abj1 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

abj2 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 Physical Laws RTT Physical Laws

abj3 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

abj4 Recall 1: Motion is Relative (to A Frame of Reference) Observer A in Frame A Observer B in Frame B  Velocity is relative: Observer A in Frame A Observer B in Frame B  Linear momentum is also relative: x’ y’ Observer B x y Observer A Particle

abj5 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. Don’t get confused by the integral expression. Similar applies to other properties of a continuum body, e.g., energy, etc. x y Observer A Particle m x y Observer A Continuum body

abj6 Control Volume Motion VS Frame of Reference Motion IFR x y x y Observer A x’ y’ Observer B MFR  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).

abj7 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). MFR x’ y’ Observer B on a moving airplane IFR x y Observer 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

abj8 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]

abj9 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. Observer A (IFR) Observer B (MFR which is also an IFR) IFR x y Observer A IFR x y Observer A MFR x’ y’ Observer B Recall the coincident CV(t) and MV(t)

abj10 IFR x y Observer A MFR x’ y’ Observer B Observer A (IFR) 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. Observer A (IFR): Observer B (MFR / IFR): Recall the coincident CV(t) and MV(t)

abj11 C-Mom for A Moving/Deforming CV As Observed from An Observer in IFR IFR x y Observer A Recall the coincident CV(t) and MV(t) C-Mom: Physical Laws RTT Momentum Time [Force],

abj12 C-Mom for A Moving/Deforming CV As Observed from An Observer in IFR Recall the coincident CV(t) and MV(t) Physical Laws RTT [Force], Momentum Time 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

abj13 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: RTT: 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. In MFR (moving IFR-B), we have IFR x y Observer A MFR x’ y’ Observer B Recall the coincident CV(t) and MV(t) C-Mom: Physical Laws RTT [Force], Momentum Time

abj14 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) Physical Laws RTT [Force], Momentum Time 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

abj15 and Free-Body Diagram (FBD) for the Coincident CV(t) and MV(t) 1. Concentrated/Pointed Surface Force 2. Distributive Surface Force in Fluid [Pressure p + Friction  ] Net Surface ForceNet Volume/Body Force 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. CV(t) MV(t) Pressure p Shear  2. Distributive Surface Force (in fluid part) 1.Concentrated/Point Surface Force Coincident CV(t) and MV(t) Volume/Body Force FBD

abj16 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 P x and P y 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 ( F x and F y ) acting on the MV/CV, and - whether the corresponding net force is positive or negative.

abj17 x y V1V1 V 2 = V 1 (a) (yes/no) If not, positive or negative Net F x on CV? (yes/no) If yes, F x positive or negative (b) (yes/no) If not, positive or negative Net F y on CV? (yes/no) If yes, F y positive or negative V1V1 V 2 > V 1 V1V1 V 2 = V 1 V1V1  V1V1

abj18 Example: Cart with Guide Vane

abj19 C-Mom for A Moving/Deforming CV As Observed from An Observer in A Translating Frame of Reference with Respect to IFR

abj20 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 MFR x’ y’ Observer B Physical Laws (IFR) RTT (MFR) Kinematics of Relative Motion ???

abj21 Position Vectors: Velocity Vectors: Acceleration Vectors: Kinematics of Relative Motion: Translating Reference Frame (RF) with Acceleration Observer A IFR x y MFR x’ y’ Observer B

abj22 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 Observer A IFR x y MFR x’ y’ Observer B

abj23 Kinematics of Relative Motion: Relation between Time Rates of Change of Linear Momenta of The Two Reference Frames (Short Version.) 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 y MFR x’ y’ Observer B

abj24 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  2. Distributive Surface Force (in fluid part) 1.Concentrated/Point Surface Force Coincident CV(t) and MV(t) Volume/Body Force FBD Newton’s Second Law of Motion: Relation between Linear Momenta: RTT: Thus, we have [Force], Momentum Time

abj25 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) [Force], Momentum Time 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

abj26 Special Case:: Moving IFR, MFR that moves at constant velocity with respect to another IFR In this case, the C-Mom reduces down to that of the moving IFR that we derived earlier.

abj27 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 IFR x y Observer A MFR x’ y’ Observer B on a moving airplane Balloon jet in an airplane If the CV is stationary and non-deforming in MFR, we have Hence,

abj28 C-Mass for A Moving/Deforming CV As Observed from An Observer in A Moving Frame of Reference (MFR) with Respect to IFR

abj29 C-Mass for A Moving/Deforming CV As Observed from An Observer in A Moving Frame of Reference (MFR) with Respect to IFR RTT (in MFR) Regardless of frame of reference (in classical mechanics), we have the physical law of conservation of mass C-Mass in MFR 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.