Simple Nature - Light and Matter

Discussion QuestionsA I’m not making this up. XS Energy Drink has ads that read like this:All the “Energy” ... Without the Sugar! Only 8 Calories!” Comment onthis.2.5 OscillationsLet’s revisit the example of the stretched spring from the previoussection. We know that its energy is a form of electrical energyof interacting atoms, which is nice conceptually but doesn’t helpus to solve problems, since we don’t know how the energy, U, dependson the length of the spring. All we know is that there’s anequilibrium (figure a/1), which is a local minimum of the functionU. An extremely important problem which arises in this connectionis how to calculate oscillatory motion around an equilibrium,as in a/4-13. Even if we did special experiments to find out how thespring’s energy worked, it might seem like we’d have to go throughjust as much work to deal with any other kind of oscillation, suchas a sapling swinging back and forth in the breeze.Surprisingly, it’s possible to analyze this type of oscillation in avery general and elegant manner, as long as the analysis is limitedto small oscillations. We’ll talk about the mass on the spring forconcreteness, but there will be nothing in the discussion at all thatis restricted to that particular physical system. First, let’s choosea coordinate system in which x = 0 corresponds to the position ofthe mass where the spring is in equilibrium, and since interactionenergies like U are only well defined up to an additive constant, we’llsimply define it to be zero at equilibrium:a / The spring has a minimumenergylength, 1, and energy isrequired in order to compress orstretch it, 2 and 3. A mass attachedto the spring will oscillatearound the equilibrium, 4-13.U(0) = 0Since x = 0 is an equilibrium, U(x) must have a local minimumthere, and a differentiable function (which we assume U is) has azero derivative at a local minimum:dUdx (0) = 0There are still infinitely many functions that could satisfy thesecriteria, including the three shown in figure b, which are x 2 /2,x 2 /2(1 + x 2 ), and (e 3x + e −3x − 2)/18. Note, however, how all threefunctions are virtually identical right near the minimum. That’sbecause they all have the same curvature. More specifically, eachfunction has its second derivative equal to 1 at x = 0, and the secondderivative is a measure of curvature. We write k for the secondderivative of the energy at an equilibrium point,. d 2 Udx 2 (0) = k Section 2.5 Oscillations 113