The gravitational interactions of irregular shapes are rarely discussed in the compulsory schooling system and sometimes even ignored at the university level. This omission is due to the complexities encountered in extending Newton's law of gravitation to bodies that are not spherical. However, a deep understanding of the link between the gravity and the shape is quite important to interpret some basic facts of nature. In this paper, we show how simple concepts can be used to create a more general algorithm that has been implemented in matlab to compute the gravity of irregular bodies. Shapes are described in terms of Standard Tessellation Language files, the standard format for 3D printing. This approach to teaching allows students to model physical bodies, and the 3D representation of complex problems can help students acquire a more complete understanding of physics.

1.
E.
Baldy
, “
A new educational perspective for teaching gravity
,”
Int. J. Sci. Educ.
29
(
14
),
1767
1788
(
2007
).
2.
Leontyna
Brizova
,
Kelsey
Gerbec
,
Jiří
Šauer
, and
Jan
Slegr
, “
Flat Earth theory: An exercise in critical thinking
,”
Phys. Educ.
53
,
045014
(
2018
).
3.
T. G. G.
Chanut
,
S.
Aljbaae
, and
V.
Carruba
, “
Mascon gravitation model using a shaped polyhedral source
,”
Mon. Not. R. Astron. Soc.
450
(
4
),
3742
3749
(
2015
).
4.
T.
Fukushima
, “
Precise and fast computation of the gravitational field of a general finite body and its application to the gravitational study of asteroid eros
,”
Astron. J.
154
(
4
),
1
15
(
2017
).
5.
R.
Furfaro
,
R.
Barocco
,
R.
Linares
,
F.
Topputo
,
V.
Reddy
,
J.
Simo
, and
L.
Le Corre
, “
Modeling irregular small bodies gravity field via extreme learning machines and Bayesian optimization
,”
Adv. Space Res.
67
(
1
),
617
638
(
2021
).
6.
Y.
Jiang
and
H. X.
Baoyin
, “
Periodic orbit families in the gravitational field of irregular-shaped bodies
,”
Astron. J.
152
(
5
),
1
11
(
2016
).
7.
R. S.
Park
,
R. A.
Werner
, and
S.
Bhaskaran
, “
Estimating small-body gravity field from shape model and navigation data
,”
J. Guid. Control Dyn.
33
(
1
),
212
221
(
2010
).
8.
M.
Szilvasi-Nagy
and
G.
Matyasi
, “
Analysis of STL files
,”
Math. Comput. Modell.
38
(
7–9
),
945
960
(
2003
).
9.
L.
Casas
, “
Teaching periodicity and aperiodicity using 3D-printed tiles and polyhedra
,”
J. Appl. Crystallogr.
53
,
1583
1592
(
2020
).
10.
S.
Ford
and
T.
Minshall
, “
Invited review article: Where and how 3D printing is used in teaching and education
,”
Addit. Manuf.
25
,
131
150
(
2019
).
11.
See <github.com/eduMindmap/Gravity/tree/main/GRAPE> to download the MATLAB package GRAPE presented in the manuscript.
12.
See supplementary material at https://www.scitation.org/doi/suppl/10.1119/10.0005404 for additional information about the verification of the GRAPE algorithm with spheres, cubes, and cylinders, and for the resolution of the hollow sphere problem.
13.
R.
Borghi
, “
On Newton's shell theorem
,”
Eur. J. Phys.
35
(
2
),
028003
(
2014
).
14.
C.
Schmid
, “
Newton's superb theorem: An elementary geometric proof
,”
Am. J. Phys.
79
(
5
),
536
539
(
2011
).
15.
H. D.
Young
,
R. A.
Freedman
,
A. L.
Ford
, and
F. W.
Sears
,
University Physics: With Modern Physics
, 11th ed. (
Pearson Addison Wesley
,
San Francisco
,
2004
).
16.
See <https://www.openscad.org/> to download OpenSCAD and for a practical guide on how to use it.
17.
See <https://www.mathworks.com/matlabcentral/fileexchange/37856-inpolyhedron-are-points-inside-a-triangulated-volume> for “
MATLAB Central File Exchange: “Inpolyhedron” MATLAB Function by Sven
” (last accessed
April 13, 2021).
18.
R. A.
Werner
and
D. J.
Scheeres
, “
Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia
,”
Celestial Mech. Dyn. Astron.
65
(
3
),
313
344
(
1996
).
19.
See
<https://www.thingiverse.com/thing:4367315> to download the STL file of the asteroid 4769 Castalia.
20.
K. H.
Glassmeier
,
H.
Boehnhardt
,
D.
Koschny
,
E.
Kuhrt
, and
I.
Richter
, “
The Rosetta mission: Flying towards the origin of the solar system
,”
Space Sci. Rev.
128
(
1–4
),
1
21
(
2007
).
21.
F. L.
Johansson
,
A. I.
Eriksson
,
N.
Gilet
,
P.
Henri
,
G.
Wattieaux
,
Mggt
Taylor
,
C.
Imhof
, and
F.
Cipriani
, “
A charging model for the Rosetta spacecraft
,”
Astron. Astrophys.
642
,
1
12
(
2020
).
22.
A.
Probst
,
A.
San Martin
,
A.
Zimmer
,
S.
Broschart
, and
R.
Forstner
, “
Vertical controlled landing on 67P/Churyumov-Gerasimenko
,”
Planet. Space Sci.
192
,
1
10
(
2020
).
23.
See <http://open.esa.int/rosetta-3d-model/> to download the STL file of the comet 67P/Churyumov-Gerasimenko.
24.
S. C.
Hu
,
J. H.
Ji
,
D. C.
Richardson
,
Y. H.
Zhao
, and
Y.
Zhang
, “
The formation mechanism of 4179 Toutatis' elongated bilobed structure in a close Earth encounter scenario
,”
Mon. Not. R. Astron. Soc.
478
(
1
),
501
515
(
2018
).
25.
S. J.
Ostro
,
R. S.
Hudson
,
R. F.
Jurgens
,
K. D.
Rosema
,
D. B.
Campbell
,
D. K.
Yeomans
,
J. F.
Chandler
,
J. D.
Giorgini
,
R.
Winkler
,
R.
Rose
,
S. D.
Howard
,
M. A.
Slade
,
P.
Perillat
, and
I. I.
Shapiro
, “
Radar images of asteroid-4719 Toutatis
,”
Science
270
(
5233
),
80
83
(
1995
).
26.
See
<https://echo.jpl.nasa.gov/> for additional information about the asteroid 4179 Toutatis.
27.
See <https://www.thingiverse.com/thing:2698995> to download an example of STL file of a dinosaur.
28.
Walter J.
Wild
, “
Euler three body problem
,”
Am. J. Phys.
48
(
4
),
297
301
(
1980
).
29.
E.
Segre
and
W. H. Freeman and Company
,
From Falling Bodies to Radio Waves: Classical Physicists and Their Discoveries
, 1th ed. (
W.H. Freeman & Co
.,
New York
,
1984
).
30.
R.
Brazil
, “
Fighting flat-Earth theory
,”
Phys. World
33
(
7
),
35
39
(
2020
).
31.
O.
Kuzii
and
A.
Rovenchak
, “
What the gravitation of a flat Earth would look like and why thus the Earth is not actually flat
,”
Eur. J. Phys.
40
(
3
),
1
11
(
2019
).
32.
A. R.
Landrum
,
A.
Olshansky
, and
O.
Richards
, “
Differential susceptibility to misleading flat earth arguments on YouTube
,”
Media Psychol.
24
,
136
165
(
2021
).
33.
B.
Sherwood
, “
Incorrect predictions made by a popular flat-earth model
,”
Am. J. Phys.
88
(
6
),
430
430
(
2020
).
34.
See <https://www.thingiverse.com/thing:1464596> to download the STL file of a flat planet.
35.
James
Chappell
,
Mark
Chappell
,
Azhar
Iqbal
, and
Derek
Abbott
, “
The gravity field of a cube
,”
Phys. Int.
3
(
2
),
50
57
(
2013
).
36.
W.
Dittrich
, “
The eastward displacement of a freely falling body on the rotating Earth: Newton and Hooke's debate of 1679
,”
Ann. Phys.
522
(
8
),
601
607
(
2010
).
37.
J. C.
Huang
,
J. H.
Ji
,
P. J.
Ye
,
X. L.
Wang
,
J.
Yan
,
L. Z.
Meng
,
S.
Wang
,
C. L.
Li
,
Y.
Li
,
D.
Qiao
,
W.
Zhao
,
Y. H.
Zhao
,
T. X.
Zhang
,
P.
Liu
,
Y.
Jiang
,
W.
Rao
,
S.
Li
,
C. N.
Huang
,
W. H.
Ip
,
S. C.
Hu
,
M. H.
Zhu
,
L. L.
Yu
,
Y. L.
Zou
,
X. L.
Tang
,
J. Y.
Li
,
H. B.
Zhao
,
H.
Huang
,
X. J.
Jiang
, and
J. M.
Bai
, “
The ginger-shaped asteroid 4179 Toutatis: New observations from a successful flyby of Chang'e-2
,”
Sci. Rep.
3
,
1
6
(
2013
).
38.
M. D.
Pintado
, “
Gravity in Bruno, Kleper and Newton. Distribution of matter and cosmological implications
,”
Claridades-Rev. Filos.
8
,
23
48
(
2016
).

Supplementary Material

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