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Math Educ Res Appl, 2015(1), 1

Received: 2014-11-12

Accepted: 2015-02-06

Online published: 2015-05-25

DOI:10.15414/meraa.2015.01.01.18-22 Original paper

The centre of gravity in technical practice

Vladimír Matušek1, Eva Matušeková2

1 Slovak University of Agriculture, Faculty of Economics and Management, Department of

Mathematics, Tr. A. Hlinku 2, 949 76 Nitra, Slovak Republic

2 Slovak University of Agriculture, Faculty of Economics and Management, Department of

Languages, Tr. A. Hlinku 2, 949 76 Nitra, Slovak Republic

ABSTRACT

The aim of this paper is to show the different methods of determination of a position of a centre of

gravity in education: derivation of a formula for calculating the centre of gravity of a trapezoid and a

derivation of a formula for calculating the volume of a truncated cylinder using gravity. The centre of

gravity can be determined graphically, by calculation and experimentally. We use the calculation of

a position of the centre of gravity by means of applying mathematics in engineering branches.

KEYWORDS: centre of gravity, application, trapezoid, truncated cylinder

JEL CLASSIFICATION: I21, J25

INTRODUCTION

The concept of "a centre of mass" in the form of the "a centre of gravity" was first introduced

by the ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse. He

worked with simplified assumptions about gravity that amount to a uniform field, thus

arriving at the mathematical properties of what we now call the centre of mass [1].

Archimedes showed that the torque exerted on a lever by weights resting at various points

along the lever is the same as what it would be if all of the weights were moved to a single

point their centre of mass. In work on floating bodies he demonstrated that the orientation of a

floating object is the one that makes its centre of mass as low as possible. He developed

mathematical techniques for finding the centres of mass of objects of uniform density of

various well-defined shapes. Later mathematicians who developed the theory of the center of

mass include Pappus of Alexandria, Guido Ubaldi, Francesco Maurolico, Federico

Commandino, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin, John

Corresponding author: Vladimír Matušek, Slovak University of Agriculture, Faculty of Economics and

Management, Department of Mathematics, Tr. A. Hlinku 2, 949 76 Nitra, Slovak Republic

E-mail: vladimir.matusek@uniag.sk

Mathematics in Education, Research and Applications (MERAA), ISSN 2453-6881

Math Educ Res Appl, 2015(1), 1

Slovak University of Agriculture in Nitra :: Department of Mathematics, Faculty of Economics and Management :: 2015 19

Wallis, Louis Carré, Pierre Varignon, and Alexis Clairaut. Newton's second law is

reformulated with respect to the center of mass in Euler's first law

3

.

MATERIAL AND METHODS

The experimental determination of the center of mass of a body uses gravity forces on the

body and relies on the fact that in the parallel gravity field near the surface of the earth the

center of mass is the same as the center of gravity.

The center of mass of a body with an axis of symmetry and constant density must lie on this

axis. Thus, the center of mass of a circular cylinder of constant density has its center of mass

on the axis of the cylinder. In the same way, the center of mass of a spherically symmetric

body of constant density is at the center of the sphere. In general, for any symmetry of a body,

its center of mass will be a fixed point of that symmetry [2].

The term centre of gravity is introduced to pupils at elementary schools for the first time.

Further information about the problem they get at secondary schools and universities. Table 1

shows different methods of teaching centre of gravity in connection with the particular type of

schools.

Table 1 Centre of gravity at different type of schools description of the table

Type of school

Subject

Method

Achievement of results

elementary school

physics

Graphic

geometric average of symmetry

secondary school

physics

Graphic

plane system of forces

mathematics

Calculus

analytical geometry

university

SUA in Nitra

mathematics

Calculus

binary integral

statics

Graphic

plane system of forces

RESULTS AND DISCUSSION

Calculate the coordinates of the centre of gravity of an isosceles trapezoid

ABCD

shown in

the Fig. 1,

21212 ,,,,0,0,,0 ddDccCBaA

. Obviously, the centre of gravity lies

on the line segment

MN

, at a distance

CG

x

from the side

AB

.

A coordinate

T

y

can be found out easily, if we calculate the midpoint of

AB

or the midpoint

of

CD

. Next, we concentrate on the calculation of the coordinates

CG

x

.

In the calculation we use a double integral, therefore, we must determine the straight line of

BC

and

AD

.

We create them by applying the knowledge of an analytic geometry and subsequently we get

.:

,:

2

22

2

ax

vad

yAD

x

v

c

yBC

Next, the coordinate

CG

x

will be calculated in accordance with the formula

D

CG dxdyx

S

x1

.

Mathematics in Education, Research and Applications (MERAA), ISSN 2453-6881

Math Educ Res Appl, 2015(1), 1

Slovak University of Agriculture in Nitra :: Department of Mathematics, Faculty of Economics and Management :: 2015 20

Fig. 1 The coordinates of the centre of gravity of a isosceles trapezoid

ABCD

Then, an elementary area D is given by

.

0

2

222 ax

vad

yx

v

c

vx

The area S of a trapezoid can be calculated by a definite integral, so we get

,

2)( 222 vcda

S

then we have

D

vax

vad

x

v

c

CG dxdyx

vcda

dxdyx

S

x

0

222

2

22

2

)( 21

.

After adjusting we get

222

222 )(2

3cda cda

v

xCG

.

From the Fig.1 we can see that

.

22

2cdc

aa

After substituting the previously given and its substituent modification, we can derive

a coordinate

T

x

in the form of

.

)(3 )2( ca cav

xCG

The above consideration can be generalized:

The center of area (center of mass for a uniform lamina) lies along the line joining the

midpoints of the parallel sides, at a perpendicular distance x from the longer side a.

The situation is illustrated in the Fig. 2.

Mathematics in Education, Research and Applications (MERAA), ISSN 2453-6881

Math Educ Res Appl, 2015(1), 1

Slovak University of Agriculture in Nitra :: Department of Mathematics, Faculty of Economics and Management :: 2015 21

Fig. 2 Trapezoid

ABCD

- coordinate

CG

x

Source: our own

In terms of gravity and its applications, a task of calculating the volume of a truncated

cylinder seems to be really interesting. Let`s suppose that the cylindrical body has a projection

in the plane (x, y) and is bounded by a plane

0 dczbyax

from above. The cylindrical

body is shown in the Fig. 3.

Fig. 3 Truncated cylinder

Let`s express the plane

0 dczbyax

by the function

rqypxz

, where

c

d

r

c

b

q

c

a

p ,,

and calculate the volume of body

.)(),( dydxrdxdyyqdxxpdydxrqypxdydxyxfV

DDDDD

Mathematics in Education, Research and Applications (MERAA), ISSN 2453-6881

Math Educ Res Appl, 2015(1), 1

Slovak University of Agriculture in Nitra :: Department of Mathematics, Faculty of Economics and Management :: 2015 22

We adjust the given terms so that the formula calculating the centre of gravity of the shape

acts in the last expression.

,PrSqSpV xy

where

xy SS and

are the first moments of area with respect to the axis x (y) and P is

a volume of an elementary area D, that is the volume of the base of the cylinder.

After some further adjustments we get

dxdyrdxdyy

P

qdxx

P

pPV

DDD

11

.

Furthermore, from the definition of gravity and geometric shape, it is obvious that

CG

D

xdydxx

P

1

,

CG

D

ydxdyy

P

1

,

,Pdxdy

D

then

.

CGCGCGCGCG zPrqypxPPrqypxPV

CONCLUSIONS

Calculation of the solids centre of gravity plays a very important part in the educational

process not only at elementary and secondary schools, but predominantly at technical

universities. In this paper, we derive the formula for calculating the centre of gravity of a

trapezoid, if sizes of its sides are given, next we derived the formula to calculate the volume

of a truncated cylinder using gravity. It is important for teachers to have some kind of

experience when explaining students a centre of gravity in different subjects (mathematics,

physics, technical mathematics, statics, etc.)

REFERENCES

[1] Goldstein, H., Poole, Ch., Safko, J. (2001). Classical Mechanics. 3rd Edition. Addison Wesley.

[2] Miškin, A. (1975). Introductory Mathematics for Engineers. Oxford: Wiley.

[3] Wikipedia. (2014). Center of mass. [cit. 2014-10-14]. Retrieved from

http://en.wikipedia.org/wiki/Center_of_mass.

Reviewed by

1. Doc. Jozef Rédl, PhD., Department of Machine Design, Faculty of Engineering, Slovak University of

Agriculture in Nitra, Tr. A. Hlinku 2, 94976 Nitra

2. Doc. Pavol Findura, PhD., Department of Machines and Production Systems, Faculty of Enginnering, Slovak

University of Agriculture, Tr. A. Hlinku 2, 94976 Nitra