DYNAMICS. Planar Kinetics of a Rigid Body (Impulse and Momentum)

Transcription

1 DYNAMICS Planar Kinetics of a Rigid Body (Impulse and Momentum) by: Dr. Mohd Hasnun Arif HASSAN Faculty of Manufacturing Engineering mhasnun@ump.edu.my

2 Impulse and Momentum Method Aims To introduce the Principle of Impulse and Momentum. To explain about the conservation of momentum. To learn about eccentric impact. Expected Outcomes Students are able to use the principle of impulse and momentum to solve planar kinetics problem. Students are able to apply the conservation of momentum in solving problems. Students are able to determine the velocity of rigid body experiencing eccentric impact. References Engineering Mechanics: Dynamics 12 th Edition, RC Hibbeler, Prentice Hall

3 Contents Principle of Linear Impulse and Momentum Conservation of Momentum Eccentric Impact

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5 ҳ ۳ ۳ ) ۳ Ú< 7Ú< Z Xҳ ҳ OQ? O Q? 7 ҳ@ ) ZXҳ Z ۳ Z % ҳ ҳ 7 ) 1 ) 1 Principle of Angular Impulse &, N b D JZ B S3b, E 2 b HD3EVZD b T Momentum ҳ OQ? ۳ ZXҳ ۳ b Z T Ú< 7 ) JZ B S3b, E 2 b HD3EVZD 1 Ú % OQ? ۳ ZXҳ ۳ Z Ú< 7 ) ҳ 2 W* [Ȥ 1ҳ ҳ p pҳ (ҳ ( ҳ 2 ҳ pʥҳ ҳpҳ Ġ ҳ 2 ҳ Ġ ҳpҳ pʥҳ F F F F F F ҳ 2 ҳ F F pʥҳ 56ʵ Ġ ҳpҳ 08Q-! Σ ҳ 2 ҳ Ġ ҳpҳ pʥҳ T % 17 ҳ. ҳ ҳ ) 08Qҳ NO^ -EQ0: NQ ) -ҳ ) ҳ _X 7ҳ œj 17 ҳ. ҳ ҳ 08Q-! Σ - _ ҳ. ҳ ҳ ) ) Σҳ ) ҳ : NQ X 7ҳ 08Q-! 17 7 ҳ. ҳ ҳ ) ҳ ) ҳ - _X 7ҳ ) : NQ à ҳ. ҳ ҳ ) ҳ ) ҳ ) Q X _ 7ҳ 08o-/ Σ _X 7ҳ ) 08o NO -8Q-! -8Q0/ 7 F Angular Impulse - F Angular Momentum

6 Linear and Angular Momentum Linear momentum Angular momentum Identical to the kinetic moments about point P

7 Linear and Angular Momentum (cont d)

8 Principle of Impulse and Momentum L " + I "&' = L ' Principle of Linear Impulse and Momentum / 0 (H + ) " + - M + dt = H + ' / 1 Principle of Angular Impulse and Momentum

9 Principle of Impulse and Momentum For a system of rigid bodies (more than one rigid bodies)

10 Principle of Impulse and Momentum When to use the principle of impulse and momentum? To solve problems involving force, velocity and time. When to use the work and energy method? To solve problems involving force, velocity and displacement. When to use the force and acceleration method? To solve problems involving force and acceleration.

11 Conservation of Momentum Conservation of Linear Momentum If there is no/negligible net external linear impulse m v 7 " = m v 7 ' Conservation of Angular Momentum If there is no/negligible net external angular impulse H + " = H + '

12 Conservation of Momentum It may be possible to apply conservation of linear momentum when the linear impulses are small or nonimpulsive (small forces acting over very short periods of time e.g., weight of a body).

13 Conservation of Momentum It may be possible to apply conservation of linear momentum when the linear impulses are small or nonimpulsive (small forces acting over very short periods of time e.g., weight of a body). W h F The weight of the hammer and nail are small relative to the force, F. The impact duration is very short. They are considered non-impulsive. W n

14 Conservation of Momentum It may be possible to apply conservation of linear momentum when the linear impulses are small or nonimpulsive (small forces acting over very short periods of time e.g., weight of a body). W h W n Considering both rigid bodies as one unit, there is no external impulse. The force, F cancel out each other. The weights are non-impulsive. Linear momentum is conserved. The angular momentum can be conserved while the linear momentum is not. Such cases occur whenever the external forces creating the linear impulse pass through either the center of mass of the body or a fixed axis of rotation.

15 Eccentric Impact O Line of impact Eccentric impact occurs when the line connecting the mass centers of the two bodies does not coincide with the line of impact. A C B Often occurs when one or both of the bodies are constrained to rotate about a fixed axis. Plane of impact

16 Eccentric impact In general, a problem involving the impact of two bodies requires determining the two unknowns (v A ) 2 and (v B ) 2, assuming (v A ) 1 and (v B ) 1 are known. The first equation generally involves application of the conservation of angular momentum to the two bodies. The second equation can be obtained using the definition of the coefficient of restitution, e, which is a ratio of the restitution impulse to the deformation impulse. Coefficient of restitution

17 Planar Kinetics of a Rigid Body (Impulse and Momentum) No great discovery was ever made without a bold guess. Sir Isaac Newton