Comparing this with Eq. (5.1) we see that
kDm!^2 ; (5.7)or
!D
r
k
m: (5.8)
In Eq. (5.2), the amplitudeAdepends on how far the particle was displaced from equilibrium before being
released; the phase constantıjust depends on when we choose timet D 0 ; but the angular frequency!
depends on the physical parameters of the system: the stiffness of the springkand the mass of the particle
m.
5.1 Energy
The kinetic energyKof a particle of massmmoving with speedvis defined to be the work required to
accelerate the particle from rest to speedv; this is found to be
KD^12 mv^2 : (5.9)From Hooke’s law, the potential energyUof a simple harmonic oscillator particle at positionxcan be shown
to be
UD^12 kx^2 : (5.10)Thetotalmechanical energyEDKCUof a simple harmonic oscillator can be found by observing that
whenxD ̇A,wehavevD 0 , and therefore the kinetic energyKD 0 and the total energy is all potential.
Since the potential energy atxD ̇AisUDkA^2 =2(by Eq. (5.10)), the total energy must be
ED^12 kA^2 : (5.11)Since total energy is conserved, the energyEis constant and does not change throughout the motion, although
the kinetic energyKand potential energyUdo change.
In a simple harmonic oscillator, the energy sloshes back and forth between kinetic and potential energy,
as shown in Fig. 5.3. At the endpoints of its motion (xD ̇A), the oscillator is momentarily at rest, and the
energy is entirely potential; when passing through the equilibrium position (xD 0 ), the energy is entirely
kinetic. In between, kinetic energy is being converted to potential energy or vice versa.
We can find the velocityvof a simple harmonic oscillator as a function of positionx(rather than timet)
by writing an expression for the conservation of energy:
EDKCU (5.12)
1
2 kA(^2) D 1
2 mv
(^2) C 1
2 kx
(^2) (5.13)
Solving forv,wefind
v.x/D ̇A
r
k
m
r
1
x^2
A^2
: (5.14)
This can be simplified somewhat by using Eq. (5.8) to give
v.x/D ̇A!r
1 x^2
A^2; (5.15)
whereA!is, by inspection of Eq. (5.3), the maximum speed of the oscillator (the speed it has while passing
through the equilibrium position).