Laboratory 10
Simple Harmonic Motion
Introduction
Any physical system governed by a
restoring force
, described (at least approximately) by a
quadratic potential well
, and which executes
periodic motion
, can be described as a
simple
harmonic oscillator
. Simple harmonic oscillators occur in many felds, From engineering and
physics to geology and biology. ThereFore, understanding this concept is critical to analyzing a
broad range oF theoretical models describing realworld systems.
One oF the simplest examples oF a simple harmonic oscillator is a mass oscillating due to a spring’s
restoring Force. A biochemical extension oF this consists oF two oxygen atoms (masses) oscillating
under the in±uence oF the interatomic Force (spring) between them. The atoms’ positions ±uctuate
through their equilibrium position as they are attracted to and repelled by each other. Although in
teratomic Forces are generally complex, in the region near equilibrium they are wellcharacterized
by the linear relationship
F
=
−
k
(
x
−
x
0
)
as long as the atoms’ excursions From their equilibrium
separation,
x
0
, are not too large. Both a mass in a massspring system and the oxygen atoms in the
biological system will undergo oscillatory motion at some natural Frequency that is a Function oF
the system itselF and does not depend on the amplitude oF the oscillation.
During this laboratory, we will investigate simple harmonic motion through the measurement oF
the natural oscillation Frequencies oF three diFFerent systems.
Theoretical Background: Simple Harmonic Motion
Simple harmonic motion
is sinusoidal motion that results when an object is subject to a restoring
Force oF the Form
°
F
=
−
k
°
x
,
where
°
x
is the object’s displacement From its
equilibrium position
 the position where it expe
riences no net Force. This Formula says that no matter which direction the displacement is in, the
restoring Force will act to bring the object back toward its equilibrium position. A graph oF Force
vs. displacement For simple harmonic motion is shown in ²igure 1.
Using Newton’s second law,
°
F
=
m
°
a
, we obtain the equation oF motion
ma
=
−
kx
,
(1)
85
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentFigure 1:
These two graphs are defning Features oF simple harmonic motion. The Force law is
linear with respect to the displacement,
x
, and the potential energy Function is a parabola. You can
think oF simple harmonic motion as a situation where a ball sitting at the bottom oF the potential
energy “well” is displaced slightly. What would happen? It would roll back and Forth periodically,
and its
x
position would be a sine curve.
where
x
is the object’s position and
a
is its acceleration. ±or those Familiar with calculus,
a
is the
second derivative oF
x
with respect to time, so
m
d
2
x
dt
2
=
−
kx
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Gerson
 Force, Simple Harmonic Motion, Angular frequency

Click to edit the document details