The
pendulum
of
the
hand
of
statistics
and
engineering
Abstract.
-
In
the
following
work,
you
will
find
the
research
carried
out
to
understand
in
a
more
feasible
way
the
analysis
of
the
simple
pendulum
since
this
study
helps
to
understand
many
aspects
found
in
everyday
life,
such
as
the
operation
of
a
clock.
For
this,
a
model
was
made
to
carry
out
all
the
necessary
measures
to
process
the
data
obtained.
It
was
observed
that
there
is
a
variation
of
the
average
time
depending
on
its
length,
so
applying
statistical
principles,
where
taking
several
samples,
it
was
possible
to
visualize
a
natural
phenomenon,
applying
an
analysis
based
on
engineering
for
gravity
calculation,
analytically
and
graphically.
Keywords:
Gravity,
statistics,
physics,
measurement,
pendulum.
ISSN-E:
2737-6419
Athenea
Journal,
Vol.
4,
Issue
11,
(pp.
15-22)
Almache E.
et al.
The pendulum of the hand of statistics and engineering
Almache
Eguez
Giovanny
https://orcid.org/0000-0002-7373-6794
giovanny.almache@udla.edu.ec
Universidad
de
Las
Américas,
Facultad
de
ingeniería
y
ciencias
aplicadas,
carrera
de
Ingeniería
en
Electrónica
y
Automatización
Quito-Ecuador
Resumen:
En
el
siguiente
trabajo
encontrarás
las
investigaciones
realizadas
para
entender
de
una
manera
más
factible
el
análisis
del
péndulo
simple,
ya
que
este
estudio
ayuda
a
entender
muchos
aspectos
que
se
encuentran
en
la
vida
cotidiana,
como
el
funcionamiento
de
un
reloj,
para
lo
cual
se
realizó
un
modelo
que
permitió
realizar
todas
las
mediciones
necesarias
para
luego
procesar
los
datos
obtenidos.
Se
observó
que
existe
una
variación
del
tiempo
promedio
como
consecuencia
de
su
duración,
para
lo
cual,
aplicando
principios
estadísticos,
donde
tomando
varias
muestras,
fue
posible
visualizar
un
fenómeno
natural,
aplicando
un
análisis
basado
en
ingeniería,
para
el
cálculo
de
la
gravedad,
de
manera
analítica
y
gráfica.
Palabras
clave:
Gravedad,
estática,
física,
medición,
péndulo.
El
péndulo
de
la
mano
de
la
estadística
y
la
ingeniería
15
R
e
c
e
i
v
e
d
(
1
3
/
0
8
/
2
0
2
2
)
,
A
c
c
e
p
t
e
d
(
0
5
/
0
1
/
2
0
2
3
)
B
o
r
j
a
C
a
m
p
u
z
a
n
o
D
a
v
i
d
https://orcid.org/0000-0001-7741-0284
david.borja.campuzano@udla.edu.ec
Universidad
de
Las
Américas,
Facultad
de
ingeniería
y
ciencias
aplicadas,
carrera
de
Ingeniería
en
Electrónica
y
Automatización
Quito-Ecuador
Millán
Márquez
Katherin
https://orcid.org/0000-0001-8873-9210
katherin.millan@udla.edu.ec
U
n
i
v
e
r
s
i
d
a
d
d
e
L
a
s
A
m
é
r
i
c
a
s
,
F
a
c
u
l
t
a
d
d
e
ingeniería
y
ciencias
aplicadas,
carrera
de
Ingeniería
en
Electrónica
y
Automatización
Quito-Ecuador
Tarambis
Velasco
Michael
https://orcid.org/0000-0002-2395-6342
michael.tarambis@udla.edu.ec
U
n
i
v
e
r
s
i
d
a
d
d
e
L
a
s
A
m
é
r
i
c
a
s
,
F
a
c
u
l
t
a
d
d
e
ingeniería
y
ciencias
aplicadas,
carrera
de
Ingeniería
en
Electrónica
y
Automatización
Quito-Ecuador
https://doi.org/10.47460/athenea.v4i11.50
I. INTRODUCTION
The
pendulum
is
a
severe
body
that
can
oscillate
suspended
from
a
point
by
a
thread
or
rod.
This
has
a
mass,
so,
in
turn,
it
generates
a
force
that
attracts
it
toward
the
gravitational
center.
There
are
several
variations
of
the
system.
These
can
be
formed
from
different
materials.
However,
they
are
all
governed
by
the
same
principle,
oscillating
and
performing
in
certain
circumstances,
movements
considered
periodic
or
quasi-
periodic
[1].
In
this
sense,
the
pendulum
represents
the
oscillatory
movements
of
physics.
The
pendulum
describes
a
circular
trajectory.
However,
the
arc
it
generates
will
have
the
radius
of
the
length
of
the
thread,
b
e
i
n
g
t
h
e
p
e
n
d
u
l
u
m
a
t
t
h
e
m
o
s
t
s
i
g
n
i
f
i
c
a
n
t
a
n
g
l
e
a
t
w
h
i
c
h
t
h
e
w
e
i
g
h
t
w
i
l
l
b
e
t
h
r
o
w
n
;
t
h
i
s
h
a
s
p
o
t
e
n
t
i
a
l
e
n
e
r
g
y
,
which
will
be
converted
into
kinetic
energy
until
reaching
the
equilibrium
point
(when
it
has
an
angle
of
90°
with
the
horizontal),
When
it
begins
to
rise
to
the
greater
angle
it
becomes
potential
again.
One
of
the
most
valuable
tools
for
analyzing
the
behavior
of
simple
pendulum
variables
is
descriptive
s
t
a
t
i
s
t
i
c
s
,
w
h
i
c
h
c
o
n
t
r
i
b
u
t
e
s
s
i
g
n
i
f
i
c
a
n
t
l
y
t
o
m
o
s
t
e
n
g
i
n
e
e
r
i
n
g
w
o
r
k
.
D
e
s
c
r
i
p
t
i
v
e
s
t
a
t
i
s
t
i
c
s
i
s
a
b
r
a
n
c
h
u
s
e
d
t
o
summarize
and
present
data
clearly
and
concisely.
In
simple
pendulum-based
work,
measurements
of
pendulum
swing
times
can
be
summarized
by
descriptive
statistics
and
presented
as
tables,
graphs,
and
statistics
as
mean
median,
and
standard
deviation.
This
allows
for
a
better
understanding
of
the
data
and
a
more
accessible
interpretation
of
experimental
results.
In
this
work,
the
behavior
of
gravity
in
the
simple
pendulum
was
analyzed
with
statistical
applications.
For
this,
an
experimental
practice
of
the
pendulum
was
carried
out
to
take
the
necessary
data
to
evaluate
the
severity
values
later
and
make
the
respective
calculations
of
errors
and
statistical
analysis.
Ten
length
measurements
and
three-time
values
have
been
considered
for
each
case
to
optimize
calculation
procedures.
This
work
consists
of
4
sections;
in
the
first,
the
fundamentals
of
the
subject
of
study
have
been
described;
in
the
second,
the
theoretical
elements
that
support
this
research
will
be
raised;
in
the
third
section,
we
will
proceed
to
explain
the
methodological
processes
of
the
experiment.
Finally,
the
results
and
conclusions
are
presented.
II. DEVELOPMENT
Ideally,
a
simple
pendulum
has
a
mass
(m)
suspended
from
a
wire
of
length
(l),
inextensible
and
without
mass
[1].
The
mass
moves
through
an
angle
θ
with
the
vertical
axis.
In
our
case,
this
angle
should
not
exceed
15°,
and
the
oscillation,
without
imparting
an
initial
speed,
is
allowed
to
move
only
under
the
restrictions
imposed
by
gravity
and
rope.
16
I
S
S
N
-
E
:
2
7
3
7
-
6
4
1
9
Athenea
Journal,
Vol.
4,
Issue
11,
(pp.
15-22)
Fig. 1.
Simple pendulum in balanced position.
Almache E.
et al.
The pendulum of the hand of statistics and engineering
Therefore,
the
particle's
motion
is
confined
within
an
arc
of
radius
(θ)
in
the
plane.
The
only
forces
acting
on
this
mass
are
the
weight
(
)
and
the
string's
tension
(
).
17
I
S
S
N
-
E
:
2
7
3
7
-
6
4
1
9
Athenea
Journal,
Vol.
4,
Issue
11,
(pp.
15-22)
Fig. 2.
Physical relationship of a simple module
Some
movements
are
constant
in
physics
and
everyday
life,
such
as
tides,
heartbeats,
and
clocks.
Still,
this
last
is
the
most
thought
for
analyzing
a
simple
pendulum
because
it
is
the
most
obvious
example
of
periodic
movement.
Periodic
motion
is
the
movement
of
a
body
from
one
side
to
the
other
along
a
fixed
path,
returning
to
each
position
and
velocity
after
a
fixed
time
interval.
A
simple
pendulum
consists
of
an
insignificant
mass
suspended
from
a
string
[2].
Any
periodic
motion
can
be
thought
of
as
the
result
of
a
set
of
simultaneous
simple
harmonic
oscillatory
movements.
For
this
reason,
simple
harmonic
motion
is
the
basis
for
studying
all
periodic
motion
and,
therefore,
all
periodic
phenomena.
The
actual
period
and
frequency
are
obtained
in
current
practice,
and
the
pendulum
formula
obtains
the
theoretical
frequency.
where
T
=
period
(s),
l
=
length
and
g
=
gravity
m/s.
For
laboratory
analysis,
single
pendulum
oscillators
are
idealized
as
natural
systems
with
less
than
fifteen-
degree
angles.
Due
to
their
relative
vibration
relative
to
equilibrium,
they
generate
mechanical
energy
dissipated
in
the
form
of
kinetic
energy.
Based
on
equilibrium
calculations,
kinetic
energy
at
its
peak
is
idealized
to
identify
its
other
measurements,
such
as
amplitude,
frequency,
period,
rapidity,
and
quality
[3].
Gravity
and
oscillation
are
two
fundamental
parts
of
this
study.
However,
it
is
known
that
gravity
is
the
force
that
attracts
objects
to
the
earth's
center.
While
oscillation
is
the
movement
of
an
object
between
two
certain
positions,
these
oscillations
can
vary
depending
on
the
length
of
the
thread
or
material
that
holds
it
suspended.
Due
to
this,
it
was
decided
that
the
best
way
to
check
everything
described
above
would
be
to
make
a
model
that
is
functional,
that
allows
and
perform
the
analysis
of
the
oscillation
and
the
change
that
exists
in
it
according
to
the
length
of
the
thread
that
helps
to
keep
the
dough
suspended.
This
model
aims
to
verify
that
what
is
in
theory
can
be
put
into
practice.
On
the
other
hand,
you
want
to
implement
automation
to
it
in
a
way
that
facilitates
the
collection
of
data
and
the
necessary
calculation
with
them.
Almache E.
et al.
The pendulum of the hand of statistics and engineering
A.
Descriptive
statistics
in
engineering
Statistics
is
essential
to
engineering,
providing
a
solid
foundation
for
decision-making
and
problem-solving.
In
experiment
design,
statistics
is
used
to
plan,
design,
and
analyze
experiments
to
determine
relationships
between
variables
and
to
optimize
processes.
For
example,
engineers
use
statistical
techniques
in
chemical
p
r
o
c
e
s
s
e
n
g
i
n
e
e
r
i
n
g
t
o
o
p
t
i
m
i
z
e
r
e
a
c
t
i
o
n
c
o
n
d
i
t
i
o
n
s
a
n
d
m
a
x
i
m
i
z
e
p
r
o
c
e
s
s
e
f
f
i
c
i
e
n
c
y
.
I
n
a
d
d
i
t
i
o
n
,
i
n
q
u
a
l
i
t
y
c
o
n
t
r
o
l
,
s
t
a
t
i
s
t
i
c
s
a
r
e
u
s
e
d
t
o
c
o
n
t
r
o
l
a
n
d
i
m
p
r
o
v
e
t
h
e
q
u
a
l
i
t
y
o
f
p
r
o
d
u
c
t
s
a
n
d
p
r
o
c
e
s
s
e
s
t
h
r
o
u
g
h
s
t
a
t
i
s
t
i
c
a
l
t
e
c
h
n
i
q
u
e
s
s
u
c
h
a
s
p
r
o
c
e
s
s
c
o
n
t
r
o
l
a
n
d
p
r
o
c
e
s
s
c
a
p
a
b
i
l
i
t
y
.
T
h
i
s
a
l
l
o
w
s
e
n
g
i
n
e
e
r
s
t
o
d
e
t
e
c
t
and
correct
quality
issues
early,
reducing
costs
and
increasing
customer
satisfaction.
In
maintenance
engineering,
statistics
are
used
to
plan
and
schedule
preventive
and
predictive
maintenance
of
machinery
and
equipment,
helping
to
reduce
costs
and
increase
asset
availability.
In
short,
statistics
is
a
valuable
tool
for
engineering
as
it
allows
engineers
to
collect,
analyze,
and
use
data
to
make
informed
decisions
and
solve
problems
in
various
fields.
T
h
e
m
a
i
n
e
r
r
o
r
s
t
o
a
n
a
l
y
z
e
i
n
t
h
i
s
p
a
p
e
r
a
r
e
:
1.
Absolute
error:
the
difference
exists
between
the
measurements'
theoretical
and
practical
values
obtained
when
making
the
measurements
[1].
2.
Systematic
error:
it
varies
predictably;
this
means
you
have
an
idea
of
the
error
that
will
come
out
[2].
3.
Zero
error:
this
error
is
one
of
the
most
common
on
a
day-to-day
basis;
many
times,
it
is
due
to
a
factory
error;
this
happens
when
the
equipment
is
zero;
it
marks
a
value
that
it
should
not.
4.
Non-linearity
error
happens
when
the
results
do
not
generate
a
straight
line
but
have
a
nonlinear
trend
[3].
5.
Standard
deviation:
determines
the
variation
between
the
data
and
the
mean;
when
it
is
low,
the
data
is
concentrated
near
the
mean.
A
high
deviation
indicates
that
the
data
is
distributed
over
a
broader
range
[4].
6.
Variance:
is
an
indicator
of
how
uneven
the
data
are
around
the
mean;
the
higher
the
average,
the
greater
the
dispersion
of
the
data
and
the
less
representative
the
mean
[7].
III. METHODOLOGY
For
the
experimental
data,
the
data
collection
was
carried
out
with
the
prepared
model
and
a
body
of
7
grams
of
weight
at
15
degrees
of
inclination.
The
model
includes
a
design
to
build
a
model
in
a
physical
or
a
simple
oscillatory
pendulum.
It
has
a
wooden
base
to
keep
it
stable,
a
wooden
bar
1m
30cm
high
with
a
crossbar
at
the
tip
to
hold
a
string
with
a
weight
attached
as
presented
in
the
fig.
3.
18
ISSN-E:
2737-6419
Athenea
Journal,
V
o
l
.
2
7
,
N
ú
m
.
1
1
8
,
(
p
p
.
1
5
-
2
2
)
Almache E.
et al.
The pendulum of the hand of statistics and engineering
19
ISSN-E:
2737-6419
Athenea
Journal,
Vol.
27,
Núm.
118,
(pp.
15-22)
Fig. 2.
Physical relationship of a simple module
After
making
the
model
as
presented
in
the
schematic
diagram,
a
weight
of
7
grams
was
incorporated
with
a
diameter
of
3
cm
attached
to
a
string
of
initial
length
of
10
cm,
which
is
varied
for
different
lengths,
increasing
from
5
cm
to
5
cm
after
each
measurement,
in
turn,
to
determine
the
time
the
pendulum
was
positioned
at
an
angle
of
15°,
where
the
weight
will
be
released
to
start
with
the
taking
of
time
until
a
complete
oscillation
ends,
to
later
continue
with
the
analysis
of
the
average
and
obtain
a
more
accurate
measurement.
For
the
theoretical
data,
data
were
taken
from
a
simulator
at
the
University
of
Colorado,
where
the
weight
of
the
body
was
prepared
as
0.10
kg;
due
to
the
minimum
mass
limitation
of
the
system,
the
same
inclination
was
adjusted
and
taken
at
a
slow
speed
for
greater
precision
of
the
time
in
which
the
cycle
is
completed.
Once
the
data
was
obtained,
it
was
entered
into
an
Excel
table.
With
the
experimental
data
collected
at
three
different
times,
an
average
of
the
three
values
was
obtained
(Table
1)
Table 1.
Physical relationship of a simple module
Almache E.
et al.
The pendulum of the hand of statistics and engineering
IV. RESULTS
Once
the
experiment
was
performed,
the
following
results
were
found:
Regarding
the
calculation
of
errors,
it
can
be
seen
in
Table
2.
That
the
absolute
error
is
very
little
because
the
measures
taken
did
not
significantly
differ
from
one
to
the
other.
Therefore,
it
is
considered
that
the
values
are
the
most
accurate
possible.
For
absolute
error
calculations,
the
formula
is
implemented:
20
ISSN-E:
2737-6419
Athenea
Journal,
V
o
l
.
2
7
,
N
ú
m
.
1
1
8
,
(
p
p
.
1
5
-
2
2
)
The
absolute
error
presented
a
range
from
0.0133s
to
0.0233s,
representing
the
study.
On
the
other
hand,
the
zero
error
was
0.68s.
On
the
other
hand,
in
Table
2,
you
can
see
the
nonlinearity
errors
obtained
when
evaluating
the
values
in
the
formulas
found
and
the
period
and
severity
calculations;
these
values
help
to
understand
the
system's
b
e
h
a
v
i
o
r
.
The
absolute
error
presented
a
range
from
0.0133s
to
0.0233s,
representing
the
study.
On
the
other
hand,
the
zero
error
was
0.68s.
On
the
other
hand,
in
Table
2,
you
can
see
the
nonlinearity
errors
obtained
when
evaluating
the
values
in
the
formulas
found
and
the
period
and
severity
calculations;
these
values
help
to
understand
the
system's
behavior.
Table 2.
Physical relationship of a simple module
Almache E.
et al.
The pendulum of the hand of statistics and engineering
21
For
calculating
the
lines,
equations
(3)
to
(6)
were
used,
allowing
the
use
of
both
theoretical
and
experimental
data.
F
o
r
t
h
e
c
a
l
c
u
l
a
t
i
o
n
o
f
g
r
a
v
i
t
y
,
e
q
u
a
t
i
o
n
s
(
5
)
a
n
d
(
6
)
h
a
v
e
b
e
e
n
c
o
n
s
i
d
e
r
e
d
,
w
h
e
r
e
t
h
e
l
e
n
g
t
h
s
a
n
d
t
i
m
e
s
o
f
t
h
e
experiment
are
related.
As
can
be
seen,
the
gravity
calculations
closely
resemble
the
theoretical
value
of
9.81
.
For
the
comparison
of
the
theoretical
and
practical
values,
figure
4
was
made.
The
theoretical
and
experimental
data
show
a
practically
identical
similarity
between
both.
ISSN-E:
2737-6419
A
t
h
e
n
e
a
J
o
u
r
n
a
l
,
Vol.
27,
Núm.
118,
(pp.
15-22)
Fig. 4
. Length as a function of time, (a) theoretical valúes, (b) experimental values.
It
was
observed
that
analytical
gravity
has
a
value
very
close
to
the
theoretical
value
of
gravity,
while
gravity
a
n
a
l
y
z
e
d
g
r
a
p
h
i
c
a
l
l
y
p
r
e
s
e
n
t
s
a
n
a
v
e
r
a
g
e
e
r
r
o
r
o
f
0
.
0
0
0
1
1
6
6
7
t
o
t
h
e
t
h
e
o
r
e
t
i
c
a
l
v
a
l
u
e
o
f
g
r
a
v
i
t
y
;
t
h
i
s
r
e
f
l
e
c
t
s
that
the
experiment
was
performed
consistently
and
fairly
accurately.
However,
it
would
be
prudent
to
repeat
the
tests
with
a
more
consistent
mass,
a
thread
with
less
resistance,
and
a
more
accurate
stopwatch
to
reduce
e
r
r
o
r
v
a
l
u
e
s
a
n
d
i
m
p
r
o
v
e
t
h
e
q
u
a
l
i
t
y
o
f
t
h
e
p
r
o
c
e
s
s
p
e
r
f
o
r
m
e
d
.
CONCLUSIONS
Throughout
the
research
and
the
implementation
of
the
theory,
it
was
understood
that
in
many
opportunities,
the
theoretical
result
is
very
similar
to
the
practical
one.
However,
it
must
be
taken
into
account
that
these
values
may
differ
if,
when
taking
the
measurements,
the
forecast
of
having
precision
in
them
needs
to
be
taken.
Statistics
is
a
great
tool
that
facilitates
calculations
and
allows
you
to
predict
the
results
obtained
throughout
the
studies.
On
the
other
hand,
it
is
confirmed
that
statistics
and
engineering
can
shake
hands
when
conducting
studies.
Statistics
is
a
great
ally
when
doing
research
because
it
helps
predict
results
and
know
if
the
results
being
obtained
are
adequate.
The
practical
formulations
in
the
understanding
of
concepts
of
physics
and
statistics
are
beneficial
for
teaching
and
learning
in
engineering
careers
since
they
allow
interaction
with
the
theoretical
context
and
deepen
the
concepts
achieving
more
significant
learning.
.
Almache E.
et al.
The pendulum of the hand of statistics and engineering
22
ISSN-E:
2737-6419
Athenea
Journal,
Vol.
27,
Núm.
118,
(pp.
15-22)
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Almache E.
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The pendulum of the hand of statistics and engineering