Simple Nature - Light and Matter

The minimum velocity required for this to happen is called escapevelocity. For speeds above escape velocity, the orbits areopen-ended hyperbolas, rather than repeating elliptical orbits. Infigure i, Pioneer’s hyperbolic trajectory becomes almost indistinguishablefrom a line at large distances from the sun. The motionslows perceptibly in the first few years after 1974, but later thespeed becomes nearly constant, as shown by the nearly constantspacing of the dots.The gravitational fieldWe got the energy equation U = −Gm 1 m 2 /r by integratingg ∝ 1/r 2 and then inserting a constant of proportionality to makethe proportionality into an equation. The opposite of an integral isa derivative, so we can now go backwards and insert a constant ofproportionality in g ∝ 1/r 2 that will be consistent with the energyequation:dU = m 2 g 1 drg 1 =1 dUm 2 dr= 1 (d− Gm )1m 2m 2 dr r( )d 1= −Gm 1dr r= Gm 1r 2This kind of inverse-square law occurs all the time in nature. Forinstance, if you go twice as far away from a lightbulb, you receive1/4 as much light from it, because as the light spreads out, it islike an expanding sphere, and a sphere with twice the radius hasfour times the surface area. It’s like spreading the same amount ofpeanut butter on four pieces of bread instead of one — we have tospread it thinner.Discussion QuestionsA A bowling ball interacts gravitationally with the earth. Would itmake sense for the gravitational energy to be inversely proportional to thedistance between their surfaces rather than their centers?2.3.5 The shell theoremNewton’s great insight was that gravity near the earth’s surfacewas the same kind of interaction as the one that kept the planetsfrom flying away from the sun. He told his niece that the ideacame to him when he saw an apple fall from a tree, which made himwonder whether the earth might be affecting the apple and the moonin the same way. Up until now, we’ve generally been dealing withgravitational interactions between objects that are small comparedto the distances between them, but that assumption doesn’t apply to102 Chapter 2 Conservation of Energy