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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 0° 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

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. "

& measured in Newton will be zero, as shown in equation

* 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:

* 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;

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• 1 steel plate of length (100.08(29:4cm and width

(30.08(29:4cm;

• 2 hinges

• 1 wooden stand height (130,08(29:4cm and width

(50.08(29:4cm; with 10 separate holes (10.08(29:4cm

• 1 steel rod length (50.08(29:4cm

• 1 wooden box without lid length (15,080.1) cm; width

(8.080,1) cm and height (3,080.1) cm and its base is placed a

steel plate of (15.0 0 0.1) cm and width (8.0 0.1) cm.80.1) cm

and width (8.080.1 cm

• 1 ruler of (30.0829:4+;

• 1 brush

• 1 Paper towels

• 1 1 liter cooking oil

• 2 smartphones preferably of the latest generation with the

highest number of sensors inside with Phyphox application

installed.

• 1 laptop computer.

• 1 pencil.

Figure 2. Materials and equipment used in the experiment.

Prepared by the author.

Figure 3. Phyphox application used in the experimentation.

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Materials and methods

First of all, the inclined plane is assembled with the above-

mentioned materials: wooden boards, steel plate, hinges, wooden

support, steel rod. Then, with the materials wood box and steel

surface, the box is completed. Then with the wooden support and the

rod, the inclined plane is placed with a certain angle of inclination,

as shown in Figure 1. After them, the smart phone is placed inside

the box with the Phyphox application to measure the angle of

inclination, in this case it starts with the angle of (11.6680.01)°.

After that with a brush oil is placed on the entire surface of the

inclined plane and on the surface of the box, to reduce the friction

effect. Then the laptop is synchronized with the waxed Phyphox

application that will measure the acceleration, for this both the

smartphone and laptop must be connected to the same wifi network.

Now place the smartphone inside the box, then place it on the top of

the inclined plane and indicate a mark with the pencil on the front of

the box to put it back in the same position for the next measurements.

Finally the box is released and the measurement of the acceleration

of the box is activated from the laptop with the smartphone while

sliding on the inclined plane, the measurement is deactivated when

the box reaches the bottom of the inclined plane. The process is

repeated 3 times with the same angle of inclination to measure the

acceleration, to reduce random errors. After that the angle of

inclination will be increased different from the initial one by

selecting 7 different values and the whole process is repeated

measuring in each case 3 times the acceleration of the box.

Results

The raw data obtained in the experiment are shown below.

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Table 1: Raw data of tilt angle (°) vs. acceleration of the enclosed

smartphone case (a)

Angle of inclination

θ/°

8<= * 8>9 >?@

Acceleration of the case including the smart phone

a/ms-2

8<A * 8>9 >?BC

#𝟐

11.66°

0,3

0,16

0,2

16.23°

1,11

0,99

0,82

21.13°

1,52

1,51

1,63

26.07°

2,73

2,14

2,47

31.29°

3,31

3,39

3,81

37.08°

4,53

4,26

4,03

42.67°

5,03

5,01

5,1

49.33°

5,81

5,79

5,9

When performing the analysis of the accelerations measurements, it

can be observed that there are groups of data for the same inclination

angle that have low precision and other groups of data that have high

precision, reason for which it was decided to select from each group

of accelerations, the 2 most precise ones. In the case of the first group

of acceleration data, we will obtain the mean and the uncertainty, as

shown in equations 6 and 7 as follows:

7D *

%,'()*%,+%))

* 29:E(5F

(Equation 6)

G7D *

%,+%#)%,'(

* 292H(5F

(Equation 7)

This process will be repeated with each of the groups of

accelerations found in the data table. Now we will obtain the

uncertainty of sinq , considering the first angle with its respective

uncertainty according to equation 8:

< -./ ! * <! I +,- ! * 292:

+,- ::9KK

* 2922LE (Equation 8)

This process is repeated for each of the tilt angles. The table of the

processed values is shown below:

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Table 2. Processed data of plane tilt angle and average acceleration

of the case including the smartphone with their respective

uncertainties.

Next, we proceed to construct the block mass graph including the

coins as a function of their submerged height. The uncertainties of

the measurements will be included in the graph, the line of best fit,

the line of maximum and minimum slope to determine the

uncertainty of the slope, as shown in Figure 4:

Figure 4. Plot of the acceleration of the box including the

smartphone as a function of the sine of the tilt angle, with its

uncertainties, the line of best fit, the lines of maximum and minimum

slope.

Angle of

inclination

θ/°

8<=

* 8>9 >?@

Sine of tilt

angle

MNO =

Uncertainty

of the sine of

angleq

<=JPQM =4

Average

acceleration

SBC

#𝟐

Average

acceleration

uncertainty

8<A

SBC

#𝟐

11,66°

0,20

0,01

0,18

0,02

16,23°

0,28

0,01

1,05

0,06

21,13°

0,36

0,01

1,52

0,01

26,07°

0,44

0,01

2,60

0,13

31.29°

0,52

0,01

3,35

0,04

37,08°

0,60

0,01

4,15

0,12

42,67°

0,68

0,01

5,02

0,01

49,33°

0,76

0,01

5,80

0,01

a = 10,094sen(θ) - 1,886

R² = 0,997

a= 9,690sen(θ) - 1,661

R² = 1,000

a = 10,407sen(θ) - 2,006

R² = 1,000

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Aceleración a / ms^-2

sen(θ)

Aceleración vs. sen(θ)

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From the graph it can be observed that the uncertainties of the

vertical and horizontal axis are small with respect to the scale used.

When plotting the line of best fit it can be observed that there is a

linear behavior between both variables, since the line does not pass

through the origin, this may be an indication of some type of

systematic error. Additionally, the line of best fit passes through

most of the experimental points, however, there are indications of

random errors. As already mentioned in the theoretical framework,

the slope (p) in Figure 4 is the acceleration of gravity (g), therefore,

equation 9 is obtained:

T * 6 (Equation 9)

From the graph we obtain the value of the slope, in this case:

T * 6 * :292LU(5F

(Equation 10)

With the help of the line of maximum and minimum slope, the

uncertainty is obtained. GTin this case the value is:

G6 * 29HHV(5F

Therefore, the acceleration of gravity with its uncertainty is

6 * J:292LU 8 29HHV45F

Discussion

According to the initial hypothesis and the results obtained in the

graph, it can be concluded that the hypothesis was not fulfilled, since

it was shown that the relationship between the acceleration of the

box including the smartphone and the sine of the angle of inclination

has a linear behavior and not a direct proportion. A possible reason

why experimentally a direct proportionality between both variables

was not obtained is that there is a small friction between the two

contact surfaces since there are possibilities that the oil has not filled

the imperfections of both surfaces. Based on graph 4, it can be seen

that the correlation coefficient (R2) is 0.997, which means that there

is a strong linear relationship between the acceleration of the case

including the smartphone and the sine of the tilt angle. The value

obtained for the acceleration of gravity using the didactic

methodological proposal was (:292LU 8 29HHV45F

a value very

close to that found by (Garcia et al., 2008) where the acceleration of

gravity was (9.796 ± 0.004) m.s-2. It can also be observed that its

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accuracy is high, since the percentage of error was 2.952%,

additionally the range of theoretical values is within and very close

to the range of experimental values.

The small frictional force between the two surfaces in contact

directly affects the accuracy of the gravity acceleration value and is

a source of systematic error, which is why the line of best fit was

deviated from the origin.

It is also observed the influence of errors in taking measurements

due to the delay time that exists in the human being between

activating and deactivating the acceleration reading, thus showing a

deviation in the measurements as a random error of the

experimentation.

For further research, in order to reduce errors and improve the

didactic methodological strategy, the following recommendations

shown in Table 3, below, should be considered:

Table 3. Table of weaknesses and recommendations designed by the

Author

Weaknesses

Recommendations

Friction between

contact surfaces.

- Friction could be further reduced by

polishing the surfaces and then placing the

oil on both surfaces.

Reading errors in

acceleration

measurements.

- A simple experiment could be used to

measure a person's reaction time

https://creandoconciencia.org.ar/enciclope

dia/conduccion-racional/reaccion-y-

control/medicion-del-tiempo-de-

reaccion.pdf and select the person with the

shortest reaction time.

As for the strengths of the experiment, the measurement of the tilt

angle using the other smartphone. The proposed didactic

methodological strategy yields gravity acceleration values very close

to the real value.

References

Andrés Z., M. M., Pesa, M. A., & Meneses, J. (2006). Experimental

activity in physics: Vision of university students. Paradígma,

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http://www.letteraturaitaliana.net/pdf/Volume_6/t333.pdf

Garcia, D., Larregain, P., & Machado, A. (2008). Measurement of

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https://users.exa.unicen.edu.ar/catedras/fisexp1/files/2008-

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