King Fahd University of Petroleum & Minerals

1. King Fahd University of Petroleum & Minerals Mechanical Engineering Dynamics ME 201 BY Dr. Meyassar N. Al-Haddad Lecture # 22

2. Principle of Linear Impulse and Momentum Initial momentum + Sum of all Impulse = Final momentum

3. 0 Conservation of Linear Momentum for a system of Particles Conservation of linear momentum equation

4. Impulse Impulse & Average Force

5. Impact • Impact occurs when two bodies collide with each other during a very short period of time. • Types of impact: • Central impact • Oblique impact Line of impact Plane of impact

6. Coefficient of restitution e Coefficient of restitution e is defined as the ratio of the restitution impulse to the deformation impulse. Coefficient of restitution e is defined as the ratio of relative velocity after impact to the relative velocity before impact Coefficient of restitution e is according to the impact velocity, material, size and shape of the colliding body, Coefficient of restitution e: range between 0-1 Coefficient of restitution e:is defined along the line of impact only Elastic impact e = 1(re-bounce with same velocity) Plastic impact e = 0 (couple or stick together and move with common velocity)

7. Procedure for Analysis • Identify the intial velocity ““you may use” • T1+ V1 = T2+ V2 • Apply the conservation of momentum along the line of impact, you will get one equation with two unknown velocity • Use the coefficient of restitution to obtain a second equation • Solve both equation for final velocities after the impact

8. Four unknowns Oblique Impact Central Impact : one Dimension Oblique Impact : Two Dimension

9. Procedure for Analysis • Establish x-axis as line of impact • Establish y-axis as plane of impact • Resolve the velocity components • along x, and y as • Apply the conservation of momentum along the line of impact • Use the coefficient of restitution to obtain a second equation • Solve both equation for final velocities along x-axis after the impact • The momentum is conserved along the plane of impact; so

10. ANGULAR MOMENTUM • For a point object the angular momentum is v m Units - kg.m2/s or sl.ft2/s It is a vector. Here the vector is pointing toward you. Using right-hand rule r

11. Angular Impulse and Momentum Principles

12. Conservation of Angular Momentum

13. Example 15-3 W=50 Ib P=(20t) Ib v2=? t=2 sec. v1=3 ft/s mk=0.3

14. Example 15-3 From rest vB=? t=6 sec. Block A Block B

15. Problem m=12 Mg Fy=150 kN V=? h=? t=6 s Start from rest

16. Problem m = 28 Mg At rest V = ? t = 4 s F = 4 - 0.01 t2

17. Example 15-4 mA=15 Mg mB=12 Mg Couple together V2=? After coupling Favg = ? In 0.8 s

18. Example 15-5 mC =1200-Ib mp = 8-Ib vp=1500ft/s t = 0.03 s. vc2= ? Favg= ? Favg

19. Example 15-7 mp = 800 kg mH = 300 kg From rest Impulse = ? Couple together Impulse Impulse =

20. Example 15-9

22. Example 15-11

23. Momentum of particle A,B is conserved along the y axis, since no impulse acts on particle A,B

24. Example 15-13

25. Problem m = 400 g v1= 2 m/s M = 0.6 N.m v2=? t = 3 s

26. Thank you