you of a spring. The potential energy for a spring is given by the parabola U( x)

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1 Small oscillations The theoy of small oscillations is an extemely impotant topic in mechanics. Conside a system that has a potential enegy diagam as below: U B C A x Thee ae thee points of stable equilibium, A, B, and C. The geneal coodinate x can be thought of as a position o angle. The foce on a paticle at position x in this potential is given by du F By definition a paticle located at any of the stable equilibium points dx x expeiences no foce, and the slope of the potential enegy is zeo at these positions. What happens when the paticle is distubed fom one of this equilibium points? If the paticle is pushed to the ight, it expeiences a estoing foce to the left. The behavio should emind you of a sping. The potential enegy fo a sping is given by the paabola U( x) sping kx. This paabolic potential does not look globally much like that pictued above. Howeve, locally paabolas appoximate the potential nea the stable equilibia. U B C A x

2 If the best paabolas can be detemined then the potential at the equilibium point can be appoximated by that of a sping. The question educes to finding an appoximate mathematical desciption of the potential enegy nea equilibium points. To do this, ecall that a Taylo seies expansion is such a local appoximation to a function. Specifically nea a point x : n n f ( x ) f ( x ) f ( x ) f ( x) f ( x)... f ( x) 6 n! Conside the potential enegy U nea an equilibium point x. Expanding to second ode U( x ) U( x) U( x) U ( x). Conside each tem, U( x ) is the constant value of the potential enegy at the local minimum. Since only diffeences in potential enegy ae physically significant this constant tem is not impotant. The next tem is U( x ) and is equal to zeo. This is because at the equilibium point the fist deivative of the potential enegy vanishes. The last tem ( ) U x is vey inteesting and impotant. It is the equation of a paabola in, the displacement fom the equilibium point. This is just what we needed. Nea an equilibium point the potential enegy is a constant plus a quadatic dependence in the displacement, U( x ) U( x) U( x). This is the equation fo the potential enegy of a sping with a sping constant, o stiffness, equal to U ( x ). Look at the figue of the potential enegy and points, A, B, and C. Although all of these ae stable equilibia, the cuvatue, o equivalently U ( x ), vaies with each. Point C has the highest cuvatue, followed by A and then B, equivalently U ( x ) U ( x ) U ( x ). Again, this cuvatue plays the ole of the sping stiffness k. C A B The angula fequency of oscillations fo a sping is given by k m. Theefoe the angula U( x ) fequency of oscillations about a stable equilibium point is. Such hamonic m behavio is a geneal featue of small amplitude oscillations about any stable equilibium point fo an abitay potential enegy. As the amplitude of oscillations inceases, the paabolic appoximation to the potential enegy becomes less accuate and the behavio becomes anhamonic (peiodic, but not sinusoidal).

3 Let s conside a specific example, let U() [ ] ELand [ ] E. Always sketch the potential: L whee and,. Dimensionally The ed cuves show the two pats of the potential; thee is a linea attactive piece and a epulsive pat, thei sum geneates the total potential. Fist the equilibium point must be found. This is done by setting U( ) and solving fo. The deivative is given by U() and setting equal to zeo and solving yields. Note that dimensionally this combination yields a length. Evaluating the potential at U ( ) gives the minimum value of the potential U ( ). We can now label these on the potential enegy cuve.

4 U ( ) In ode to find the fequency of small oscillations, we need to find the cuvatue fom the second deivative of the potential enegy, U() evaluating at the equilibium point U( ). Once again you should check the units to make sue that you answe can be coect, the dimensions should be E L. We ae now eady to look at the best appoximate hamonic (paabolic) potential, U( ) U ( ~ H ) U( ) U( ), fo this above case U( ) U H ( ~ ) plot of U point.. The plots ae done with and and so a ( ) H will show the best hamonic appoximation nea the equilibium

5 U ( ) The ed cuve is the hamonic appoximation to the actual potential. When the total enegy is just slightly above the minimum, the potential enegy is well appoximated by the hamonic potential. This is actually a common occuence in natue and so this small amplitude oscillation appoximation is vey impotant. The last thing to find is the fequency of these small oscillations. Using the hamonic appoximation the sping constant k is equal to the cuvatue evaluated at the equilibium point, k U( ) U ( ), theefoe the angula fequency can be found fom o m m 4 m m. The fequency is f 4 m. This pocedue may be conducted nea any point of stable equilibium fo any potential enegy pofile, independent of the natue of the undelying inteactions. This povides the tue context fo the impotance of the hamonic oscillato poblem: nea stable equiliba all systems behave as if they wee simple hamonic oscillatos.