Revista Científica Interdisciplinaria Investigación y Saberes

2023, Vol. 13, No. 2 e-ISSN: 1390-8146

Published by: Universidad Técnica Luis Vargas Torres

How to cite this article (APA):

Guerrero, M., Bajaña, O., Aguirre, J., Jurado, R. (2023) Didactic proposal

for determining the density of a liquid, Revista Científica Interdisciplinaria Investigación y Saberes, 13(2)

55-65

Didactic proposal for determining the density of a liquid

Propuesta didáctica para determinar la densidad de un líquido

Marcos Francisco Guerrero Zambrano

Magister En Enseñanza De La Fisica, Universidad Estatal de Milagro, mguerreroz@unemi.edu.ec

https://orcid.org/0000-0002-5617-6836

Oscar Alonso Bajaña Calle

Magister En Educación Mención Enseñanza De La Matemática, Unidad Educativa Dr Miguel Encalada

Mora, alonso.bajana@educacion.gob.ec, https://orcid.org/0000-0001-7618-8968

Juan Patricio Aguirre Mateus

Magister En Enseñanza De La Fisica, Universidad Estatal de Milagro, jaguirrem1@unemi.edu.ec

https://orcid.org/0000-0003-1245-0925

Rodrigo Salomón Jurado Echeverría

Magister En Educacion Mención En Enseñanza De La Matematica, State University of Milagro

rjuradoe@unemi.edu.ec, https://orcid.org/0009-0000-5464-4256

This article proposes an alternative didactic methodology to

determine the density of a liquid using the relationship between the

mass of a body and its submerged height. An experiment is

performed by placing a floating wooden cube in a transparent fish

tank with water. Copper coins of equal mass are added to increase

the total mass of the body, and the submerged height is measured

with a tape measure. Although no direct proportionality was found

between mass and submerged height according to the theoretical

model, a linear relationship was observed, influenced mainly by the

absorption of water in the wooden cube.

Keywords:

Mass, submerged height, density, alternative

methodology, theoretical model.

Abstract

Received 2022-07-23

Revised 2022-10-12

Published 2023-02-23

Corresponding Author

Marcos Francisco Guerrero

mguerreroz@unemi.edu.ec

Pages: 55-65

https://creativecommons.org/lic

enses/by-nc-sa/4.0/

Distributed under

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

Resumen

Este artículo propone una metodología didáctica alternativa para

determinar la densidad de un líquido utilizando la relación entre la

masa de un cuerpo y su altura sumergida. Se realiza un experimento

colocando un cubo de madera flotante en una pecera transparente

con agua. Se agregan monedas de cobre de igual masa para

aumentar la masa total del cuerpo, y se mide la altura sumergida con

una cinta métrica. Aunque no se encontró una proporcionalidad

directa entre la masa y la altura sumergida según el modelo teórico,

se observó una relación lineal, influenciada principalmente por la

absorción de agua en el cubo de madera.

Palabras clave:

Masa, altura sumergida, densidad, metodología

alternativa, modelo teórico.

Introduction

Research into the factors that influence the relationship between the

mass of a body and its submerged height has made significant

advances in recent years. The interest in understanding the forces

acting on water and how we can harness them to float without

difficulty has driven numerous studies in this area.

Questions about why bodies sink and how much they sink have

aroused the curiosity of scientists and have prompted extensive

research. The main objective of these studies is to reveal the different

factors that influence the buoyancy of bodies, including density,

buoyancy, submerged volume, mass, submerged height and others.

These variables have a relationship in several sectors or areas of

application, for example: at the University of Costa Rica and the

National Metrology Center of Mexico (CENAM), an investigation was

carried out to determine the incidence of these variables in the

measurement of the density of liquids using the hydrostatic weighing

method. In this study, it was possible to obtain the density of a

standard solid without the need to know its volume. This

methodology represents an important advance in the accurate

determination of density (Hernández Sánchez, 2014) (Centeno et al.,

2004). In addition, at the National Institute of Metrology Research of

the Republic of Cuba (INIMET), during the calibration process of

density meters, solutions with different density values are prepared.

These solutions are essential to verify the accuracy of the measuring

instruments used in the determination of density (Valdivia-Medina et

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

al., 2001). (Valdivia-Medina et al., 2010).. Another study conducted at

the National University of Colombia, performed density

measurements of water and homeopathic medicines using a Vibrating

Tube Density Meter. This approach has allowed obtaining accurate

and reliable results in the determination of the density of various

substances. (Pineda Garcia, 2010). In the field of agronomy, a study

was carried out in the Scientific Journal UDO Agrícola where three

methods were compared to determine bulk density and solidity in

soils. The methods evaluated included the use of hydrometers, the

Uhland method with free fall and the Uhland method with forced fall.

These methods provided valuable information on the density and

solidity of soils, which is essential for crop analysis and planning.

(Hossne García & Cedeño Campos, 2012)..

These advances in research have contributed significantly to our

understanding of the factors that influence the relationship between

the mass of a body and its submerged height. More accurate

measurement methods and tools have been developed, allowing us

to obtain reliable results in determining the density of liquids and

other materials.

Regarding didactic proposals to determine the density of different

bodies, the author Fuertes (2016), describes some methods such as:

hydrostatic, geometric and margin of indeterminacy, however, there

is limited research regarding didactic proposals for determining the

density of a liquid, this is because, currently many teachers use other

types of resources for teaching science such as videos and

simulations, and not promoting face-to-face experimentation.

Nowadays, experimentation in the learning process is important for

the student. (Fuentes, 2016).

Due to the importance of the aforementioned research, emphasis

should be placed in the educational context and especially in science

teaching, where these variables sometimes teachers focus more on

the definition than on the structuring and deepening of a concept that

demonstrates the relationship between the physical phenomenon

and the variable. The latter is evident with the density variable, which

is one of the most misunderstood concepts despite the fact that it is

addressed from the early stages of the study of the physical sciences

(Martínez-Borreguero et al., 2018)..

In the teaching of physical sciences, the problem of how to transmit

knowledge to students has always arisen, for which methodologies

must be structured to allow the student to develop skills from

experience.

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

In conclusion, a remarkable process has been observed in many

investigations on how to determine the density of a fluid or a solid

using different methods, one more complex than others, however, for

this research the didactic methodological process is promoted for

student learning through experimentation, for this it is proposed to

determine the density of a liquid based on the relationship the mass

of the submerged body in terms of submerged looseness.

THEORETICAL FRAMEWORK

Archimedes of Syracuse mentioned "Every body immersed in a fluid

experiences an upward force called thrust, equal to the weight of the

fluid displaced by the body" (Falco, Franceschelli, & Marco, 2001).

We apply this principle when we place an object in water; the body

sinks if its weight is greater than the weight of the displaced fluid,

while the object floats when its weight is less than or equal to the

weight of the displaced fluid (Falco, Franceschelli, & Marco, 2001).

Considering the experimental didactic proposal, two forces act on the

object, the thrust force and the gravitational force. !

% and the

gravitational force !

% both measured in Newtons, which in this

situation have equal magnitude and opposite direction, as shown in

Figure 1.

Vector description of the forces acting on the object.

Prepared by the author.

Since the submerged object is in equilibrium, therefore, the sum of

forces measured in Newton will be zero, as shown in equation 1. !

(

measured in Newton will be zero, as shown in equation 1.

(

(Equation 1)

Considering the positive upward reference system we have equation

two:

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

(Equation 2)

Where, the thrust is equal to the product of the density of the fluid

measured in grams per cubic centimeter, the acceleration of

gravity

measured in meters per square second and the volume

occupied by the object in the fluid measured in cubic centimeters.

measured in cubic centimeters. In the case of weight it is equal to the

product of the mass of the object measured in grams and the

acceleration of gravity measured in meters per second squared.

measured in grams and the acceleration of gravity. Therefore,

replacing the above in equation 2, we have equation 3, as shown

below:

-2/20+1/)*

(Equation 3)

Since the object to be considered is a cubic block of wood, the

volume occupied by the object in the fluid is equal to the product of

the area of the base

measured in centimeters squared and the

submerged height

measured in centimeters. Therefore, replacing

the above in equation 3, we have equation 4, as shown below:

-2/2324+1/)*

(Equation 4)

Now clearing the submerged height and simplifying the acceleration

of gravity from equation 4 gives equation 5, as shown below:

"#$

(Equation 5)

After having cleared the formula, it is observed that the submerged

height is directly proportional to the mass of the object, therefore, if

one of them increases in magnitude, the other also increases in the

same proportion; which allows to obtain 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 submerged height, in the horizontal axis

goes the mass of the body and the slope of the graph would be the

inverse of the product of the density by the area. Then, our didactic

experimental proposal consists of increasing the mass of the object

by placing from one to seven identical coins and then observing the

submerged height of the object.

Determine the relationship between the submerged height and the

mass of the object including coins by means of a two-dimensional

graph to calculate the density of water from its slope.

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

Table 1.

Experimental variables.

Independent

Dependent

Controlled

Uncontrolled

• Mass of

the object

including

coins

• Submerged

height of the

object.

• Water density

• Me the coins.

• Temperature of

the agasas of

each d

• and the

environment.

• Friction with

water.

• Water

absorption in

the object.

How does the submerged height of the object affect when the mass

of the object is varied with coins?

In this didactic experimental proposal, the submerged height is

expected to be directly proportional to the mass of the object

including the coins, i.e., if the mass of the object using the coins

increases, the height at which the object sinks into the fluid will

increase proportionally with the same amount.

LIST OF MATERIALS AND EQUIPMENT

§ A block-shaped wooden object, measuring (12, 10 ± 0.01) cm long,

(8, 10 ± 0.01) cm wide and (3.50 ± 0.01) cm high with a mass of

(137.00± 0.01) g

§ A transparent fish tank measuring (23.00 ± 0.01) cm long, (1.40 ± 0.01)

cm wide and (3.50 ± 0.01) cm high.

§ Measuring tape (17.00 ± 0.01) cm

§ copper coins, each with a mass of (15,00 ±0,01) g

§ A balance of precision ±0.01g

§ 3.75 liters of water for the fish tank.

Figure 2.

Measurement of the mass of the wood block including

coins. Prepared by the author.

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

Figure 3.

Fish tank with the wooden block including the coins with

the tape measure attached to take the data. Prepared by the author.

Methodology

Initially with the wooden block, first we measure the length and width

of its base to obtain its area in square centimeters, second we

measure the mass in grams by means of a balance and finally we cut

the tape measure and stick it on the wooden block, in such a way that

the 0 of the tape measure is at the lower edge as shown in figure 3.

Now place 3.75 liters of water inside the transparent fish tank, and

submerge the wooden block to obtain the submerged height

measured in centimeters. Then a coin is placed on the block and its

combined mass is measured again in grams, and finally the new

submerged height is measured. Then another coin is added to the

block of wood and the new mass is measured again to measure the

submerged height, and the process is repeated until the seven coins

are completed. For each mass value, 4 submerged heights will be

measured from the tape measure attached to the block.

Results

The raw data obtained in the experiment are shown below.

Table 2.

Raw data of block mass including coin and their submerged

heights.

Mass of the block

including coin m/g

± Δm = 0, 01 g

Submerged height of the block

h/cm

± Δh = ±0,01cm

137, 00

1, 38

1, 40

1, 42

1, 45

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

152, 00

1, 60

1, 61

1, 65

167, 00

1, 81

1, 73

1, 77

1, 69

182, 00

1, 85

1, 87

1, 86

1, 89

197, 00

1, 99

2, 02

1, 97

2, 01

212, 00

2, 11

2, 18

2, 14

2, 16

227, 00

2, 34

2, 30

2, 31

242, 00

2, 45

2, 48

2, 39

2, 42

When performing the analysis of the submerged height

measurements, it can be observed that there are groups of data for

the same mass that have low accuracy and other groups of data for

the same mass that have high accuracy, which is why it was decided

to select from each group of masses, the 3 most accurate submerged

heights. In the case of the first group of submerged heights data, we

will obtain the mean and the uncertainty, as shown in equations 6 and

7 as follows:

)

%&'()*%&+,)*)%&+-)

)678*9:;

(Equation 6)

<4)

%&+-.)%&'(

)*7*=9:;

(Equation 7)

This process will be repeated with each of the groups of submerged

heights found in the data table. The table of processed values is

shown below:

Table 3.

Processed data of block mass including coins and average

submerged height, with their respective uncertainties.

Mass m/g

±∆m=± 0.01 g

Average height

ℎ

/cm

Uncertainty of

average height

±∆h/cm

137

1,40

0,02

152

1,61

0,01

167

1,73

0,04

182

1,86

0,01

197

2,01

0,02

212

2,16

0,02

227

2,30

0,01

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:

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

Figure 4.

Plot of the mass curve of the block including coins and

average submerged height, with its uncertainties, the line of best fit,

the line of maximum and minimum slope.

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

vertical axis are too small with respect to the scale used, which is why

they are not visualized in the graph. 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 inverse of the

product of the submerged cross-sectional area of the wood block with

the density of the water, therefore, equation 8 is obtained:

"#$

(Equation 8)

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

>)*7**?@689A

2:;

From equation 8, we clear the density and we obtain equation 9:

/#$

(Equation 9)

Now the value of slope and submerged area are replaced in equation

9 to obtain the density value:

*7**?@689A

2:;

?B7*6:;

)67*C*9A2:;

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

uncertainty is obtained.

in this case the value is:

<>)*7***6?B9A:;

Now propagating errors from equation 9 we obtain equation 10:

)

(Equation 10)

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

By subtracting the uncertainty of the density from equation 10,

equation 11 is obtained:

<-)-

F (Equation 10)

Then replacing values in equation 10 we obtain:

G-)67*C*9A:;

*7***6?B9/I1

*7**?@689/I1

*7**C9I1

?B7*69I1

)*7*=68A:;

Therefore, the density of water with its uncertainty is

-),67*C*9K*7*=68.A:;

Conclusions

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 demonstrated that the relationship between the mass of the

wood block including the coins and its height submerged in the water

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 variation in the mass of the

wood block due to the absorption of water. In our case, the wood

block, having retained water, increased its mass causing the

submerged height to be affected. Based on graph 1, it can be seen

that the correlation coefficient (R2) is 0.9988, which means that there

is a strong linear relationship between the mass of the block and its

submerged height.

The value obtained for the density of water using the didactic

methodological proposal was (1.050±0.021) g.cm-3, a value very

close to that found in the Workshop in Hydrometer Calibration, where

the density of water, under normal conditions is (0.998±0.010) g.cm-

3. It can also be observed that its accuracy is low, since the percentage

of error was 5.210 %, additionally the range of values obtained

experimentally presents a range quite close to the range of the

theoretical value, therefore, it means that the value obtained

experimentally has a low accuracy. The variation of the mass of the

body by absorption of water appeared as a random error that caused

disformity in the measurements and a low precision at the beginning

of the experiment.

The influence of readability errors in the measurements due to the

refraction of light on the face of the tank is also observed, thus

Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2

showing a deviation in the measurements as a systematic error of the

experimentation.

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Didactic proposal to determine the density of a liquid.

Revista Científica Interdisciplinaria Investigación y Saberes , / 2023/ , Vol. 13, No. 2