Chapter 14 SYSTEMS OF PARTICLES

The effective force of a particle Pi of a given system is the product miai of its mass mi and its acceleration ai with respect to a newtonian frame of reference centered at O. The system of the external forces acting on the particles and the system of the effective forces of the particles are equipollent; i.e., both systems have the same resultant and the same moment resultant about O :

Fi =miaii =1

n

i =1

n

(ri x Fi ) =(ri x miai)i =1

n

i =1

n

Sharif University-Aerospace Dep. Fall 2004

The linear momentum L and the angular momentum Ho about point O are defined as

L =mivii =1

n

Ho =(ri x mivi)i =1

n

It can be shown that

F = L Mo = Ho

. .

This expresses that the resultant and the moment resultant about O of the external forces are, respectively, equal to the rates of change of the linear momentum and of the angular momentum about O of the system of particles.

The mass center G of a system of particles is defined by a position vector r which satisfies the equation

mr =mirii =1

n

where m represents the total mass mi. Differentiating both

F = ma

.

i =1

n

members twice with respect to t, we obtain

L = mv L = ma.

where v and a are the velocity and acceleration of the mass center G. Since F = L, we obtain

Therefore, the mass center of a system of particles moves as if the entire mass of the system and all the external forces were concentrated at that point.

x

y

z

Ox’

y’

z’

Gr’i Pi

miv’i Consider the motion of the particles of a system with respect to a centroidal frame Gx’y’z’ attached to the mass center G of the system and in translation with respect to the newtonian frame Oxyz. The angular momentum of the system about its mass center G is defined as the

sum of the moments about G of the momenta miv’i of the particles in their motion relative to the frame Gx’y’z’. The same result is obtained by considering the moments about G of the momenta mivi of the particles in their absolute motion. Therefore

HG = (r’i x mivi) = (r’i x miv’i) i =1

n

i =1

n

miv’i

x

y

z

Ox’

y’

z’

Gr’i Pi

HG = (r’i x mivi) i =1

n

i =1

n= (r’i x miv’i)

We can derive the relation

MG = HG .

which expresses that the moment resultant about G of the external forces is equal to the rate of change of the angular momentum about G of the system of particles.

When no external force acts on a system of particles, the linear momentum L and the angular momentum Ho of the system are conserved. In problems involving central forces, the angular momentum of the system about the center of force O will also be conserved.

miv’i The kinetic energy T of a system of particles is defined as the sum of the kinetic energies of the particles.

T = mivi2

i = 1

n12

x

y

z

Ox’

y’

z’

Gr’i Pi

Using the centroidal reference frame Gx’y’z’ we note that the

kinetic energy of the system can also be obtained by adding the kinetic energy mv2 associated with the motion of the mass center G and the kinetic energy of the system in its motion relative to the frame Gx’y’z’ :

12

T = mv 2 + miv’ii = 1

n12

212

miv’i

x

y

z

Ox’

y’

z’

Gr’i Pi

T = mv 2 + miv’ii = 1

n12

212

The principle of work and energy can be applied to a system of particles as well as to individual particles

T1 + U1 2 = T2

where U1 2 represents the work of all the forces acting on the particles of the system, internal and external.

If all the forces acting on the particles of the system are conservative, the principle of conservation of energy can be applied to the system of particles

T1 + V1 = T2 + V2

x

y

O

(mAvA)1

(mBvB)1

(mCvC)1

x

y

O

(mAvA)2

(mBvB)2

(mCvC)2

x

y

t1

t2

Fdt

t1

t2

MOdtO

The principle of impulse and momentum for a system of particles can be expressed graphically as shown above. The momenta of the particles at time t1 and the impulses of the external forces from t1 to t2 form a system of vectors equipollent to the system of the momenta of the particles at time t2 .

x

y

O

(mAvA)1

(mBvB)1

(mCvC)1

x

y

O

(mAvA)2

(mBvB)2

(mCvC)2

If no external forces act on the system of particles, the systems of momenta shown above are equipollent and we have

L1 = L2 (HO)1 = (HO)2

Many problems involving the motion of systems of particles can be solved by applying simultaneously the principle of impulse and momentum and the principle of conservation of energy or by expressing that the linear momentum, angular momentum, and energy of the system are conserved.

(m)vA

mivi

AS

B

(m)vB

mivi

AS

B

For variable systems of particles, first consider a steady stream of particles, such as a stream of water diverted by a fixed vane or the flow of air through a jet engine. The principle of impulse and momentum is applied to a system S of particles during a time interval t, including particles which enter the system at A during that time interval and those (of the same mass m) which leave the system at B. The system formed by the momentum (m)vA of the particles entering S in the time t and the impulses of the forces exerted on S during that time is equipollent to the momentum (m)vB of the particles leaving S in the same time t.

S F t

M t

(m)vA

mivi

AS

B

(m)vB

mivi

AS

B

Equating the x components, y components, and moments about a fixed point of the vectors involved, we could obtain as many as three equations, which could be solved for the desired unknowns. From this result, we can derive the expression

F = (vB - vA)dmdt

where vB - vA represents the difference between the vectors vB and vA and where dm/dt is the mass rate of flow of the stream.

S F t

M t

Consider a system of particles gaining mass by continually absorbing particles or losing mass by continually expellingparticles (as in the case of a rocket). Applying the principle of impulse and momentum to the system during a time interval t, we take care to include particles gained or lost during the time interval. The action on a system S of the particles being absorbed by S is equivalent to a thrust

v

mv

m

va

(m) va

u = va - v

m

SS

F t

S

(m + m)

(m + m)(v + v)

P = udmdt

P = udmdt

where dm/dt is the rate at which mass is being absorbed, and u is the velocity of the particles relative to S. In the case of particles being expelled by S , the rate dm/dt is negative and P is in a direction opposite to that in which particles are being expelled.

v

mv

m

va

(m) va

u = va - v

m

SS

F t

S

(m + m)

(m + m)(v + v)