Simple Nature - Light and Matter

a/The spring has a minimum-

energy length, 1, and energy is

required in order to compress or

stretch it, 2 and 3. A mass at-

tached to the spring will oscillate

around the equilibrium, 4-13.

b/Three functions with the

same curvature atx=0.

2.5 Oscillations

Let’s revisit the example of the stretched spring from the previ-

ous section. We know that its energy is a form of electrical energy

of interacting atoms, which is nice conceptually but doesn’t help

us to solve problems, since we don’t know how the energy,U, de-

pends on the length of the spring. All we know is that there’s an

equilibrium (figure a/1), which is a local minimum of the function

U. An extremely important problem which arises in this connec-

tion is how to calculate oscillatory motion around an equilibrium,

as in a/4-13. Even if we did special experiments to find out how the

spring’s energy worked, it might seem like we’d have to go through

just as much work to deal with any other kind of oscillation, such

as a sapling swinging back and forth in the breeze.

Surprisingly, it’s possible to analyze this type of oscillation in a

very general and elegant manner, as long as the analysis is limited

tosmall oscillations. We’ll talk about the mass on the spring for

concreteness, but there will be nothing in the discussion at all that

is restricted to that particular physical system. First, let’s choose

a coordinate system in whichx= 0 corresponds to the position of

the mass where the spring is in equilibrium, and since interaction

energies likeUare only well defined up to an additive constant, we’ll

simply define it to be zero at equilibrium:

U(0) = 0

Sincex = 0 is an equilibrium,U(x) must have a local minimum

there, and a differentiable function (which we assumeU is) has a

zero derivative at a local minimum:

dU

dx

(0) = 0

There are still infinitely many functions that could satisfy these
criteria, including the three shown in figure b, which arex^2 /2,
x^2 /2(1 +x^2 ), and (e^3 x+e−^3 x−2)/18. Note, however, how all three
functions are virtually identical right near the minimum. That’s
because they all have the same curvature. More specifically, each
function has its second derivative equal to 1 atx= 0, and the sec-
ond derivative is a measure of curvature. We writekfor the second
derivative of the energy at an equilibrium point,

d^2 U

dx^2

(0) =k.

Physically,kis a measure of stiffness. For example, the heavy-duty

springs in a car’s shock absorbers would have a high value ofk. It

is often referred to as the spring constant, but we’re only using a

spring as an example here. As shown in figure b, any two functions

that haveU(0) = 0, dU/dx= 0, and d^2 U/dx^2 =k, with the same

value ofk, are virtually indistinguishable for small values ofx, so if

Section 2.5 Oscillations 115