We implement an experimental system based on optical levitation of a silicone oil droplet to demonstrate a damped driven harmonic oscillator. The apparatus allows us to control all the parameters present in the differential equation that theoretically describes such motion. The damping coefficient and driving force can be manipulated in situ by changing the pressure in the apparatus and by applying a variable electric field. We present two different experimental procedures. First, a transition from the overdamped to underdamped regimes is demonstrated by gradually lowering the air pressure. The characteristic resonance associated with an underdamped driven harmonic oscillator is observed by studying how the amplitude of the oscillation varies as a function of the driving force. Second, in order to observe qualitative differences between the overdamped and underdamped regimes of a harmonic oscillator, three driving functions (sine, square, and sharp delta pulses) were separately applied, both at atmospheric pressure and under vacuum conditions. Our overall aim is to design a hands-on apparatus that is easy to use and that allows undergraduate and graduate students to observe and manipulate the basic physical processes associated with a damped driven harmonic oscillator.

1.
Frank
Oppenheimer
, “
Forced harmonic oscillator
,”
Am. J. Phys.
31
,
13
24
(
1963
).
2.
Francis J.
McCormack
, “
Smooth transitions between the three damping cases for the harmonic oscillator
,”
Am. J. Phys.
63
,
1151
(
1995
).
3.
Dennis
Ugolini
,
Hanna
Rafferty
,
Max
Winter
,
Carsten
Rockstuhl
, and
Antje
Bergmann
, “
LIGO analogy lab—A set of undergraduate lab experiments to demonstrate some principles of gravitational wave detection
,”
Am. J. Phys.
87
,
44
56
(
2019
).
4.
Marcello
Carla
and
Samuele
Straulino
, “
Measurements on a guitar string as an example of a physical nonlinear driven oscillator
,”
Am. J. Phys.
85
,
587
595
(
2017
).
5.
Patrick
Shea
,
Brandon P.
van Zyl
, and
Rajat K.
Bhaduri
, “
The two-body problem of ultra-cold atoms in a harmonic trap
,”
Am. J. Phys.
77
,
511
515
(
2009
).
6.
Phillip
Gould
, “
Laser cooling of atoms to the Doppler limit
,”
Am. J. Phys.
65
,
1120
1123
(
1997
).
7.
Michael J.
Ruiz
, “
The human body's response to glucose and three physical models
,”
Am. J. Phys.
55
,
641
645
(
1987
).
8.
R.
Hauko
,
D.
Andreevski
,
D.
Paul
,
M.
Sterk
, and
R.
Repnik
, “
Teaching of the harmonic oscillator damped by a constant force: The use of analogy and experiments
,”
Am. J. Phys.
86
,
657
662
(
2018
).
9.
Christopher C.
Jones
, “
A mechanical resonance apparatus for undergraduate laboratories
,”
Am. J. Phys.
63
,
232
236
(
1995
).
10.
David
Faiman
and
Archibald W.
Hendry
, “
Electromagnetic decays of baryon resonances in the harmonic-oscillator model
,”
Phys. Rev.
180
,
1572
1577
(
1969
).
11.
Wenhua
Zhang
and
Kimberly L.
Turner
, “
Application of parametric resonance amplification in a single-crystal silicon micro-oscillator based mass sensor
,”
Sens. Actuators, A
122
,
23
30
(
2005
).
12.
Kenneth G.
Holt
,
Joseph
Hamill
, and
Robert O.
Andres
, “
Predicting the minimal energy costs of human walking
,”
Med. Sci. Sports Exercise
23
,
491
498
(
1991
).
13.
M.
Alonso
and
E. J.
Finn
,
Fundamental University Physics, Volume I, Mechanics
, 1st ed. (
Addison-Wesley Publishing Company
,
Reading, MA
,
1976
).
14.
D.
Halliday
,
R.
Resnick
, and
J.
Walker
,
Principles of Physics
, 9th ed. (
Wiley
,
Hoboken, NJ
,
2011
).
15.
R. D.
Knight
,
Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
, 1st ed. (
Pearson Education
,
San Francisco, CA
,
2016
).
16.
A.
Giambattista
,
B. M. C.
Richardson
, and
R. C.
Richardson
,
College Physics
, 1st ed. (
McGraw-Hill Higher Education
,
New York, NY
,
2007
).
17.
J. C.
Zamora
,
F.
Fajardo
, and
J.—A.
Rodríguez
, “
Oscillator experiments with periods between the simple pendulum and a rigid rod
,”
Am. J. Phys.
77
,
169
172
(
2009
).
18.
D.
Kharkongor
and
Mangal C.
Mahato
, “
Resonance oscillation of a damped driven simple pendulum
,”
Eur. J. Phys.
39
,
065002
(
2018
).
19.
A.
Ashkin
, “
History of optical trapping and manipulation of small-neutral particle, atoms, and molecules
,”
IEEE J. Sel. Top. Quantum Electron.
6
,
841
856
(
2000
).
20.
A.
Ashkin
and
J. M.
Dziedzic
, “
Optical levitation by radiation pressure
,”
Appl. Phys. Lett.
19
,
283
285
(
1971
).
21.
Stephen P.
Smith
,
Sameer R.
Bhalotra
,
Anne L.
Brody
,
Benjamin L.
Brown
,
Edward K.
Boyda
, and
Mara
Prentiss
, “
Inexpensive optical tweezers for undergraduate laboratories
,”
Am. J. Phys.
67
,
26
35
(
1999
).
22.
John
Bechhoefer
and
Scott
Wilson
, “
Faster, cheaper, safer optical tweezers for the undergraduate laboratory
,”
Am. J. Phys.
70
,
393
400
(
2002
).
23.
M. S.
Rocha
, “
Optical tweezers for undergraduates: Theoretical analysis and experiments
,”
Am. J. Phys.
77
,
704
712
(
2009
).
24.
D. C.
Appleyard
,
K. Y.
Vandermeulen
,
H.
Lee
, and
M. J.
Lang
, “
Optical trapping for undergraduates
,”
Am. J. Phys.
75
,
5
14
(
2007
).
25.
Oscar
Isaksson
,
Magnus
Karlsteen
,
Mats
Rostedt
, and
Dag
Hanstorp
, “
An optical levitation system for a physics teaching laboratory
,”
Am. J. Phys.
86
,
135
142
(
2018
).
26.
J.
Mas
,
A.
Farre
,
J.
Cuadros
,
I.
Juvells
, and
A.
Carnicer
, “
Understanding optical trapping phenomena: A simulation for undergraduates
,”
IEEE Trans. Educ.
54
,
133
140
(
2011
).
27.
A.
Ashkin
and
J. M.
Dziedzic
, “
Optical levitation in high vacuum
,”
Appl. Phys. Lett.
28
,
333
335
(
1976
).
28.

A treatment with equivalent results can be found in Ref. 13. Otherwise, it is easy to derive Eqs. (5) and (6) by plugging Eq. (4) into Eq. (1). For Eq. (5) make t = 0 and for Eq. (6)t=π/2ω and α=α+(π/2).

29.
A.
Ashkin
, “
Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime
,”
Biophys. J.
61
,
569
582
(
1992
).
30.
E.
Hetch
,
Optics
, 4th ed. (
Addison-Wesley
,
San Francisco, CA
,
2002
).
31.
Thomas R.
Lettieri
,
Wilhelmina D.
Jenkins
, and
Dennis A.
Swyt
, “
Sizing of individual optically levitated evaporating droplets by measurement of resonances in the polarization ratio
,”
Appl. Opt.
20
,
2799
2805
(
1981
).
32.
A.
Chiasera
,
Y.
Dumeige
,
P.
Féron
,
M.
Ferrari
,
Y.
Jestin
,
G.
Nunzi Conti
,
S.
Pelli
,
S.
Soria
, and
G. C.
Righini
, “
Spherical whispering-gallery-mode microresonators
,”
Laser Photonics Rev.
4
,
457
482
(
2010
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.