Simple Nature - Light and Matter

aa / Example 10.to either side of equilibrium will increase it, whereas the unstableequilibrium represents a maximum.Note that we are using the term “stable” in a weaker sense thanin ordinary speech. A domino standing upright is stable in the sensewe are using, since it will not spontaneously fall over in response toa sneeze from across the room or the vibration from a passing truck.We would only call it unstable in the technical sense if it could betoppled by any force, no matter how small. In everyday usage, ofcourse, it would be considered unstable, since the force required totopple it is so small.An application of calculus example 10⊲ Nancy Neutron is living in a uranium nucleus that is undergoingfission. Nancy’s nuclear energy as a function of position can beapproximated by U = x 4 − x 2 , where all the units and numericalconstants have been suppressed for simplicity. Use calculusto locate the equilibrium points, and determine whether they arestable or unstable.⊲ The equilibrium points occur where the U is at a minimum ormaximum, and minima and maxima occur where the derivative(which equals minus the force on Nancy) is zero. This derivativeis dU/ dx = 4x 3 − 2x, and setting it equal to zero, we have x =0, ±1/ √ 2. Minima occur where the second derivative is positive,and maxima where it is negative. The second derivative is 12x 2 −2, which is negative at x = 0 (unstable) and positive at x = ±1/ √ 2(stable). Interpretation: the graph of U is shaped like a roundedletter ‘W,’ with the two troughs representing the two halves of thesplitting nucleus. Nancy is going to have to decide which half shewants to go with.4.1.6 Proof of Kepler’s elliptical orbit lawKepler determined purely empirically that the planets’ orbitswere ellipses, without understanding the underlying reason in termsof physical law. Newton’s proof of this fact based on his laws ofmotion and law of gravity was considered his crowning achievementboth by him and by his contemporaries, because it showed that thesame physical laws could be used to analyze both the heavens andthe earth. Newton’s proof was very lengthy, but by applying themore recent concepts of conservation of energy and angular momentumwe can carry out the proof quite simply and succinctly. Thissubsection can be skipped without losing the continuity of the text.The basic idea of the proof is that we want to describe the shapeof the planet’s orbit with an equation, and then show that this equationis exactly the one that represents an ellipse. Newton’s originalproof had to be very complicated because it was based directly onhis laws of motion, which include time as a variable. To make anystatement about the shape of the orbit, he had to eliminate time262 Chapter 4 Conservation of Angular Momentum