Lecture 6.1 : Conservation of Linear Momentum (C-Mom)

1. 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 • Moving IFR (with respect to another IFR) [Moving Frame of Reference (MFR) that moves at constant velocity with respect to another IFR] • C-Mom for A Moving/Deforming CV As Observed From An Observer in A Translating Frame of Reference (MFR) with Respect to IFR • Example: Velocities in The Net Convection Efflux Term • C-Mass for A Moving/Deforming CV As Observed from An Observer in A Moving Frame of Reference (MFR) with Respect to IFR

2. Physical Laws RTT Physical Laws RTT Very Brief Summary of Important Points and Equations [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) • C-Mom for A Moving/Deforming CV As Observed From An Observer in A Translating Frame of Reference (MFR) with Respect to IFR

3. C-Mass in MFR Very Brief Summary of Important Points and Equations [2] C-Mass for A Moving/Deforming CV As Observed from An Observer in A Moving Frame of Reference (MFR) with Respect to IFR

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

5. Continuum body m Particle y Observer A y x Observer A x • 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. Recall 2: Linear Momentum of A Particle VS of A Continuum Body • Don’t get confused by the integral expression. • Similar applies to other properties of a continuum body, e.g., energy, etc.

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

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

8. C-Mom for A Moving/Deforming CV As Observed from An Observer in IFR • Stationary IFR • Moving IFR (with respect to another IFR) [Moving Frame of Reference (MFR) that moves at constant velocity with respect to another IFR]

9. y’ y y Observer B MFR x’ x x IFR IFR • 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 Observer A Observer A (IFR) Observer B (MFR which is also an IFR) Recall 3: Newton’s Second Law Recall the coincident CV(t) and MV(t)

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

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

12. 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 Hence, and the C-Mom becomes [Force], Momentum Physical Laws Time RTT C-Mom for A Moving/Deforming CV As Observed from An Observer in IFR

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

14. 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 Hence, and the C-Mom becomes [Force], Momentum Physical Laws Time RTT 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.]

15. Pressure p Coincident CV(t) and MV(t) CV(t) 2. Distributive Surface Force (in fluid part) MV(t) Shear t FBD • Concentrated/Point Surface Force Volume/Body Force • Keys • Recognize various types of forces. • Know how to find the resultant of various types of forces (e.g., pressure, etc.). • Sum all the external forces. Net Surface Force Net Volume/Body Force 1. Concentrated/Pointed Surface Force 2. Distributive Surface Force in Fluid [Pressure p + Friction t ] and Free-Body Diagram (FBD) for the Coincident CV(t) and MV(t)

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 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 CVshown 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.

17. V2 = V1 V2 > V1 V1 V2 = V1 V1 V1 V1 V2 = V1 • (yes/no) If not, positive or negative • Net Fx on CV? (yes/no) If yes, Fxpositive or negative • (b) (yes/no) If not, positive or negative • Net Fy on CV? (yes/no) If yes, Fypositive or negative q V2 = V1 V1 y x

18. Example: Cart with Guide Vane

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

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

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

22. y’ Observer B y x’ MFR Observer A x IFR Kinematics of Relative Motion: Relation between Linear Momenta of The Two Reference Frames Momentum for an identified mass [ MV(t) ] as observed in IFR-A: Momentum for an identified mass [ MV(t) ] as observed in MFR-B: 

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

24. Coincident CV(t) and MV(t) Pressure p CV(t) 2. Distributive Surface Force (in fluid part) MV(t) Shear t FBD • Concentrated/Point Surface Force Volume/Body Force Newton’s Second Law of Motion: Relation between Linear Momenta: RTT: Thus, we have [Force], Momentum Time C-Mom for A Moving/Deforming CV As Observed from An Observer in A Translating Frame of Reference with Respect to IFR

25. 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 Hence, and the C-Mom becomes [Force], Momentum Time C-Mom for A Moving/Deforming CV As Observed from An Observer in A Translating Frame of Reference with Respect to IFR

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

27. Balloon jet in an airplane y’ y Observer B on a moving airplane x x’ MFR IFR • 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 If the CV is stationary and non-deforming in MFR, we have Hence, Observer A Example: Velocities in The Net Convection Efflux Term

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

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