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Sinergias educativas
October - December Vol. 8 - 4- 2023
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eISSN: 2661-6661
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Approved: January 12 , 2023
Didactic proposal to determine the
acceleration of gravity
Propuesta didáctica para determinar la aceleración de
la gravedad
Marcos Francisco Guerrero Zambrano
*
Rodrigo Salomón Jurado Echeverría*
Juan Patricio Aguirre Mateus*
Luis Javier Aguirre Mateus*
Abstract
This paper proposes an alternative didactic methodology to
determine the acceleration of gravity using a smart phone inside a
box with a steel base and sliding on an inclined plane with a steel
base. The experiment is performed by placing oil on the steel surface
of the inclined plane, then the smartphone is placed on the box and
released from the top of the inclined plane and with the help of the
Phyphox application the angle of inclination of the plane and the
acceleration of the box is measured. The process is repeated by
increasing the angle of inclination and in each case the acceleration
of the box is measured. Although no direct proportionality was found
between the box acceleration and the sine of the tilt angle, a linear
* Magister En Enseñanza De La Fisica,
Universidad Estatal de Milagro
mguerreroz@unemi.edu.ec
https://orcid.org/0000-0002-5617-6836
* Magister En Educacion Mención En
Enseñanza De La Matematica, Universidad
Estatal de Milagro
rjuradoe@unemi.edu.ec
https://orcid.org/0009-0000-5464-4256
* Magister En Enseñanza De La Fisica,
Universidad Estatal de Milagro
jaguirrem1@unemi.edu.ec
https://orcid.org/0000-0003-1245-0925
* Magister En Enseñanza De La Física,
Instituto Superior Tecnológico Babahoyo
laguirre@istb.edu.ec
https://orcid.org/0000-0002-5770-1014
Article
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relationship was observed, mainly influenced by the friction between
the two steel surfaces.
Keywords: Angle of inclination, acceleration, acceleration of
gravity, alternative methodology, theoretical model.
RESUMEN
Este artículo propone una metodología didáctica alternativa para
determinar la aceleración de la gravedad utilizando un teléfono
inteligente que se encuentra en el interior de una caja con base de
acero y que se desliza sobre un plano inclinado con base de acero.
Se realiza el experimento colocando aceite sobre la superficie de
acero del plano inclinado, luego el teléfono inteligente se coloca
sobre la caja y se suelta desde la parte superior del plano inclinado y
con ayuda de la aplicación Phyphox se mide el ángulo de inclinación
del plano y la aceleración de la caja. Se repite el proceso
aumentando el ángulo de inclinación y en cada caso se mide la
aceleración de la caja. Aunque no se encontró una proporcionalidad
directa entre la aceleración de la caja y el seno del ángulo de
inclinación, se observó una relación lineal, influenciada
principalmente por el rozamiento existente entre las dos superficies
de acero.
Palabras clave: Ángulo de inclinación, aceleración, aceleración de
la gravedad, metodología alternativa, modelo teórico.
Introduction
The acceleration of the Earth's gravity is the acceleration that a body
undergoes in interaction with the gravitational field of the planet.
The acceleration of gravity discovered by Galileo Galilei in 1604 and
the formulation of the law of universal gravitation proposed by Sir
Isaac Newton in 1687 provided the basis for what is known as the
acceleration of gravity and its relation to bodies. For Suwanpayak et
al., (2018) at different points on the planet the acceleration of gravity
tends to vary, this is mainly due to the Earth's gravitational field.
For that the experimental determination of the acceleration of gravity
in different locations has been an object of study with respect to the
fallible methods that can be used and that approximate the standard
gravity or gravity at sea level. It should be noted that the
experimental method, which, through Galileo Galieli, proclaimed by
certain historians as the father of experimental physics, Papp, (1961)
argues that:
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"Galileo does not use experience to find the law, he invokes it only
to verify it, already found by deductive reasoning. One does not err
by overemphasizing this characteristic of the Galilean method ... The
revolutionary methodological innovation introduced by the great
Italian in science ... does not consist in the apotheosis of experience
with contempt for deductive speculation, but, as we have said, in the
masterly synthesis of the triple method of his precursors -
philosophical, mathematical and empirical - in one and indivisible
unity."(Papp, 1961)
Experimental methods must count on the elements that Galileo
traversed in his indivisible unity, which demonstrates the true sense
and purpose of experimental research. Related to the determination
of the acceleration of gravity, it is evident the postulation and
implementation of several methods that supported in the
mathematical component and supported in the theory are used to
determine the acceleration of gravity, for example, the principle of
free fall refers to the use of an intelligent timer to detect the time t
for a metal ball falling between a point 1 to a distant point 2, the
difference between these heights is known as distance h. In contrast,
the simple pendulum method takes into consideration the principle
of simple harmonic motion, where a metal ball of mass m, connected
to a string of length L, and using a digital timer to detect the time of
oscillation of the ball of 10 revolutions with five repetitions by
modifying the angle between and 10°. On the other hand, the rigid
pendulum technique consists of a rigid body undergoing a fixed-
axis rotation around a fixed point. In this experiment, a 100 cm long
metal ruler with holes for the rotation points is used, in addition, a
digital stopwatch was used to detect the time taken for the metal ruler
to complete 10 revolutions, which was estimated from five
repetitions, in 5 cm phases, from 5 to 95 cm. In addition, Atwood's
machine consists of a pulley with two weights of unequal mass and
such difference produces a net force which accelerates both hanging
masses, and the time between the distance from a point 1 to a distant
point 2 is taken. (Suwanpayak et al., 2018).
The methods mentioned above have a mathematical analysis
component, which provides experimental values that can be
compared with the theoretical value and determine their uncertainty
and percentage error.
In addition, there are other types of methods that make use of
smartphones or smartphones, for example, through the use of
applications I can use the principle of free fall because these devices
through the use of sensors, allow to observe, measure and record the
data that are being taken. (Kuhn & Vogt, 2013)..
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Likewise Martínez Pérez, (2015) mentions about the use of these
mobile devices for the determination of the acceleration of gravity
by means of the simple pendulum:
... ICTs are equipped with sensors such as accelerometers,
gyroscopes, barometers, etc. and, in the last two years, several
researches have shown their feasibility to be used in experiences
inside the physics classroom or in your laboratory, in acoustic,
optical and mechanical phenomena.... (Martínez Pérez, 2015)
The relevance of the aforementioned research highlights the
emphasis needed in science teaching. Teachers must structure and
deepen concepts, demonstrating the relationship between physical
phenomena and variables (Martínez-Borreguero et al., 2018)..
In physical science teaching, the question of how to transmit
knowledge to students arises. It is essential to structure
methodologies that encourage the development of skills through
experience.
At present it does not seem to make sense to speak of the
experimental method, but rather of an experimental activity that is
part of a body of knowledge, and includes a diversity of methods.
Science seeks theories that effectively solve problems, which may
be empirical or conceptual; the progress of science seems to occur to
the extent that more problems are solved or eluded. In this sense,
changes are gradual, accepting the coexistence of rival programs,
and although there is a bidirectional relationship between theory and
methods, progress in each field may not be simultaneous (Andrés Z.
et al., 2006)..
In conclusion, it has been observed that there are multiple methods
used to determine the acceleration of gravity, as well as the evolution
of this measurement has been evidenced, since it comes from using
timers or stopwatches to use smartphones with sensors that allow
data collection, the latter is the case of this research, which seeks to
determine the acceleration of gravity by placing an object in a plane
tilting based on its resultant force.
The fundamentals of dynamics were set out in Galileo Galilei's book
"Dialogo sopra i due massimi sistemi del mondo" which means
"Dialogue on the two main systems of the world", where the notion
of the concept of inertia was implicit and explained. In addition,
Galileo's experiments with inclined planes had made it possible to
establish mathematical relationships between the kinematic
variables for motion with constant velocity and acceleration.
(Galilei, 1632). On the other hand, Isaac Newton in his book
"Philosophiae naturalis principia mathematica" which means
"Mathematical principles of natural philosophy", mentions his three
laws of which we will mention two that will be used in the didactic
strategy proposed...:
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Newton's First Law states "All bodies persevere in their state of rest
or uniform motion in a straight line, unless they are forced to change
that state by an external force".
Newton's second law: "The change of motion is proportional to the
impressed motive force, and is in the same direction as the straight
line in which that force is impressed" (Newton, 1686). (Newton,
1686).
Both laws we apply it to the box moving along the inclined plane of
angle !and with the indicated reference system, as shown in figure
1.
Figure 1. Vector description of the forces acting on the box and the
reference system used.
Considering the experimental didactic proposal, there are two forces
acting on the object, the force of Normal "#
$
$
%
& force exerted by the
inclined plane on the box and the weight "'
$
%
&(which is the
gravitational force exerted by the earth on the box, both forces
measured in Newtons, as shown in figure 1. '
$
$
%
(as '
!
$
$
$
%
()(('
"
$
$
$
$
%
respectively, according to the indicated reference system, as shown
in equations 1 and 2:
'
!
* ' +,- ! (Equation 1)
'
"
* ' -./ ! (Equation 2)
In addition, it is observed that Newton's First Law is fulfilled on the
vertical axis and Newton's Second Law is fulfilled on the horizontal
axis. "
0
1
%
& measured in Newton will be zero, as shown in equation
3.
0
1
%
* 2
$
%
(Equation 3)
Considering the reference frame of figure 1 we have equation 4:
# 3 '
"
* 2 (Equation 4)
Where, the vertical component of the weight ('
"
4measured in
Newton, is equal to the product of the mass (m) of the box including
the smartphone measured in kilograms and the cosine of the tilt angle
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(!). Therefore, replacing the above in equation 4, we have equation
5, as shown below:
# 3 56 +,- ! * 2 (Equation 5)
Now analyzing the horizontal axis, it is observed that Newton's
Second Law is fulfilled, so that the sum of forces is equal to mass
times acceleration, as shown in Equation 6:
"
0
1
%
* 57%&(Equation 6)
Considering the reference frame of figure 1, we have equation 7:
'
!
* 57(Equation 7)
Where, the horizontal component of the weight ('
!
4measured in
Newton, is equal to the product of the mass (m) of the box including
the smartphone measured in kilograms and the sine of the angle of
inclination (!). Therefore, replacing the above in equation 7, we
have equation 8, as shown below:
56 -./ ! * 57 (Equation 8)
Analyzing equations 5 and 8, it follows that to determine the
acceleration of the box including the smartphone, only equation 8 is
needed, since the friction is neglected, therefore, simplifying the
mass m of this equation, equation 9 is obtained:
7 * 6 -./ ! (Equation 9)
From the obtained result it is observed that the acceleration of the
box including the smartphone is directly proportional to the sine of
the angle of inclination, therefore, if the sine of the angle of
inclination increases in magnitude, the acceleration of the box
including the smartphone will increase in the same proportion;
which allows obtaining a theoretical model that can be verified with
the experimental model obtained in practice. If we compare it with
the equation of direct proportion we notice that in the vertical axis
goes the acceleration of the box including the included smartphone,
in the horizontal axis goes the sine of the angle of inclination and the
slope of the graph would be the acceleration of gravity. Then, our
didactic experimental proposal consists of increasing the sine of the
angle of inclination and then obtaining the acceleration of the box
with the smartphone included. In this didactic experimental
proposal, the sine of the inclination angle is expected to be directly
proportional to the acceleration of the box including the smartphone
while sliding down the inclined plane, i.e., if the angle of inclination
is increased, the acceleration of the box including the smartphone
will increase proportionally because increasing the angle will
increase its speed.
LIST OF MATERIALS AND EQUIPMENT
1 inclined plane made with:
2 wooden boards of length (100,08(29:4cm and width
(30.08(29:4cm;