bei48482_FM

 x  0 (5.62)

There are various ways to write the solution to Eq. (5.62). A common one is

xAcos (2t ) (5.63)

where

  (5.64)

is the frequency of the oscillations and Ais their amplitude. The value of , the phase

angle, depends upon what xis at the time t0 and on the direction of motion then.

The importance of the simple harmonic oscillator in both classical and modern

physics lies not in the strict adherence of actual restoring forces to Hooke’s law, which

is seldom true, but in the fact that these restoring forces reduce to Hooke’s law for

small displacements x. As a result, any system in which something executes small

vibrations about an equilibrium position behaves very much like a simple harmonic

oscillator.

To verify this important point, we note that any restoring force which is a func-

tion of xcan be expressed in a Maclaurin’s series about the equilibrium position

x0 as

F(x)Fx 0    

x 0

x     

x 0

x^2   

x 0

x^3 ...

Since x0 is the equilibrium position, Fx 0 0. For small xthe values of x^2 , x^3 ,...

are very small compared with x, so the third and higher terms of the series can be

neglected. The only term of significance when xis small is therefore the second one.

Hence

F(x)  

x 0

x

which is Hooke’s law when (dFdx)x 0 is negative, as of course it is for any restoring

force. The conclusion, then, is that alloscillations are simple harmonic in character

when their amplitudes are sufficiently small.

The potential-energy function U(x) that corresponds to a Hooke’s law force may be

found by calculating the work needed to bring a particle from x0 to xxagainst

such a force. The result is

U(x)

x

0

F(x) dxk

x

0

xdx kx^2 (5.65)

which is plotted in Fig. 5.10. The curve of U(x) versus xis a parabola. If the energy

of the oscillator is E, the particle vibrates back and forth between xAand x

A, where Eand Aare related by E^12 kA^2. Figure 8.18 shows how a nonparabolic

potential energy curve can be approximated by a parabola for small displacements.

1

2

dF

dx

d^3 F

dx^3

1

6

d^2 F

dx^2

1

2

dF

dx

k

m

1

2 

Frequency of

harmonic oscillator

k

m

d^2 x

dt^2

Harmonic

oscillator

188 Chapter Five

Energy

E

• A 0+A
  x

U = 12 kx^2

Figure 5.10The potential energy

of a harmonic oscillator is pro-

portional to x^2 , where xis the

displacement from the equilib-

rium position. The amplitude A

of the motion is determined by

the total energy Eof the oscillator,

which classically can have any

value.

bei48482_ch05.qxd 1/17/02 12:17 AM Page 188