1. Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca ENGI 1313 Mechanics I Lecture 12:3D Particle Equilibrium

2. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 2 Chapter 3 Objectives to introduce the concept of the free-body diagram for a particle. to show how to solve particle equilibrium problems using the equations of equilibrium

3. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 3 Lecture 12 Objectives to examine and apply Chapter 3 objectives in 3D space

4. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 4 Note on Tutorial Problem Set #3 Revised Problem Set  Problem 6 with spring deleted  Problem added on Dot Product

5. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 5 Example 12-01 A plate with a mass of 150 kg is supported by three cables and is in equilibrium. Find the tension force in each cable.

6. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 6 Example 12-01 (cont.) What is known?  Plate mass  Coordinate geometry What is needed?  Convert mass to weight (force)  Determine cable forces Cartesian force vectors  Magnitude, direction and sense

7. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 7 Example 12-01 (cont.) Draw FBD at A Define Cartesian Force Vectors F AD F AB F AC A F A = W= 150 kg (9.806 m/s 2 ) = 1471 N

8. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 8 Example 12-01 (cont.) Combine Like Terms  x, y and z directions F AD F AB F AC A F A = W= 150 kg (9.806 m/s 2 ) = 1471 N

9. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 9 Example 12-01 (cont.) Multiply  F x by 1.5 and add to  F y F AD F AB F AC A F A = W= 150 kg (9.806 m/s 2 ) = 1471 N

10. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 10 Example 12-01 (cont.) Multiply  F x by 3 and add to  F z F AD F AB F AC A F A = W= 150 kg (9.806 m/s 2 ) = 1471 N

11. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 11 Pulley Systems Assumptions  In this course for analysis of all pulley systems Weightless Zero friction Tension cables Fixed Pulley  Class 1 Fixed axle  Used to change direction of the pull force  Mechanical advantage of 1

12. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 12 Pulley Systems (cont.) Moveable Pulley  Class 2 Floating axle  Used to multiply forces  Mechanical advantage of 2

13. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 13 Pulley Systems (cont.) Compound Pulley  Combination of fixed and moveable pulley system

14. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 14 Example 12-02 The "scale" consists of a known weight W which is suspended at A from a cord of total length L. Determine the weight w at B if A is at a distance y for equilibrium. Neglect the sizes and weights of the pulleys. CD

15. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 15 Example 12-02 (cont.) Examine Pulley System  Known weight, W 1  Find weight w 2 at B for equilibrium position y w 2 = ? CD W1W1 W1W1 W1W1

16. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 16 Example 12-02 (cont.) Draw FDB at Point B w2w2 B  F BD = W 1 F BC = W 1 w 2 = ? W1W1 W1W1 W1W1  How to determine  ? CD

17. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 17 Example 12-02 (cont.) Determine   Total cable length, L  Triangle geometry Neglect pulley size and weight w2w2 W1W1 d/2 (L-y)/2 h  CD 

18. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 18 Example 12-03 The joint of a space frame is subjected to four member forces. Member OA lies in the x–y plane and member OB lies in the y–z plane. Determine the forces acting in each of the members required for equilibrium of the joint.

19. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 19 Example 12-03 (cont.) Draw 3D FBD at Point O  Define position and unit vectors for F 1 x z y F1F1  45  F2F2  40  F3F3 F4F4 O

20. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 20 Example 12-03 (cont.) Draw 3D FBD at Point O  Define position and unit vectors for F 2 x z y F1F1  45  F2F2  40  F3F3 F4F4 O

21. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 21 Example 12-03 (cont.) Draw 3D FBD at Point O  Define position and unit vectors for F 3 and F 4 x z y F1F1  45  F2F2  40  F3F3 F4F4 O

22. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 22 Example 12-03 (cont.) Unit and Force Vectors x z y F1F1  45  F2F2  40  F3F3 F4F4 O

23. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 23 Example 12-03 (cont.)  F x Equilibrium x z y F1F1  45  F2F2  40  F3F3 F4F4 O

24. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 24 Example 12-03 (cont.)  F z Equilibrium x z y F1F1  45  F2F2  40  F3F3 F4F4 O

25. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 25 Example 12-03 (cont.)  F y Equilibrium x z y F1F1  45  F2F2  40  F3F3 F4F4 O

26. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 26 Classification of Textbook Problems Hibbeler (2007)

27. ENGI 1313 Statics I – Lecture 12© 2007 S. Kenny, Ph.D., P.Eng. 27 References http://en.wikipedia.org Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_eng mech_1