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Lecture 6.1 :
Conservation of Linear Momentum (C-Mom)
1. Recalls
2. Control Volume Motion
VS
Frame of Reference Motion
3. Conservation of Linear Momentum
1. C-Mom for A Moving/Deforming CV As Observed From An Observer in An Inertial Frame of
Reference (IFR)
1. Stationary IFR
2. Moving IFR (with respect to another IFR)
[Moving Frame of Reference (MFR) that moves at constant velocity with respect to
another IFR]
2. C-Mom for A Moving/Deforming CV As Observed From An Observer in A Translating Frame
of Reference (MFR) with Respect to IFR
4. Example:
Velocities in The Net Convection Efflux Term
5. C-Mass for A Moving/Deforming CV As Observed from An Observer in A Moving
Frame of Reference (MFR) with Respect to IFR
abj
1
Very Brief Summary of Important Points and Equations [1]
1.
C-Mom for A Moving/Deforming CV As Observed From An Observer in An Inertial Frame of Reference (IFR)

Stationary IFR

F


dPMV (t )
dt



dPCV (t )

dt

CS (t )

 
V ( V f / s  dA),

Force,
dm  dQ
 Momentum 
 Time 


Physical Laws

PV (t ) 
:
RTT

V
(

dV
)
,
V (t ) is MV (t ) or CV (t ),




V f / s  V f  Vs
V (t )

Moving IFR (with respect to another IFR)

F


 (t )
dPMV
dt



 (t )
dPCV

dt

CS (t )



V ( V f / s  dA),

Force,
dm  dQ
 Momentum 
 Time 


Physical Laws

PV (t ) 
:
RTT

V (t ) is MV (t ) or CV (t ),
 V ( dV ),



V f / s  V f  Vs
V (t )
2.
C-Mom for A Moving/Deforming CV As Observed From An Observer in A Translating Frame of Reference (MFR) with
Respect to IFR

abj

F


arf ( dV )

MV (t ) CV (t )
:




 (t )
dPCV

V ( V f / s  dA),

dt
CS (t )


PV (t ) 


V ( dV ),
V (t )
Force,
dm  dQ
V (t ) is MV (t ) or CV (t ),
 Momentum 
 Time 



 2
V f / s  V f  Vs
Very Brief Summary of Important Points and Equations [2]
3.
C-Mass for A Moving/Deforming CV As Observed from An Observer in A Moving Frame of Reference
(MFR) with Respect to IFR
C-Mass in MFR
0

:
dM MV (t )
dt
M V (t ) 

 dV ,
dM CV (t )

dt


 V f / s  dA
CS (t )
V (t ) is MV (t ) or CV (t )
V (t )
abj
3
Recall 1:
Motion is Relative (to A Frame of Reference)

V
Particle

V

x

x
y’
x’
y
Observer A

x

Velocity is relative:
B
Observer A in Frame A
Observer B in Frame B


aB
 dx
V 
dt
 dx 
V 
dt
Linear momentum is also relative:
Observer A in Frame A
abj
Observer B
Observer B in Frame B


P  mV


P  mV 
4
Recall 2:
Linear Momentum of A Particle VS of A Continuum Body
Continuum body
Particle m

V


P  mV

x
dm  dV
 
V ( x, t )
 
dP  Vdm


P   Vdm

x
y
y
Observer A
Observer A
x
x
 
dP  Vdm 


P  mV

P
MV (t )


Vdm
MV (t )


P  mV
 Particle
[ Mass  Velocity]
• Don’t get confused by the integral
expression.
 Continuum Body
 Conceptually, linear momentum is linear momentum.
[MassVelocity]
 Dimensionally, it must be
• Similar applies to other properties of
a continuum body, e.g., energy, etc.
 Hence, it is not much different from that of a particle; it is still
Linear Momentun  Mass  Velocity
[ Mass  Velocity]
 The difference is that different parts of a continuum body may have different velocity.
 The question simply becomes how we are going to sum all the parts to get the total.
abj
 
dP  Vdm


P


Vdm
MV (t )
[ Mass  Velocity]
5
Control Volume Motion VS
Frame of Reference Motion
CV (t  dt)
CV (t  dt)
CV (t )
CV (t )
y
y
Observer A
IFR
x
IFR
x
 Control volume and frame of reference are two different things.
 They need not have the same motion.
Motion of The Frames
 IFR
=
Inertial frame of reference. Observer A in IFR uses unprimed coordinates
 MFR
=
Moving frame of reference. This frame is moving relative to IFR.

x

Observer B in MFR uses primed coordinates x  .
Motion of CV
 In general, CV can be moving and deforming relative to both frames.
Example:
abj
A balloon jet (CV) launched in an airplane appears moving and
deforming to both observer B in the airplane (MFR) and observer A on
the ground (IFR).
6
Example:
Notation:
Control Volume Motion VS Frame of Reference Motion
Unprimed and Primed Quantities
 
Example:
A balloon jet (CV) launched in an
V ( x , t )
CV (t )
airplane appears moving and deforming relative to
CV (t  dt)
both observer B in the airplane (MFR) and observer A
on the ground (IFR).

x
 
V ( x, t )

x
y’
MFR
y
Observer B on a moving airplane
x’
Observer A
IFR

x
B

aB
Unprimed Quantity: Quantity that is defined and relative to the IFR.
 
V ( x, t )

aB
e.g.
Primed Quantity:
abj
e.g.
=
velocity field as observed and described from IFR
=
acceleration of the origin of MFR as observed from IFR
Quantity that is defined and relative to the MFR.
 
V ( x , t )
=
velocity field as observed and described from MFR
7
C-Mom for A Moving/Deforming CV As Observed from An Observer in IFR
1. Stationary IFR
2. Moving IFR (with respect to another IFR)
[Moving Frame of Reference (MFR) that moves at constant velocity with
respect to another IFR]
abj
8
Recall 3:
Newton’s Second Law
CV (t )
 
V ( x, t )
CV (t  dt)
Recall the coincident CV(t) and MV(t)
 
V ( x , t )

x
CV (t )

x
y
y
Observer A
IFR
y’
Observer B
Observer A
x

x
 
V ( x, t )
CV (t  dt)
IFR
x
MFR
x’
 
  


VB  0, aB  0; B , B  0
Newton’s Second Law for An Observer in IFR (IFR can be moving at constant velocity relative to another IFR)
•
 

V ( x , t ) [ P ] must be the velocity [and linear momentum] as observed from IFR.
• The IFR can be moving at constant velocity relative to another IFR, e.g., Case MFR of Observer B.
Observer A (IFR)
Observer B

F

F

(MFR which is also an IFR)
abj



dPMV (t )
,
dt

 (t )
dPMV
,
dt


 
N  P  mV ,   V , PMV (t ) 

 V (dV )
MV (t )




 (t ) 
N  P  mV ,   V , PMV

V
 (dV )
MV (t )
9
Recall the coincident CV(t) and MV(t)
 
V ( x , t )
 
V ( x, t )
CV (t  dt)
CV (t )
Observer A (IFR):

x
y

Observer B (MFR / IFR): PV (t ) 
Observer B
Observer A
IFR
MFR
x
x’
Observer A (IFR)
Observer B

F
(MFR which is also an IFR)
V (t )



dPMV (t )
,
dt

 (t )
dPMV
,
dt


 
N  P  mV ,   V , PMV (t ) 


 V (dV )
MV (t )







 (t ) 
N  P  mV ,   V , PMV
Time deriative of linear momentum
• The (same) MV(t) is subjected to the same net force
observed.

 V (dV )
MV (t )
• Both A and B use the same form of physical laws.



V ( dV )
 
  


VB  0, aB  0; B , B  0

F

F


V ( dV )
V (t )
y’

x

PV (t ) 

F


 dPMV (t ) dPMV
 (t ) 





dt
dt


regardless of from what frame the MV(t) is
• abj
However, A and B observe different velocity and linear momentum as shown in the box above.
10
C-Mom for A Moving/Deforming CV As Observed from An Observer in IFR
Recall the coincident CV(t) and MV(t)
CV (t )
CV (t  dt)

x
y
 
V ( x, t )
Observer A
IFR


C-Mom: N  P  mV ,
x

 V
[Force],
Physical Laws

F



Net external force

dPMV (t )
dt



Time rate of change
of linear momentum of MV (t )


dPCV (t )
dt




Time rate of change
of linear momentum of CV (t )
Momentum
Time

 
V
(

V

d
A
)
f /s
 

CS (t )
dm  dQ




,
Net convectionefflux
of linear momentum through CS (t )
RTT
:
abj

PV (t ) 

V
 ( dV ),
V (t )
V (t ) is MV (t ) or CV (t ),



V f / s  V f  Vs
11
C-Mom for A Moving/Deforming CV As Observed from An Observer in IFR
[Force],
Physical Laws

F



dPMV (t )
dt




Net external force

Time rate of change
of linear momentum of MV (t )

PV 
:

dPCV (t )
dt




V
 ( dV ),

 
V
(

V

d
A
)
f /s
 

CS ( t )
dm  dQ





Time rate of change
of linear momentum of CV (t )
Momentum
Time
Net convectionefflux
of linear momentum through CS ( t )
RTT



V f / s  V f  Vs
V (t ) is MV (t ) or CV (t ),
V (t )
Recall the coincident CV(t) and MV(t)
SPECIAL CASE:
Stationary and Non-Deforming CV in IFR


If the CV is stationary and non-deforming in IFR, we have Vs  0






Hence, V
f / s  V f  Vs  V f V

abj

F

and the C-Mom becomes

dPMV (t )
dt


dPCV (t )

dt

CS (t )
  
V ( V  dA)



dm  dQ
12
C-Mom for A Moving/Deforming CV As Observed from An Observer in A
Moving IFR
[MFR that moves at constant velocity wrt another IFR.]
 
V ( x , t )
 
V ( x, t )
CV (t  dt)
CV (t )

x

x
y
In MFR (moving IFR-B), we have



N  P  mV ,   V 

PV (t ) 
y’


V ( dV ),
V (t ) is MV (t ) or CV (t )
V (t )
Observer B
Observer A
IFR
x
Physical Laws:
RTT:
MFR
x’
 
  


VB  0, aB  0; B , B  0


 (t )
dPMV
F 
dt

 (t )
dPMV

dt
C-Mom:
Note:
RTT can be applied in any one frame of
reference so long as all the quantities in the RTT are
with respect to that frame of reference.

 (t )
dPCV

dt

CS (t )



V ( V f / s  dA),

dm  dQ
Physical Laws

F


Net external force


 (t )
dPMV
dt



Time rate of change
of linear momentum of MV (t )
[Force],


 (t )
dPCV
dt



Time rate of change
of linear momentum of CV (t )
abj
Recall the coincident CV(t) and MV(t)
RTT



V f / s  V f  Vs

Momentum
Time



V ( V f / s  dA)

CS (t )
  dQ
dm



Net convectionefflux
of linear momentum through CS (t )
13
C-Mom for A Moving/Deforming CV As Observed from An Observer in A
Moving IFR
[MFR that moves at constant velocity wrt another IFR.]
[Force],
Physical Laws

F



 (t )
dPMV
dt




Net external force

Time rate of change
of linear momentum of MV (t )

PV 
:

 (t )
dPCV
dt




Time rate of change
of linear momentum of CV (t )

V
 ( dV ),
Momentum
Time





V
(

V

d
A
)
f /s
 

CS ( t )
  dQ
dm


Net convectionefflux
of linear momentum through CS (t )
RTT



V f / s  V f  Vs
V (t ) is MV (t ) or CV (t ),
V (t )
Recall the coincident CV(t) and MV(t)
SPECIAL CASE:
Stationary and Non-Deforming CV in MFR


If the CV is stationary and non-deforming in MFR, we have Vs  0




 

Hence, V   V   V   V  V  and the C-Mom becomes
f /s
f
s
f

abj

F


 (t )
dPMV
dt


 (t )
dPCV

dt

CS (t )

 
V ( V   dA)


dm  dQ
14

F


and Free-Body Diagram (FBD) for the Coincident CV(t) and MV(t)
Net external force
Coincident CV(t) and MV(t)
CV(t)
MV(t)
Pressure p
2. Distributive Surface Force
Shear t
(in fluid part)
1.
Concentrated/Point Surface Force
Keys
FBD

Fi
Volume/Body Force


gdm  g ( dV )

F

1.
Recognize various types of forces.
2.
Know how to find the resultant of
various types of forces (e.g.,
pressure, etc.).
3.
Sum all the external forces.
Net Surface Force

 FS
1. Concentrated/Pointed Surface Force


F 


Fi
2. Distributive Surface Force in Fluid [Pressure p + Friction t ]
abj

FS 


FB
Net Volume/Body Force

mg 

 FB

g
 ( dV )
CV  MV
15
Recall: Past Example of RTT for Linear Momentum
Example 3: Finding The Time Rate of Change of Property N of an MV
By The Use of A Coincident CV and The RTT
Problem:
Given that the velocity field is steady and the flow is incompressible
1. state whether or not the time rate of change of the linear momenta Px and Py of the material
volume MV(t) that instantaneously coincides with the stationary and non-deforming control
volume CV shown below vanishes;
2. if not, state also
- whether they are positive or negative, and
- whether there should be the corresponding net force (Fx and Fy ) acting on the MV/CV,
and
- whether the corresponding net force is positive or negative.
abj
16
y
V2 = V1
x
V2 = V1
V1
V2 > V1
V1
V1
V2 = V1
V1
V2 = V1
(a)
V1
dPMV , x
dt
(b)
 0?
(yes/no)
If not,
dPMV , x
positive or negative
dt
Net Fx on CV? (yes/no)
If yes,
dPMV , y
If not,
dt
abj
q
 0?
(yes/no)
Net Fy on CV? (yes/no)
Fx
dPMV , y
positive or negative
positive or negative
dt
If yes,
Fy
positive or negative
17
Example:
abj
Cart with Guide Vane
18
C-Mom for A Moving/Deforming CV As Observed from An Observer
in A Translating Frame of Reference with Respect to IFR
abj
19
Some Issue in The Formulation of
C-Mom for A Moving/Deforming CV As Observed from An Observer
in A Translating Frame of Reference with Respect to IFR
Kinematics of Relative Motion
Physical Laws (IFR)


 dPMV
 (t )
dPMV (t )

 f
dt
dt


 dPMV (t )
 F  dt
 
V ( x, t )

 ????


???
 
V ( x , t )
RTT (MFR)






 (t )
dPMV (t ) dPCV




V ( V f / s  dA)

dt
dt
CS (t )

y

x
Observer A
IFR
abj
CV (t ), MV (t )

x
y’
dm  dQ
:
Observer B
x





V f / s  V f  Vs
x’
MFR
  

 
VB , aB  0; B ,  B  0
20


 dPMV
 (t )
dPMV (t )
 f 
dt
dt

Kinematics of Relative Motion:
Translating Reference Frame (RF) with Acceleration
 
V ( x , t )
 
a ( x , t )
 
  a ( x, t )
V ( x, t )

x
CV (t ), MV (t )

x
y
Position Vectors:
Velocity Vectors:
 
arf (a B )
 
Vrf (VB )
y’
Observer B
Observer A
IFR
x
MFR

rrf
x’
  

 
VB , aB  0; B , B  0
 

x  rrf  x 
A 
A 
A
A 

drrf

dx
d
dx 

V :

rrf  x  

;
dt
dt
dt
dt

A 
B 



drrf
dx  A dx  

:
Vrf :
, V :

B  0
dt
dt
dt
 

V  Vrf  V 

 A

A
A
A


d
V
dV
d
dV 

rf
a :

Vrf  V  

dt
dt
dt
dt



A
B

dVrf 
dV  A dV  


:
a rf :
, a :

B  0
dt
dt
dt
 

a  arf  a 



Acceleration Vectors:




abj

 ????



21
Kinematics of Relative Motion:
Relation between Linear Momenta of The Two Reference Frames
 
dP  V dm
 
dP  Vdm

x
CV (t ), MV (t )

x
y
y’
Observer B
MFR
Observer A
IFR
 
arf (a B )
 
Vrf (VB )
x’
  

 
VB , aB  0; B , B  0
x
Momentum for an identified mass [ MV(t) ] as observed in IFR-A:

PMV 

V
 dm 
MV (t )

V
dV 
 

MV (t )
dm
Momentum for an identified mass [ MV(t) ] as observed in MFR-B:

 
PMV


PMV

V
 dm 

V
dV 
 

MV (t )
MV (t )
dm



 Vdm 
Vrf  V  dm 


MV (t )
abj

PMV 

MV (t )



Vrf dm  PMV
MV (t )



Vrf dm 
MV (t )

 

V  Vrf  V 

V dm
MV (t )
22
Kinematics of Relative Motion:
Relation between Time Rates of Change of Linear Momenta of The Two
Reference Frames (Short Version.)
 
dP  V dm
 
dP  Vdm

x
CV (t ), MV (t )

y
x
 
arf (a B )
 
Vrf (VB )
y’
Observer B
Observer A
x
IFR
MFR
x’
  

 
VB , aB  0; B , B  0

 dP 

  d 

dPMV
d 
 
  Vrf dm  PMV
Vrf dm   MV


dt
dt MV (t )
dt
 dt MV (t )






 dP 
d
V
rf

dm   MV ;
System mass is independent of time.


dt
dt
Note: In some sense, this derivation is a little
 MV (t )

obscure; however, it serves our purpose for



the moment. Another line of approach is to

dPMV
dPMV

dVrf


arf dm 
;
use the volume integral.
a rf :
dt
dt
dt
MV (t )




abj
23
C-Mom for A Moving/Deforming CV As Observed from An Observer in A Translating Frame of
Reference with Respect to IFR
Coincident CV(t) and MV(t)
Pressure p
2. Distributive Surface Force
CV(t)
MV(t)
Shear t
(in fluid part)

1.
Concentrated/Point Surface Force Fi
FBD
Volume/Body Force


gdm  g ( dV )

 dPMV (t )
Newton’s Second Law of Motion:
 F  dt


 (t )
d
P
(
t
)
d
P

Relation between Linear Momenta:
MV

arf dm  MV
dt
dt

MV (t )




 (t )
d
P
(
t
)
d
P
MV
CV
RTT:

  V ( V f / s  dA)

dt
dt
CS (t )
  dQ
dm


 dPMV (t )
dP (t )

F


arf dm  MV
Thus, we have
dt
dt
MV (t )


 (t )
dPMV

[Force], Momentum
F
arf dm 
dt
MV (t ) CV (t )





Time
abj


F


arf ( dV )
MV (t ) CV (t )





 (t )
dPCV

V ( V f / s  dA),

dt
CS (t )

  dQ
dm

PV (t ) 


V ( dV )
V (t )
24
C-Mom for A Moving/Deforming CV As Observed from An Observer in A Translating Frame of
Reference with Respect to IFR

F



Net external force

 arf ( dV )
[Force],

 (t )
dPCV
dt




MV (t ) CV (t )

Time rate of change
of linear momentum of CV (t )

: PV 

V
 ( dV ),
Momentum
Time





V
(

V

d
A
)
f /s
 

CS (t )
  dQ
dm


Net convectionefflux
of linear momentum through CS (t )
V (t ) is MV (t ) or CV (t ),



V f / s  V f  Vs
V (t )
Recall the coincident CV(t) and MV(t)
SPECIAL CASE:
Stationary and Non-Deforming CV in MFR


If the CV is stationary and non-deforming in MFR, we have Vs  0




 

Hence, V   V   V   V  V  and the C-Mom becomes
f /s
f
s
f

abj

F 


arf ( dV )
MV (t ) CV (t )


 (t )
dPCV

dt

CS (t )

 

V ( V   dA)


dm  dQ
25
Special Case:


a rf  0 : Moving IFR, MFR that moves at constant velocity
with respect to another IFR

F 

0




 (t )
dPCV

f / s  dA),
 arf ( dV )  dt   V (V

MV (t ) CV (t )
CS (t )
dm  dQ

PV (t ) 

 V ( dV )
V (t )
In this case, the C-Mom reduces down to that of the moving IFR that we derived earlier.





 (t )
dPCV
f / s  dA)
 F  dt   V (V

CS (t )
dm  dQ
abj
26
Example:
Velocities in The Net Convection Efflux Term
Balloon jet in an airplane

CS (t )



V ( V f / s  dA),

CV (t )

V f / s



V f / s  V f  Vs

Vs

 V s
dm  dQ
 
 
V  V f
y’
Observer B on a moving airplane
y
Observer A
IFR

Vrf
MFR x’
x
 IFR/A sees (velocities wrt IFR/A)
 the fluid velocity (gas velocity) at the exit CS
 the velocity of the MFR/B (the airplane)
 
 
V Vf

Vrf
If the CV is stationary
and non-deforming in
 
MFR, we have Vs  0
 MFR/B sees (velocities wrt MFR/B)
 the fluid velocity (gas velocity) at the exit CS
 the velocity of the exit CS (exit control surface velocity)
 
 
V  V f

Vs
Hence,
 



 
V f / s  V f  Vs  V f V 
 An observer moving with the exit CS (not with MFR/B) sees (velocities wrt CS)
abj
 the fluid velocity (gas velocity) at the exit CS



V f / s  V f  Vs
 Vf / s  Vf  Vs 
27
C-Mass for A Moving/Deforming CV As Observed from An
Observer in A Moving Frame of Reference (MFR) with
Respect to IFR
abj
28
C-Mass for A Moving/Deforming CV As Observed from An Observer in A Moving
Frame of Reference (MFR) with Respect to IFR
dN MV (t )
dt 



SN

Source of Change in N of MV
dN CV (t )
dt 




Time rate of change
of N of MV (t )



  ( V f / s  dA)

dm  dQ


CS
Time rate of change
of N of CV (t )
Net convectionefflux
of N through CS (t )
Regardless of frame of reference (in classical mechanics), we have the physical law of conservation of mass
Physical Law:
RTT (in MFR)

0
N  M ,  1 :
C-Mass in MFR

0
:
dM MV (t )
dt
dM MV (t )
dt
dM MV (t )
dt
M V (t ) 
(for any frame of reference)
dM CV (t )


dt

 dV ,


 V f / s  dA
CS (t )
dM CV (t )

dt



 V f / s  dA
CS (t )
V (t ) is MV (t ) or CV (t )
V (t )
Note:
• Recognize also that
abj


V f / s  V f / s
.

• The same form of C-Mass – with the convection term written with the relative velocity V
f /s
valid for any frame of reference.


 V f  Vs
29
- is