lecture 6.1 : conservation of linear momentum (c-mom)
DESCRIPTION
Lecture 6.1 : Conservation of Linear Momentum (C-Mom). Recalls Control Volume Motion VS Frame of Reference Motion Conservation of Linear Momentum C-Mom for A Moving/Deforming CV As Observed From An Observer in An Inertial Frame of Reference (IFR) Stationary IFR - PowerPoint PPT PresentationTRANSCRIPT
abj 1
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 2
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
Moving IFR (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
sfsf
tV
V
tCSQdmd
sfCVMV
VVVtCVtMVtVdVVtP
Time
MomentumForceAdVV
dt
tPd
dt
tPdF
/
)(
)(
/
),(or)( is)(,)()(:
,,)()()(
Physical LawsRTT
sfsf
tV
V
tCSQdmd
sfCVMV
VVVtCVtMVtVdVVtP
Time
MomentumForceAdVV
dt
tPd
dt
tPdF
/
)(
)(
/
),(or)(is)(,)()(:
,,)()()(
RTTPhysical Laws
sfsf
tV
V
tCSQdmd
sfCV
tCVtMV
rf
VVVtCVtMVtVdVVtP
Time
MomentumForceAdVV
dt
tPddVaF
/
)(
)(
/
)()(
),(or)(is)(,)()(:
,,)()(
)(
abj 3
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
)(or)(is)(,)(:
)()(0
)(
)(
/
tCVtMVtVdVtM
AdVdt
tdM
dt
tdM
tV
V
tCS
sfCVMV
C-Mass in MFR
abj 4
Recall 1: Motion is Relative (to A Frame of Reference)
dt
xdV
Observer A in Frame A
dt
xdV
Observer B in Frame B
Velocity is relative:
VmP
Observer A in Frame A
VmP
Observer B in Frame B
Linear momentum is also relative:
x
x
x’
y’Observer B
Ba
B
V
V
x
yObserver A
Particle
abj 5
Recall 2: Linear Momentum of A Particle VS of A Continuum Body
Particle
Continuum Body
Conceptually, linear momentum is linear momentum.
Dimensionally, it must be
Hence, it is not much different from that of a particle; it is still
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.
][ VelocityMassVmP
Velocity] Mass[Velocity MassMomentunLinear
][)(
VelocityMassdmVPdmVPdtMV
][ VelocityMass
• Don’t get confused by the integral expression.
• Similar applies to other properties of a continuum body, e.g., energy, etc.
x
VmP
V
x
yObserver A
Particle m
VmP
)(tMV
dmVPdmVPd
dVdm
x
)(
),(
tMV
dmVP
dmVPd
txV
x
yObserver A
Continuum body
abj 6
Control Volume Motion VS Frame of Reference Motion
)(tCV
)( dttCV
IFRx
y
IFRx
yObserver A
x’y’ Observer B
MFR
)(tCV
)( dttCV
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.
Observer B in MFR uses primed coordinates .
Motion of CV
In general, CV can be moving and deforming relative to both frames.
Example: 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).
x
x
abj 7
Example: Control Volume Motion VS Frame of Reference Motion Notation: Unprimed and Primed Quantities
Example: A balloon jet (CV) launched in an
airplane appears moving and deforming relative to
both observer B in the airplane (MFR) and observer A
on the ground (IFR).
x
x
)(tCV
)( dttCV
MFR x’
y’
Observer B on a moving airplane
Ba
B
),( txV
),( txV
IFRx
yObserver A
Unprimed Quantity: Quantity that is defined and relative to the IFR.
e.g. = velocity field as observed and described from IFR
= acceleration of the origin of MFR as observed from IFR
Primed Quantity: Quantity that is defined and relative to the MFR.
e.g. = velocity field as observed and described from MFR
Ba
),( txV
),( txV
abj 8
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 9
Recall 3: Newton’s Second Law
Newton’s Second Law for An Observer in IFR (IFR can be moving at constant velocity relative to another IFR)
• 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.
][),( PtxV
)(
)()(,,,)(
tMV
MVMV dVVtPVVmPNdt
tPdF
Observer A (IFR)
)(
)()(,,,)(
tMV
MVMV dVVtPVVmPNdt
tPdF
Observer B
(MFR which is also an IFR)
x
)(tCV)( dttCV
),( txV
IFRx
yObserver A
x
)(tCV)( dttCV
),( txV
IFRx
yObserver A
MFRx’
y’
Observer B
0,;0,0
BBBB aV
x
),( txV Recall the coincident CV(t) and MV(t)
abj 10
x
)(tCV)( dttCV
),( txV
IFRx
yObserver A
MFRx’
y’
Observer B
0,;0,0
BBBB aV
x
),( txV
)(
)()(,,,)(
tMV
MVMV dVVtPVVmPNdt
tPdF
Observer A (IFR)
)(
)()(,,,)(
tMV
MVMV dVVtPVVmPNdt
tPdF
Observer B
(MFR which is also an IFR)
• Both A and B use the same form of physical laws.
• The (same) MV(t) is subjected to the same net force regardless of from what frame the MV(t)
is observed.
• However, A and B observe different velocity and linear momentum as shown in the box above.
dt
tPd
dt
tPdF MVMV )()(
momentumlinear of deriative Time
F
Observer A (IFR):
Observer B (MFR / IFR):
)(
)()(tV
V dVVtP
)(
)()(tV
V dVVtP
Recall the coincident CV(t) and MV(t)
abj 11
C-Mom for A Moving/Deforming CV As Observed from An Observer in IFR
x
)(tCV
)( dttCV
),( txV
IFRx
yObserver A
Recall the coincident CV(t) and MV(t)
C-Mom: VVmPN
,
sfsf
tV
V
tCS
tCSQdmd
sf
tCV
CV
tMV
MV
VVVtCVtMVtVdVVtP
AdVVdt
tPd
dt
tPdF
/
)(
)( through momentumlinear ofefflux convectionNet
)(
/
)( of momentumlinear of change of rate Time
)( of momentumlinear of change of rate Time
forceexternalNet
),(or)( is)(,)()(:
,)()()(
Physical Laws
RTT
Momentum
Time
[Force],
abj 12
C-Mom for A Moving/Deforming CV As Observed from An Observer in IFR
Recall the coincident CV(t) and MV(t)
sfsf
tV
V
tCS
tCSQdmd
sf
tCV
CV
tMV
MV
VVVtCVtMVtVdVVP
AdVVdt
tPd
dt
tPdF
/
)(
)( through momentumlinear ofefflux convectionNet
)(
/
)( of momentumlinear of change of rate Time
)( of momentumlinear of change of rate Time
forceexternalNet
),(or)(is)(,)(:
)()()(
Physical Laws
RTT
[Force], MomentumTime
SPECIAL CASE: Stationary and Non-Deforming CV in IFR
If the CV is stationary and non-deforming in IFR, we have
Hence, and the C-Mom becomes VVVVV fsfsf
/
0
sV
)(
)()()(
tCS Qdmd
CVMV AdVVdt
tPd
dt
tPdF
abj 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.]
Physical Laws:dt
tPdF MV )(
RTT:
VVmPN
,
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.
)(or)( is)(,)()()(
tCVtMVtVdVVtPtV
V
In MFR (moving IFR-B), we have
sfsf
tCSQdmd
sfCVMV VVVAdVVdt
tPd
dt
tPd
/
)(
/ ,)()()(
x
)(tCV)( dttCV
),( txV
IFRx
yObserver A
MFRx’
y’
Observer B
0,;0,0
BBBB aV
x
),( txV
Recall the coincident CV(t) and MV(t)
C-Mom:
)( through momentumlinear ofefflux convectionNet
)(
/
)( of momentumlinear of change of rate Time
)( of momentumlinear of change of rate Time
forceexternalNet
)()()(
tCS
tCSQdmd
sf
tCV
CV
tMV
MV AdVVdt
tPd
dt
tPdF
Physical Laws
RTT
[Force], MomentumTime
abj 14
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.]
Recall the coincident CV(t) and MV(t)
sfsf
tV
V
tCS
tCSQdmd
sf
tCV
CV
tMV
MV
VVVtCVtMVtVdVVP
AdVVdt
tPd
dt
tPdF
/
)(
)( through momentumlinear ofefflux convectionNet
)(
/
)( of momentumlinear of change of rate Time
)( of momentumlinear of change of rate Time
forceexternalNet
),(or)(is)(,)(:
)()()(
Physical Laws
RTT
[Force], MomentumTime
SPECIAL CASE: Stationary and Non-Deforming CV in MFR
If the CV is stationary and non-deforming in MFR, we have
Hence, and the C-Mom becomes VVVVV fsfsf
/
0
sV
)(
)()()(
tCS Qdmd
CVMV AdVVdt
tPd
dt
tPdF
abj 15
and Free-Body Diagram (FBD) for the Coincident CV(t) and MV(t)
forceexternalNet
F
BS FFF
1. Concentrated/Pointed Surface Force iF
2. Distributive Surface Force in Fluid [Pressure p + Friction ]
Net Surface Force SF
Net Volume/Body Force BF
MVCV
dVggm )(
Keys
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.
F
CV(t)MV(t)
Pressure p
Shear
iF
2. Distributive Surface Force
(in fluid part)
1. Concentrated/Point Surface Force
Coincident CV(t) and MV(t)
)( dVgdmg
Volume/Body Force
FBD
abj 16
Recall: Past Example of RTT for Linear MomentumExample 3: Finding The Time Rate of Change of Property N of an MV By The Use of A Coincident CV and The RTTProblem: 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 17
x
y
V1V2 = V1
(a) (yes/no) If not, positive or negative
Net Fx on CV? (yes/no) If yes, Fx positive or negative
(b) (yes/no) If not, positive or negative
Net Fy on CV? (yes/no) If yes, Fy positive or negative
?0, dt
dP xMV
dt
dP xMV ,
?0, dt
dP yMV
dt
dP yMV ,
V1V2 > V1
V1
V2 = V1
V1
V2 = V1
V1
V2 = V1
abj 18
Example: Cart with Guide Vane
abj 19
C-Mom for A Moving/Deforming CV As Observed from An Observer
in A Translating Frame of Reference with Respect to IFR
abj 20
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
IFR
x
y
Observer A
MFRx’
y’
Observer B
0,;0,
BBBB aV
x
),( txV
x
),( txV
)(),( tMVtCV
Physical Laws (IFR)
dt
tPdF MV )(
RTT (MFR)
sfsf
tCSQdmd
sfCVMV
VVV
AdVVdt
tPd
dt
tPd
/
)(
/
:
)()()(
????)()(
dt
tPdf
dt
tPd MVMV
Kinematics of Relative Motion
???
abj 21
Position Vectors:
Velocity Vectors:
Acceleration Vectors:
0:,::
;:
B
ABrf
A
rf
Arf
A
rf
AA
dt
xd
dt
xdV
dt
rdV
dt
xd
dt
rdxr
dt
d
dt
xdV
Kinematics of Relative Motion: Translating Reference Frame (RF) with Acceleration
xrx rf
aaa rf
0:,::
:
B
ABrf
A
rf
Arf
A
rf
AA
dt
Vd
dt
Vda
dt
Vda
dt
Vd
dt
VdVV
dt
d
dt
Vda
VVV rf
????)()(
dt
tPdf
dt
tPd MVMV
0,;0,
BBBB aV Observer A
IFR x
y x
),( txV ),( txV
)(),( tMVtCV
),( txa
),( txa
MFRx’
y’
Observer B
x
)( Brf aa
)( Brf VV
rfr
abj 22
Momentum for an identified mass [ MV(t) ] as observed in IFR-A:
Momentum for an identified mass [ MV(t) ] as observed in MFR-B:
Kinematics of Relative Motion: Relation between Linear Momenta of The Two Reference Frames
)()( tMVtMV
MV
dm
dVVdmVP
)()( tMVtMV
MV
dm
dVVdmVP
MV
tMV
rfMV PdmVP
)(
)()()()( tMVtMV
rf
tMV
rf
tMV
MV dmVdmVdmVVdmVP
VVV rf
Observer A
IFR x
y x)(),( tMVtCV
dmVPd
MFRx’
y’
Observer B
x
)( Brf aa
)( Brf VV
dmVPd
0,;0,
BBBB aV
abj 23
Kinematics of Relative Motion: Relation between Time Rates of Change of Linear Momenta of The Two Reference Frames (Short Version.)
;)(
dt
Pddma
dt
Pd MV
tMV
rfMV
dt
PddmV
dt
dPdmV
dt
d
dt
Pd MV
tMV
rfMV
tMV
rfMV
)()(
time.oft independen is mass System;)(
dt
Pddm
dt
VdMV
tMV
rf
dt
Vda rf
rf
:
Note: In some sense, this derivation is a little
obscure; however, it serves our purpose for
the moment. Another line of approach is to
use the volume integral.
Observer A
IFR
x
x
y
)(),( tMVtCV
dmVPd
MFRx’
y’
Observer B
x
)( Brf aa
)( Brf VV
dmVPd
0,;0,
BBBB aV
abj 24
C-Mom for A Moving/Deforming CV As Observed from An Observer in A Translating Frame of Reference with Respect to IFR
CV(t)MV(t)
Pressure p
Shear
iF
2. Distributive Surface Force
(in fluid part)
1. Concentrated/Point Surface Force
Coincident CV(t) and MV(t)
)( dVgdmg Volume/Body Force
FBD
Newton’s Second Law of Motion:
Relation between Linear Momenta:
RTT:
Thus, we have
dt
tPddma
dt
tPd MV
tMV
rfMV )()(
)(
)(
/ )()()(
tCSQdmd
sfCVMV AdVVdt
tPd
dt
tPd
dt
tPddma
dt
tPdF MV
tMV
rfMV )()(
)(
dt
tPddmaF MV
tCVtMV
rf)(
)()(
dt
tPdF MV )(
)()(
/
)()(
)()(,)()(
)(tV
V
tCSQdmd
sfCV
tCVtMV
rf dVVtPAdVVdt
tPddVaF
[Force], MomentumTime
abj 25
C-Mom for A Moving/Deforming CV As Observed from An Observer in A Translating Frame of Reference with Respect to IFR
Recall the coincident CV(t) and MV(t)
sfsf
tV
V
tCS
tCSQdmd
sf
tCV
CV
tCVtMV
rf
VVVtCVtMVtVdVVP
AdVVdt
tPddVaF
/
)(
)( through momentumlinear ofefflux convectionNet
)(
/
)( of momentumlinear of change of rate Time
)()(forceexternalNet
),(or)(is)(,)(:
)()(
)(
[Force], MomentumTime
SPECIAL CASE: Stationary and Non-Deforming CV in MFR
If the CV is stationary and non-deforming in MFR, we have
Hence, and the C-Mom becomes VVVVV fsfsf
/
0
sV
)()()(
)()(
)(tCS Qdmd
CV
tCVtMV
rf AdVVdt
tPddVaF
abj 26
Special Case: : Moving IFR, MFR that moves at constant velocity with respect to another IFR
0
rfa
)(
/
)()(
/
)()(
)()(
)()(,)()(
)(
tCSQdmd
sfCV
tV
V
tCSQdmd
sfCV
tCVtMV
rf
AdVVdt
tPdF
dVVtPAdVVdt
tPddVaF
0
In this case, the C-Mom reduces down to that of the moving IFR that we derived earlier.
abj 27
Example: Velocities in The Net Convection Efflux Term
IFR/A sees (velocities wrt IFR/A)
the fluid velocity (gas velocity) at the exit CS
the velocity of the MFR/B (the airplane)
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)
An observer moving with the exit CS (not with MFR/B) sees (velocities wrt CS)
the fluid velocity (gas velocity) at the exit CS sfsfsfsf VVVVVV
//
fVV
rfV
fVV
sV
sfsf
tCSQdmd
sf VVVAdVV
/
)(
/ ,)(
IFRx
yObserver A MFR x’
y’Observer B on a moving airplane rfV
)(tCV
sfV /
fVV
sV
sV
Balloon jet in an airplane
If the CV is stationary
and non-deforming in
MFR, we have
Hence,
VVVVV fsfsf
/
0
sV
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
abj 29
C-Mass for A Moving/Deforming CV As Observed from An Observer in A Moving Frame of Reference (MFR) with Respect to IFR
)( through ofefflux convectionNet
/
)( of of change of rate Time
)( of of change of rate Time
V of in Change of Source
)()()(
tCSN
CSQdmd
sf
tCVN
CV
tMVN
MV
MN
N AdVdt
tdN
dt
tdNS
)(
/)()(
:1,tCS
sfCVMV AdVdt
tdM
dt
tdMMN
RTT (in MFR)
Regardless of frame of reference (in classical mechanics), we have the physical law of conservation of mass
)(or)(is)(,)(:
)()(0
)(
)(
/
tCVtMVtVdVtM
AdVdt
tdM
dt
tdM
tV
V
tCS
sfCVMV
C-Mass in MFR
dt
tdM MV )(0 Physical Law: (for any frame of reference)
Note:
• Recognize also that .
• The same form of C-Mass – with the convection term written with the relative velocity - is
valid for any frame of reference.
sfsf VV //
sfsf VVV
/