Cycloidal paths are ubiquitous in physics. Here, we show that representative cycloidal paths in physics can be described as superpositions of translations and rotations of a point through space. Using this unifying principle, the parametric equations of the path of a point on a rolling disk are derived for rolling without slipping, rolling with frictionless slipping, and when kinetic solid-on-solid friction is present during rolling with slipping. In a similar way, the parametric equations versus time for the orbit with respect to a star of a moon in a circular orbit about a planet that is in a circular orbit about a star are derived, where the orbits are coplanar. The parametric equations versus time for the path of the magnetization vector during undamped electron-spin resonance are found using the same principle, which show that cycloidal paths can occur under specified conditions.

Topics
Electromagnetism, Physical quantities, Fundamental constants, Magnetic fields, Rotational dynamics, Electron paramagnetic resonance spectroscopy, Nuclear magnetic resonance, Multivariable calculus, Differential geometry
1.
E. A.
 
Whitman
, “
Some historical notes on the cycloid
,”
Am. Math. Mon.
 
50
(
5
),
309
–
315
(
1943
).
https://doi.org/10.1080/00029890.1943.11991383
Google Scholar
Crossref
Search ADS
 
2.
T.
 
Roidt
, “
Cycloids and paths
,” M.S. thesis, Portland State University,
2011
, 55 pp.
Google Scholar
 
3.
R. A.
 
Proctor
,
The Cycloid and All Forms of Cycloidal Curves
(
Longmans, Green, and Co
.,
London
,
1878
).
Google Scholar
 
4.
M. M.
 
McKinley
, “
Brachistochrones, tautochrones, evolutes, and tessellations
,”
Am. J. Phys.
 
47
(
1
),
81
–
86
(
1979
).
https://doi.org/10.1119/1.11679
Google Scholar
Crossref
Search ADS
 
6.
I.
 
Bruce
, “
Some properties of cycloid trajectories
,”
Am. J. Phys.
 
65
(
12
),
1164
–
1167
(
1997
).
https://doi.org/10.1119/1.18755
Google Scholar
Crossref
Search ADS
 
7.
D. J.
 
Griffiths
,
Introduction to Electrodynamics
, 3rd ed. (
Prentice Hall
,
Upper Saddle River, NJ
,
1999
).
Google Scholar
 
8.
J. S.
 
Miller
, “
Cycloid generator
,”
Am J. Phys.
 
27
(
1
),
62
(
1959
).
https://doi.org/10.1119/1.1934751
Google Scholar
Crossref
Search ADS
 
10.
G. L.
 
Baker
 and
J. A.
 
Blackburn
,
The Pendulum—A Case Study in Physics
(
Oxford U.P
.,
Oxford
,
2006
).
Google Scholar
 
11.
H.
 
Goldstein
,
C. P.
 
Poole
, and
J.
 
Safko
,
Classical Mechanics
, 3rd ed. (
Addison Wesley
,
San Francisco
,
2001
).
Google Scholar
 
12.
PhET Interactive Simulations
, “
My-Solar-System
,” University of Colorado, Boulder.
13.
G. E.
 
Pake
 and
T. L.
 
Estle
,
The Physical Principles of Electron Paramagnetic Resonance
, 2nd ed. (
Benjamin
,
Reading, MA
,
1973
).
Google Scholar
 
14.
R. H.
 
Taylor
, “
Electron spin resonance of magnetic ions in metals—An experimental review
,”
Adv. Phys.
 
24
(
6
),
681
–
791
(
1975
).
https://doi.org/10.1080/00018737500101501
Google Scholar
Crossref
Search ADS
 
15.
S. E.
 
Barnes
, “
Theory of electron spin resonance of magnetic ions in metals
,”
Adv. Phys.
 
30
(
6
),
801
–
938
(
1981
).
https://doi.org/10.1080/00018738100101447
Google Scholar
Crossref
Search ADS
 
16.
C. P.
 
Poole
, Jr.,
Electron Spin Resonance
, 2nd ed. (
Wiley
,
New York
,
1983
).
Google Scholar
 
17.
A.
 
Abragam
,
Principles of Nuclear Magnetism
(
Clarendon
,
Oxford
,
1961
).
Google Scholar
 
18.
C. P.
 
Slichter
,
Principles of Magnetic Resonance
(
Harper & Row
,
New York
,
1963
).
Google Scholar
 
19.
F.
 
Bloch
, “
Nuclear induction
,”
Phys. Rev.
 
70
(
7,8
),
460
–
474
(
1946
).
https://doi.org/10.1103/PhysRev.70.460
Google Scholar
Crossref
Search ADS
 
20.
Reference 13, pp.
14
–
25
.
21.
See, e.g.,
H. D.
 
Young
 and
R. A.
 
Freedman
,
University Physics
, 14th ed. (
Pearson Education
,
USA
,
2016
).
Google Scholar
 
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2019
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