Teodorescu P.P. Mechanical Systems, Classical Models Volume I: Particle Mechanics

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MECHANICAL SYSTEMS, CLASSICAL MODELS

2.1.3 Systems of units

The basic and derived units form a system of units. Two systems of units may differ

by the chosen basic quantities and by the units corresponding to these quantities. We

mention thus

the physical systems in which the basic quantities are the length (of unit

L ), the mass (of unit M ) and the time (of unit T ), and the technical systems to which

correspond the length,

the force (of unit

) and the time. Unlike the physical systems

of units, independent of the point in which we are on the surface of the Earth

(independent of the presence of a gravitational field), the technical systems of units

depend on the latitude of the place (depend on the presence of a gravitational field).

Among the physical systems we mention

the CGS system, in which the units are the

centimetre

, cm, the gram, g and the second, s, and the international system, SI, which

uses

the metre, m, the kilogram, kg and the second; in the latter physical system one

introduces also, as basic quantities:

the intensity of the electric current (ampere, A), the

thermodynamical temperature

(kelvin, K), the quantity of substance (mol, mol) and the

light intensity

(candle, cd). The SI system was adopted at the XIth International

Conference of Measures and Weights (Paris, 1960). The most used technical system is

the MKfS system, where one introduces the metre,

the kilogram force, kgf and the

second. We notice that in the SI system the unit of force is the newton, N (the necessary

force to induce to a mass of 1 kg an acceleration of

1 m/s in vacuum; 1 kgf ≅ 10 N).

One uses

the decimal system for space and mass, while for time remains the classical

sexagesimal system. Theoretically, the metre is defined as 1/40 000 000 of the length

of the Paris meridian. Practically, the metre is equal to

1 650 763.73 wave lengths of

the radiation which corresponds to the transition of the atom of krypton 86 between the

energy levels 2p

and 5d

in vacuum (Paris, 1960); the centimetre is the hundredth part

of the metre defined above. Before 1960, the metre was defined as the length, at a

temperature of 0

°C, of the international prototype in irradiate platinum, sanctioned by

the General Conference of Measures and Weights in 1889, preserved at the International

Bureau of Measures and Weights, at the pavilion of Breteuil (Sèvres, France). The

kilogram represents the mass of the international prototype in irradiate platinum

sanctioned at the same time and preserved in the same place as the metric prototype; the

gram is the thousandth part of the kilogram defined above. The mass of the mentioned

prototype represents theoretically the mass of a decimetre cube of distilled water at 4ºC,

at a pressure of an atmosphere; the weight of this prototype at

45º boreal latitude, at the

sea level, represents a kilogram force. Till 1960, the second was defined as the

86400th

part of the mean solar day, considered constant and defined with respect to the tropical

year (the mean interval between two consecutive passages of the Sun at the spring mean

equinox), admitted to be also constant. Observing that the latter one is not constant (it is

expressed by a polynomial formula with respect to time), after 1960 the tropical year,

which has

365.242 198 79 solar days or 31 556 925.974 7 seconds, corresponding to

1900, has been taken into consideration; the second thus defined is the ephemerides’

second. In the international system SI the second is defined with the aid of atomic

horologes; thus, a second is an interval of time equal to

9 192 631 770 oscillation

periods of radiations emitted by the transition between two hyperfine energy levels

(F=4, M

=0; F=3, M

=0) of the basic state (

1/2

) of the atom of caesium 133, in the

Newtonian model of mechanics

absence of perturbations of a magnetic field. The standard hour (implicitly the second)

is conserved by the International Bureau of the Hour in Paris.

The units of geometrical and mechanical quantities encountered in the physical

systems CGS and SI and in the technical system MKfS are given in Table 1.1. As well,

in Table 1.2 are given the decimal multiples and submultiples of these units, exception

the time, which has other units (second, minute, hour, day, year a.s.o). We notice that in

the SI the unit for pressure is the pascal (

1 Pa 1 N/m= ); as well, dyn, rad, Hz, erg, J,

W, are used for dyne, radian, hertz, erg, joule and watt, respectively.

Table 1.1

Dimensions in the

system

Units in the system

Quantity

Symbol

LMT LFT CGS SI MKfS

Length l L L cm m m

Mass m M L

-1

g kg

kgf⋅s

Time t T T s s s

Force F LMT

-2

F dyn N kgf

Area A L

Volume V L

Plane angle

,…,

1 1 rad rad rad

Period T T T s s s

Frequency f T

-1

Hz Hz Hz

Pulsation (circular

frequency)

-1

Angular velocity

-1

rad/s rad/s rad/s

Angular acceleration

-2

rad/s

Velocity

-1

cm/s m/s m/s

Acceleration a LT

-2

cm/s

m/s

Unit mass (density)

-3

M L

-1

g/cm

kg/m

kgf⋅s

Unit weight

-2

-3

F dyn/cm

N/m

kgf/m

Moment of inertia

of the mass

LFT

g⋅cm

kg⋅m

m⋅kgf⋅s

Pressure p L

-1

-2

F dyn/cm

N/m

kgf/m

Percussion (impulse

the force)

∫

dFt

LMT

-1

dyn⋅s

N⋅s

kgf⋅s

Moment of a force

(couple)

-2

dyn⋅cm

N⋅m

kgf⋅m

Impulse of the

moment of a force

∫

dMt

-1

LFT

dyn⋅cm⋅s

N⋅m⋅s

kgf⋅m⋅s

Momentum (linear

momentum)

LMT

-1

g⋅cm/s

kg⋅m/s

kgf⋅s

Moment of

momentum (angular

momentum)

-1

LFT

g⋅cm

kg⋅m

kgf⋅m⋅s

Work L L

-2

LF erg J

kgf⋅m

Energy (mechanical) E,T,V L

-2

LF erg J

kgf⋅m

Power (mechanical) P L

-3

LFT

-1

erg/s W

kgf⋅m/s

MECHANICAL SYSTEMS, CLASSICAL MODELS

The force is a derived quantity in case of a physical system of units; it is determined

by the basic law of mechanics (1.1.89), as well as by the law of universal (Newtonian)

attraction (1.1.84). The universal constant

has a value which depends univocally on

the arbitrary magnitude of the basic units; if we wish that the relation (1.1.84) be written

so as to have

= 1

, then the mass must no more be considered as a basic quantity,

being defined by means of length and acceleration. Hence, the unit of mass becomes a

derived unit, and can be expressed with the aid of units of length and time, the latter

ones remaining arbitrary, thus, the number of basic units is reduced with a unity (from

three to two). To reduce the number of basic units with a unity more, one must take into

consideration a relation between length and time, containing a universal constant (for

instance the velocity of light in vacuum); if in the considered system of units the second

remains the basic unit for time, then one obtains approximately

300 Mm as unit for

length. The unit of length becomes thus a derived unit. If we take, for instance, Planck’s

constant

h equal to unity, then we obtain a system of units without any basic unit. One

may imagine also other systems of units having this property.

Table 1.2

Submultiples Multiples

Prefix Symbol Multiplicative

factor

Prefix Symbol Multiplicative

factor

deci d 10

-1

deca da 10

centi c 10

-2

hecto h 10

mili m 10

-3

kilo k 10

micro

-6

mega M 10

nano n 10

-9

giga G 10

pico p 10

-12

tera T 10

femto f 10

-15

peta P 10

atto a 10

-18

exa E 10

2.2 Homogeneity

In what follows, after some general considerations and the introduction of

dimensional equations, we enounce the basic theorem (the theorem

Π) of dimensional

analysis; some important applications of the property of homogeneity will be also given.

2.2.1 General considerations

The laws of nature establish certain relations between various physical quantities,

i.e., between the numerical values

, ,...nn respectively; such a relation can be written

in the implicit form

(

)

,,... 0

(1.2.6)

or in the explicit form (emphasizing a numerical value)

(

)

,,...nnnϕ .

(1.2.6')

Obviously, instead to write such relations between numerical values, we may write

Newtonian model of mechanics

analogous relations between the corresponding physical magnitudes. The above

relations represent an objective truth. We may thus affirm that the laws of physical

phenomena are invariant to any change of units; in other words, if

′′

, ,...nn are new

numerical values of the same magnitudes, then we can write

(

)

′

,,... 0

nn .

This condition is fulfilled if an only if

(

)

(

)

′

12 12

, ,... , ,...

nn Cfnn ,

where

C is a constant which depends only on the ratios between the new and the old

basic quantities; but the above condition is verified only if relations of the form (1.2.6)

or (1.2.6') are homogeneous. Thus, the property of

homogeneity appears to be an

essential property in the dimensional analysis.

By a variation of the basic units takes place also a variation of the derived physical

unit; but one must express the variation of the numerical value of the derived physical

quantity as a function of the variation of the units used. A relation which expresses the

derived unit as a function of the basic units is called by Maxwell

the dimensional

equation

of the considered physical quantity. Birkhoff, Charcosset and others show that

the derived unit is expressed as a function of basic units by a

relation of monomial type,

fact considered also by Maxwell. Indeed, let be an arbitrary derived dimensional

quantity; let us suppose also that

y is a geometric quantity depending only on the

lengths

, ,...,

xx x (

(

)

, ,...,

yfxx x). Let us denote by y

′

the quantity

corresponding to the arguments

′

′′

,,...,

xx x. The numerical values of y and y

′

respectively, depend on the unit of the considered length. Let us make now the unit

times smaller; in this case, the ratio

(

)

()

(

)

()

12 1 2

, ,..., , ,...,

xx x f x x x

yfxx x fxx x

λλ λ

′

′′ ′′ ′

′

must not depend on the chosen scale for the basic units. It follows

()

(

)

()

′

′′

′′ ′

12 12

, ,..., , ,...,

xx x fxx x

fxxx fxxx

λλ λ λλ λ

(

)

()

(

)

()

′

()

λλ

ϕλ

;

hence, the ratio of numerical values of a derived geometric quantity, measured by

distinct units of length, depends only on their ratio. Noting that

()

ϕλ

(

)

()

ϕλ

MECHANICAL SYSTEMS, CLASSICAL MODELS

we obtain

(

)

()

⎛⎞

⎜⎟

⎝⎠

ϕλ λ

because for

′

121

xxλ ,

′

222

xxλ ,…,

′

xxλ we have

()

⎛⎞

′

⎜⎟

⎝⎠

⎛⎞

⎜⎟

′

⎝⎠

121

λλλ

λλ

;

differentiating with respect to

λ and taking

λλλ, we get

()

(

)

d11

() d

ϕλ

ϕλ λ λ λ

where

m is an integer or a rational number. It follows

()

ϕλ λ ,

(1.2.7)

where a multiplicative constant is taken equal to unity, because

=(1) 1ϕ . One obtains

thus the result enounced above.

2.2.2 Dimensional equations

If to a λ times variation of a basic unit does correspond a

times variation of the

derived unit, then we say that the derived unit has

the dimension m with respect to the

basic unit which caused the considered variation; thus, in a physical system, the

dimension

[

]

U of the derived unit U is given by the dimensional equation

[]

= LMTU

αβγ

(1.2.8)

where

α , β and γ are the dimensions corresponding to the basic units L, M and T.

We notice that, in such a way, one cannot determine the physical nature of a physical

quantity, but only the form in which the latter one depends on the basic units. Hence,

the dimensional equation of a physical quantity

A in the LMT system will be of the

form

[]

= LM TA

αβγ

;

(1.2.9)

such an equation is established on the basis of the equation of definition of the

considered quantity and of the relation connecting the given quantity to the chosen basic

quantities. If the numerical value of the quantity

A is equal to the product or to the

quotient of two quantities

B and C (

ABC or

/ABC) of dimensional

equations

Newtonian model of mechanics

[]

111

LMTB

αβγ

[]

222

LMTC

αβγ

then one obtains

=±

αα α

ββ β

γγ γ

;

if the numerical value of the quantity

A is a power n of the numerical value of the

quantity

B , then

nαα,

nββ,

nγγ.

In general, any physical quantity, which appears in mechanics, can be expressed by

derived units, given by relations of the form (1.2.8). In some practical problems one has

to do also with

the quantity of heat, the temperature etc.; as unit of the quantity of heat

one uses the calorie, while the temperature is measured in Celsius (eventually Réaumur

or Fahrenheit) degrees.

The measurement of various physical quantities must be made with the aid of the

most convenient units. There arises the problem of changing the units in the frame of a

chosen system of units; starting from the dimension

[

]

U of the unit of a derived

quantity, given by (1.2.8), we pass to the dimension

[]

()

′

= LMT

λμ τ

where

λ , μ , τ are numbers (non-dimensional), of the same derived quantity, which will

be thus given by

[]

′

=UU

αβγ

λμτ .

(1.2.8')

For example, 1 km/h = (10

)

(3600)

-1

m/s = (1/3.6) m/s.

A more general problem is that of changing the system of units; thus, a quantity

A is

expressed in the system LMT in the form (1.2.9), and in a new system XYZ, given by

[]

111

LMTX

αβγ

[]

222

LMTY

αβγ

[]

333

LMTZ

αβγ

in the form

[]

= XYZ

abc

A . Observing that, in both systems of units, the dimension of

the quantity

A is the same (

LMT XYZ

abcαβγ

), and taking into account the

properties mentioned above for the dimensional equations, we obtain the general

equations of transformation of the units in the form

123

abcαα α α

123

abcββ β β,

(1.2.10)

123

abcγγ γ γ.

The new system of units is correctly chosen if the determinant

MECHANICAL SYSTEMS, CLASSICAL MODELS

≡

≠

123

ααα

Δβββ

γγγ

;

(1.2.11)

for instance, if we pass from the system LMT to the system LFT for which [F] = LMT

-2

then we have (

= 1Δ )

=−a αβ,

b β ,

+2c βγ.

(1.2.12)

Let be a relation of the form

(

)

, ,..., 0

AA A

(1.2.13)

between

n physical quantities

iii

LMT

αβγ

1,2,...,in

; the matrix

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

...

αα α

ββ β

γγ γ

(1.2.14)

is called

the dimensional matrix of the considered relation. For instance, in case of the

above change of units, the dimensional matrix is

⎡⎤

−

⎢⎥

−

⎢⎥

−

⎢⎥

⎣⎦

123

αααα

ββββ

γγ γ γ

(1.2.15)

This matrix allows the study of the structure of the functional connections which can

take place between various physical quantities.

2.2.3 The Π theorem

The physical laws, obtained on theoretical or experimental way, represent certain

functional relations between various quantities, which characterize the phenomenon to

study; the system of units used has an influence on the numerical values of these

quantities, but it is not connected to the corresponding phenomenon. It follows that the

mentioned functional relations must have a certain particular structure. Let be a

dimensional quantity

A of the form

()

f , ,..., , ,...,

AAAAA A,

(1.2.16)

where

, ,...,

AA A are variable or constant independent dimensional quantities. We

suppose that the first

k quantities (

≤

kn) are dimensional independent (the

dimensional equation corresponding to such a quantity cannot be expressed as a

Newtonian model of mechanics

monomial with the aid of the dimensions of the other quantities); the index

k is less or

at the most equal to the number of basic units. In this case, the dimensional equations of

the quantities

++12

, , ,...,

AA A A are written with the aid of the dimensions of the

quantities

, ,...,

AA A, in the form

[]

[][]

[]

...

AA A A,

[]

[][]

[]

...

AAAA,…

[]

[][]

[]

...

AAA A.

Let us vary the units of the first

quantities

, ,...,

αα α times, respectively; the new

numerical values, in the new units, will be

′

111

AAα ,

′

222

AAα ,…,

′

kkk

AAα ,

′

...

AAαα α ,

′

...

AAαα α

,…,

′

...

AAαα α

It follows that

()

′

... , ,...,

Afaaaαα α

(

)

12 12

,..., , ... ,..., ...

pp qq

kk k

aa a aααααα ααα.

The function

is thus homogeneous with respect to the arbitrary scales

, ,...,

αα α;

to reduce the number of the arguments in the function

, we choose =

1/aα ,

1/aα ,…, = 1/

aα , supposing that the first k quantities do not vanish or do

not tend to infinity (the results may be applied also in these cases if the function

continuous for these values of the arguments). Noting that, by hypothesis, relation

(1.2.16) does not depend on the chosen units, one chooses a system of units with respect

to which the first

k physical quantities considered have constant numerical values,

equal to unity. The values

, , ,...,

−

ΠΠ Π

of the quantities

++12

, , ,...,

AA A A in

the new units, will be given by

Π=

...

aa a

Π=

...

aa a

,…,

−

Π=

...

aa a

(1.2.17)

where

, , ,...,

aa a a are the numerical values of the corresponding quantities in the

old units. We notice that the numerical values

, , ,...,

−

ΠΠ Π do not depend on the

initial chosen system of units; hence, they can be considered to be non-dimensional

quantities. In the new system of units, the relation (1.2.16) becomes

(

)

−

Π= Π Π

1,1,...,1, ,...,

(1.2.18)

Using the results given by Vaschy, Buckingham, Federman, Ryabouchinsky, Ehrenfest,

Afanasjeva, Bridgman etc., one may state

Theorem 1.2.1 (theorem Π). A relation of the form (1.2.16) between 1n + dimensional

physical quantities

, , ,...,

AA A A , which is independent of the chosen units, may be

MECHANICAL SYSTEMS, CLASSICAL MODELS

expressed as a relation of the form (1.2.18) between 1nk

− quantities

, , ,...,

nk−

ΠΠ Π Π , which are non-dimensional combinations of the 1n +

dimensional quantities

This theorem, often denominated as the

Buckingham theorem, is known as the

theorem of powers’ product

(explaining thus the denomination of the theorem), and is

the basic theorem of the dimensional analysis. We notice that the number

1nk+− of

the non-dimensional products, which appear in relation (1.2.18), is equal to the

difference between the number

of the physical quantities and the rank k of the

dimensional matrix of the considered relation. We can replace the system of non-

dimensional parameters

, ,...,

−

ΠΠ Π by another system of non-dimensional

parameters

, ,...,

nk−

′′ ′

ΠΠ Π

, depending on the first nk

−

parameters, the form of

function

changing in relation (1.2.18); but one cannot form more than nk−

combinations of independent powers.

It is thus seen that any physical relation between dimensional quantities can be

formulated as a relation between non-dimensional quantities; the necessity to apply the

dimensional analysis to problems of mechanical systems is put into evidence. If the

number of parameters characterizing the quantity

A is small, then the functional

dependence is much more simple. If

, that is if all the quantities

, ,...,

AA A

are independent from a dimensional point of view, then

...

ACAA A= ,

(1.2.19)

where

C is an non-dimensional constant, which is determined on a theoretical or

experimental way.

We know that one can choose the number of basic units in an arbitrary manner, but

their increasing number entails the introduction of supplementary physical constants,

while the number

1nk+− of the non-dimensional parameters remains constant; one

must judge, from case to case, if the enrichment of the number of information brought

by the increase of the number of units is useful or not.

2.2.4 Applications

The property of homogeneity, which must be verified by a physical law, leads to a

great number of applications. Thus, a non-homogeneous relation between physical

quantities cannot be correct; for instance, a relation of the form

sat

, where s is a

length,

a is an acceleration while t is the time, cannot take place. On the other hand,

the relation

2ARπ= , where A is the area of a circle of radius R is homogeneous,

but it is not correct from the point of view of the multiplicative numerical coefficient;

indeed, the condition of homogeneity is only a

necessary condition for the correctness

of a relation.

The property of homogeneity allows also to determine the nature of some physical

quantities. Let be, for instance, a relation of the form

satvts

++,

(1.2.20)

Newtonian model of mechanics

where

s is a length, while t is the time; noting that we must have

[]

/2at v t=

⎡⎤

⎣⎦

[

]

Ls==, it follows that a is an acceleration,

v is a velocity, and

s is a length.

Figure 1.19. Mathematical pendulum.

As well, with the aid of the property of homogeneity and of the Π theorem, we can

establish some formulae, abstraction making of certain multiplicative constants. Let be,

for instance, the case of

the mathematical pendulum (the motion on a circle, in a vertical

plane, of a heavy particle of mass

m , at the end of a thread of length l , fixed at the

fixed point

(Fig.1.19)); we denote by

the angle indicating the position of the

particle at the moment

t and by R the constraint force in the thread, the modulus of

which we suppose to be proportional to the modulus of the weight

mg of the particle.

We can choose as characteristic parameters the quantities

,, , ,tlmgθ , the last one

corresponding to the initial position of the particle. Hence, we write

()

,, , ,tlmgθθ θ= ,

(

)

,, , ,Rmgftlmgθ

(1.2.21)

where

θ and

are non-dimensional functions, which do not depend on the units; these

functions can be determined starting from the equations of the problem and from the

corresponding initial conditions.

Observing that

,,tlm are independent physical quantities from the dimensional

point of view, we can apply the

Π theorem, in which intervene 532−= non-

dimensional products, for instance

= and

/tglΠ= ; it follows that

(

)

θθθ=

(

)

Rmgf t

θ=

(1.2.21')

where

θΠ= , and /RmgΠ= , respectively. The formulae (1.2.21') show that the

position of the particle does not depend on the mass

m , while the tension in the thread

(the constraint force) is directly proportional to

m .

T is a characteristic interval of time, for instance the oscillation period, we may

write

()

,,,

Tlmg

ϕθ= ,

(1.2.22)