite.(b) What is the gravitational field experienced by the satellite? Giveyour answer as a percentage in relation to the gravitational field atthe earth’s surface. ⊲ Hint, p. 918 √23 Astronomers calculating orbits of planets often work in anonmetric system of units, in which the unit of time is the year,the unit of mass is the sun’s mass, and the unit of distance is theastronomical unit (A.U.), defined as half the long axis of the earth’sorbit. In these units, find an exact expression for the gravitational√constant, G.24 The star Lalande 21185 was found in 1996 to have two planetsin roughly circular orbits, with periods of 6 and 30 years. What is√the ratio of the two planets’ orbital radii?25 A projectile is moving directly away from a planet of massM at exactly escape velocity. (a) Find r, the distance from theprojectile to the center of the planet, as a function of time, t, and√also find v(t).(b) Check the units of your answer.(c) Does v show the correct behavior as t approaches infinity?⊲ Hint, p. 91826 The purpose of this problem is to estimate the height of thetides. The main reason for the tides is the moon’s gravity, and we’llneglect the effect of the sun. Also, real tides are heavily influencedby landforms that channel the flow of water, but we’ll think of theearth as if it was completely covered with oceans. Under theseassumptions, the ocean surface should be a surface of constant U/m.That is, a thimbleful of water, m, should not be able to gain or loseany gravitational energy by moving from one point on the oceansurface to another. If only the spherical earth’s gravity was present,Problems 121