Teodorescu P.P. Mechanical Systems, Classical Models Volume II: Mechanics of Discrete and Continuous Systems

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/Ufmmr=− ) being, in this case, given by

1212

mm r=Fr,

r = r , where f

is the constant of universal attraction. The equations of motion are

11 12

()

=vr

22 12

()

=−vr

(13.2.25)

Using the theorems of momentum and moment of momentum in the form (13.2.8''') and

in the form (13.2.15''), respectively, and observing that the external forces vanish

(

=R0,

′

=M0), we can state conservation theorems of momentum and of moment

of momentum, respectively; there result the first integrals

11 22 1

mm+=vvC, (13.2.26)

111222 2

() ()mm×+× =rvr vC,

(13.2.26')

where

,const=





CC . We can write the identity

1221 122221122

()( ) ( )mm m m m m m m−= + − +vv v v v.

Observing that

2112

−=



vvr, taking into account (13.2.26) and eliminating the

momentum

m v between this relation and the second equation (13.2.25), we obtain

the vector equation

12 12 1

()

ttm

⎛⎞

++ =

⎜⎟

⎝⎠



rrC0,

(13.2.27)

where

12 1 2

/( )mmm m m=+ is the reduced mass; this equation characterizes the

motion of the particle

with respect to the particle

. Analogously, we can write

21 21 1

()

ttm

⎛⎞

++ =

⎜⎟

⎝⎠



rrC0

(13.2.27')

too, this last equation characterizing the motion of the particle

P with respect to the

particle

P . In particular, if the masses

m and

m are constant in time, we find again

the classical equations (8.1.14).

The case

n = 3 (the problem of three particles) has been considered in 1932 in the

same conditions, using the above ideas, due to I.I. Plăcinţeanu. Let thus be the particles

of position vectors

r , with respect to an inertial frame of reference, and variable

masses

m , 1, 2, 3i = ; the equations of motion are

()

ii ij

mf f

=+vrr

ij j i

r =−rr

ijki≠≠≠

,, 1,2,3ijk=

(13.2.28)

13 Other Considerations on Dynamics of Mechanical Systems

183

MECHANICAL SYSTEMS, CLASSICAL MODELS

As in the preceding case, we obtain the first integrals

11 22 33 1

mmm++ =vvvC, (13.2.29)

111222333 2

() () ( )mm m×+× +× =rvrvr vC,

(13.2.29')

where

,const=





CC . We can write the identities

1221 122221122

()( ) ( )mm m m m m m m−= + − +vv v v v,

1331 133331133

()( ) ( )mm m m m m m m−= + − +vv v v v;

taking into account (13.2.28), (13.2.29), introducing the mass

123

Mm m m=++ of

the discrete mechanical system and eliminating the momenta

m v and

m v , we

find, as in the case

2n = , the equations of motion of the particles

P and

P with

respect to the particle

P in the form

(

)

(

)

22312

212 13 12

mmmmm

tMtM

⎡⎤

−− +

⎣⎦



rrr

(

)

23 2

32 1

mm m

++ =rC 0

(

)

(

)

32313

313 12 13

mmmmm

tMtM

⎡⎤

−− +

⎣⎦



rrr

(

)

23 3

23 1

mm m

++ =rC 0

(13.2.30)

Summing these two equations, we find (we have

23 32

+=rr 0)

()

(

)

112131

2 12 3 13 12 13 1

12 13

mmmmmm

mm f f

tM tM

⎡⎤

++ + − =

⎣⎦



rr r rC 0

;

(13.2.30')

this equation, together with one of the equations (13.2.30), constitute a system of two

differential equations for the problem of three particles of variable mass. Unlike the

equations (13.2.30), the equation (13.2.30') contains only two unknown vector

functions (

12 12

()t=rr and

13 13

()t=rr); we notice that we can write the latter

equation in the form of two equations

(

)

(

)

12 12 2

12 12 1

()

mm mm m

tM tM

++ =



rrCC

(

)

(

)

13 13 3

13 13 1

()

mm mm m

tM tM

++ =−



rrC C

(13.2.31)

of the form (13.2.27), where the function

()tC remains to be determined. Subtracting

one equation (13.2.30) from the other and taking into account (13.2.31), we get (we

notice that

13 12 23

−=rr r)

184

(

)

23 23

()

mm mm



rrC

(13.2.31')

Thus, the differential equations of the problem will be (13.2.31), (13.2.31'), having a

more symmetrical form, by separation of variables.

In the case of an arbitrary number

n of particles, we use the equations of motion

(

ij ij j j

r ==−rrr)

()

ii i ij

mfm

∑

, ji≠ , 1,2,...,in= ;

(13.2.32)

we obtain the first integrals

∑

(13.2.33)

()

iii

×=

∑

rvC.

(13.2.33')

Let us make a change of function and of variable

() () ()

t ϕτ τ=r

, d()dttτψ=

and introduce the notations

d/dϕϕτ

′

, d/d

′

ρρ

1,2,...,in=

; we can

calculate

()

iii

ψϕ ϕ

′′

=+v

ρρ

()()

22 2

ii i i

ψϕ ψϕ ψϕ ψϕ ψϕ

′′ ′ ′ ′ ′′

=++ ++





ρρρ

The equations of motion (13.2.32) become (

ij j i

=−

ρρρ

)

()

[

]

()

[

]

22 2

ii i i i i i i

mm m m mψ ϕ ψϕ ψ ϕ ψϕ ψ ϕ ψϕ ψϕ

′′ ′ ′ ′′ ′ ′

+++ + ++



ρρ ρ

ρρ

∑

ij≠

1,2,...,in=

;

we impose the conditions

()2 0

ψϕϕϕ

′

() 0

ψϕϕϕ

′′′

which can be written also in the form

ln( ) 2 0

ϕϕ

′

ln( ) 0

ψϕ

′′

′

We notice that we must have

2/ / constϕϕ ϕ ϕ

′′′′

, which can take place only if

kϕ = , constk = , 0ϕϕ

′′′

; in this case, it results

mψμ= , const

μ = , and

we assume that

13 Other Considerations on Dynamics of Mechanical Systems

185

MECHANICAL SYSTEMS, CLASSICAL MODELS

() ()

mt mft=

()

= ,

const

m =

1,2,...,in=

(13.2.34)

The equations of motion (13.2.32) become (

()

τ=

ρρ

, ()

ij ij

τ=

ρρ

)

()

ii ij

k ρ

′′

∑

ρρ

, ji≠ , 1,2,...,in= ,

(13.2.32')

obtaining the form of classical equations, corresponding to constant masses. It results

′′

∑

0ρ

wherefrom

()

() ()

ii ii

mmMt

kf t kf t k

===+

∑∑

r Cρρ

so that (

ρ is the position vector of the centre C)

τ +

()

MM m

∑

(13.2.35)

the centre of mass having a rectilinear trajectory; we have

11d1

() d () ()

tMtMft Mt

′

== = =

vCCρ

(13.2.35')

the velocity

()

t=vv being constant only if () const

t = , and

C being the

constant of the first integral of the momentum (13.2.33). Hence, we can state that, in

case of a discrete mechanical system

S of masses having the same variation in time

(

() ()

mt mft= ,

const

m = , 1,2,...,in= ), the centre of mass C has a rectilinear

and non-uniform motion; the motion of the centre

C is uniform only in case of constant

masses.

13.3.2.3 Motion of an Artificial Celestial Body

Let be, for instance, an artificial celestial body

B, which is launched from the Earth E

and which moves away from our planet; this body can enter in the zone of attraction of

another body of the solar system, e.g., in the attraction zone of the Sun

S. We have thus

a problem of three particles (Earth

E, Sun S and artificial celestial body B); if the mass

m of the body B can be neglected with respect to the mass

m of the Earth

186

(

mm

) and to the mass

m of the Sun (

mm

), then we can assume some

approximations of computation.

To fix the ideas, we will consider the motion of the body

B on the straight line which

joins the centre of mass

E of the Earth to the centre of mass S of the Sun, assuming that

the two celestial bodies are fixed (Fig. 13.11); the equation of motion is of the form

()

BE BS

mm mm

mFF f f

xsx

=+=− +

−

(13.2.36)

where

s is the distance from the Earth to the Sun, while x is the abscissa of the centre of

mass of the body

B with respect to the Earth E, chosen as origin. Observing that

d/d d/dvtvvx= and integrating, we obtain the first integral of the mechanical

energy

Fig. 13.11 Motion of an artificial celestial body

(

)

vf h

xsx

=++

−

hv f

xsx

⎛⎞

=− +

⎜⎟

−

⎝⎠

(13.2.36')

where

(0)xx= ,

(0)vv= , corresponding to the initial moment

0t =

. From

(13.2.36) one observes that for

x sufficiently small we have 0v <



, hence the velocity

decreases till the body

B reaches a point Q, the abscissa of which is given by

/( ) /

QQ ES

xsx mm− =

⎡⎤

⎣⎦

, hence by

==≅



=

(13.2.37)

taking into account that

310

−

≅⋅ , hence

1.732 10

−

≅⋅ ; because

2.348 10sR=⋅, where

6.38 10 cmR =⋅ is the radius of the Earth, we obtain

3105

1.729 10 s 40.579 2.590 10 cm 2.59 10 km

−

≅⋅≅ ≅⋅ =⋅

13 Other Considerations on Dynamics of Mechanical Systems

187

MECHANICAL SYSTEMS, CLASSICAL MODELS

If the initial velocity

v is too small, the velocity of the body B can vanish before

reaching the point

Q; in this case, the velocity changes of sign, so that the body returns

on the Earth. Observing that the second cosmic velocity is given by

2822

2 / 1.249700 10 m / s

II E

vfmx=≅⋅, we can write the relation (13.2.36') also in

the form

22 2

11 1 1 1

vvxv

x x sxsx

⎡

⎛⎞⎤

=+ −+ −

⎜⎟

⎢

⎥

−−

⎣

⎝⎠⎦



;

(13.2.36'')

for numerical data, it results the approximate formula

22 2 22

0.951

QII

vv v vV=− =−

(13.2.38)

where

0.951

Vv= , hence 0.975 10.902km/s

Vv≅≅ . If

vV< , then the body

B does not reach Q, returning on the Earth, while if

vV>

, then the body B reaches

Q with a non-zero velocity 0

v > and passes through this position, continuing its way

towards the Sun with a monotone increasing velocity.

If, in particular,

vV= , then the body B reaches Q with a null velocity ( 0

v = );

the point

Q represents thus a position of equilibrium, namely a labile position of

equilibrium, because an arbitrary perturbation of the position of equilibrium (towards

or towards

S), moves away the body B from this position. The moment at which the

body

B reaches Q is given by

()

∫

;

(13.2.39)

observing that

xx= is a double root of () 0vx = , we can write

()

−

∫

(13.2.39')

too, where

f (x ) is a regular function in the neighbourhood of the point

xx= . The

integral (13.2.39') is improper, so that the time

t is infinite; the body B comes near to

the position

Q, but never reaches it.

Obviously, in the above modelling, the hypothesis of rectilinearity represents an

approximation; as well, we have assumed that the masses are constant. However, the

results thus obtained are useful from a qualitative point of view.

188

13.2.3 Continuous Mechanical Systems

One meets, frequently, interesting problems where one deals with continuous

mechanical systems of variable mass, e.g.: the problem of a captive balloon, the

problem of a glacier or of an iceberg the mass of which diminishes by melting, the

problem of an airplane which flies during snowfall (the mass of the snow-flakes is

added to the mass of the airplane) etc.; more difficult are the problems in which one

must take into account the deformability of the mechanical system. In what follows, we

consider two such problems: Cayley’s problem, which has merely a historical interest,

and the problem of the winch, which has a particular practical interest, using various

approximations of calculation.

13.2.3.1 Cayley’s Problem

In 1857, Cayley considered the problem of a heavy homogeneous chain, wrapped up on

the cylinder

C, at rest on a horizontal table; we assume that the chain falls along the

vertical, due to its own weight. Let be

x the abscissa of the movable end P of the chain,

the origin being taken at the level of the table, while the

Ox-axis is directed towards the

descendent vertical (Fig. 13.12); the equation of motion of this point is

()xxt= and,

approximating the whole chain by a particle of variable mass, we can use the theorem

of momentum in the form (13.2.8''') (the Levi-Civita case). We write thus

xx x

⎡⎛ ⎞ ⎤

⎜⎟

⎢⎥

⎣⎝ ⎠ ⎦



(13.2.40)

Fig. 13.12 Cayley’s problem

where

γ is the unit weight, while ()xxt=



is the velocity of the chain (the same for all

its elements). Observing that

d/d d/dtx x=



, we can write

d( )/dxx xx x gx=

wherefrom, by integration,

()

xx x=

 ,

the integration constant vanishing (we assume that the chain begins to wrap up from the

state of rest, so that

(0) 0x = , (0) 0x = ).

We deduce (

0x ≠ )

13 Other Considerations on Dynamics of Mechanical Systems

189

MECHANICAL SYSTEMS, CLASSICAL MODELS

so that

2/3/2xgt= ; hence,

()

xt t=

()

vt t=

()

at =

(13.2.40')

the motion of the elements of the chain being uniformly accelerated.

13.2.3.2 The Winch

The winch is a simple machine formed by a homogeneous cylinder of radius R, on

which is wrapped up an inextensible and non-torsionable homogeneous cable, of own

weight

γ; we assume that the winch rotates with an angular velocity ω around a

horizontal axle passing through

O, at the end of the cable being tied a weight m=Gg,

modelled as a particle, the equation of motion of which is

()xxt= (the Ox-axis is

along the descendent vertical; Fig. 13.13). The moment of momentum with respect to

the fixed pole

O is given at the moment t by (the phenomenon being unidimensional,

we consider only the non-zero component of the moment of momentum)

KI mRvKω

′

=+ +

Fig. 13.13 Winch

where

()

IIt= is the moment of inertia of the cylinder with respect to its axis at the

moment

t, ()Rmv is the moment of momentum of the weight G, while K

′

is the

moment of momentum of the unwrapped cable (the cable

′

), given by

KRv Rvx

γγ

′

∫

We notice that

190

() ()

It It xR

⎛⎞

=−

⎜⎟

⎝⎠

where

I is the moment of inertia of the cylinder on which is wrapped the cable (at the

initial moment

0t = ), while

(/)xgRγ is the moment of inertia of the cable which

was unwrapped; assuming that the unwrapping of the cable is without sliding friction,

we have

vRω= , so that it results

()

KImRω=+

(13.2.41)

as the whole cable would be wrapped on the cylinder.

Applying the theorem of moment of momentum in the form (13.2.15'') (case

considered by Levi-Civita), we can write

()

() ( )

ImR xRmgRMMωγ+=+++



(13.2.42)

where

M is the moment of the driving couple, while M is the moment of the friction

couple of the winch with the axle about which it rotates. Supposing that

sign 2M αωβω=− −

, ,0αβ> ,

(13.2.43)

we put into evidence the Coulombian friction (case

0β = ), as well as the

hydrodynamic friction (case

0α = , where a lubricant is used). The differential

equation of the motion (13.2.42) takes the form (

//vR xRω == ,

//vR xRω ==







)

2xaxbxc+−=  , ,, constabc= , ,0ab> , (13.2.44)

with the notations

ImR

(sign)

RmgR M

ImR

αω+−

(13.2.44')

We obtain thus (the winch begins to move at the initial moment with the weight

G in

the upper position, hence with

(0) 0x = , (0) 0x = )

() e ( cosh sinh ) 1

xt a at a at

−

⎡⎤

′′ ′

=+−

⎢⎥

′

⎣⎦

aab

′

=+,

(13.2.45)

as well as

() e sinh

xt at

−

′



(13.2.45')

13 Other Considerations on Dynamics of Mechanical Systems

191

MECHANICAL SYSTEMS, CLASSICAL MODELS

() e ( cosh sinh )

xt a at a at

−

′′ ′

=−

′



(13.2.45'')

all these quantities being positive for

t > 0. The motion of the winch is thus completely

determined.

192