1. ME 221Lecture 31 ME 221 Statics www.angel.msu.edu Sections 2.2 – 2.3

2. ME 221Lecture 32 Announcements Help Sessions – ME Help Room – 1522EB – Cubicle #2 Mondays & Wednesdays – 10:15am to 5:00pm Tuesdays & Thursdays – 8:00am to 5:00pm Fridays – 8:00am to 11:00am Will begin on Tuesday, September 2 Hours also posted on Angel

3. ME 221Lecture 33 Announcements HW#1 Due on Wednesday 09/03 Chapter 1 - 1.1, 1.3, 1.4, 1.6, 1.7 Chapter 2 – 2.1, 2.2, 2.11, 2.15, 2.21 Quiz #1 on Friday 09/05

4. ME 221Lecture 34 Last Lecture Vectors, vectors, vectors Law of Cosines Law of Sines Drawing vector diagrams Chapter 1: Basics Example 1. Addition of Vectors

5. ME 221Lecture 35 Law of Cosines This will be used often in balancing forces c    b a

6. ME 221Lecture 36 Law of Sines Again, start with the same triangle    c b a

7. ME 221Lecture 37 300 lb 200 lb 45 o 25 o Example Note: resultant of two forces is the vectorial sum of the two vectors

8. ME 221Lecture 38 25 o 155 o 45 o   = 90 o +25 o -  200 lb 300 lb R 25 o 110 o 300 lb R 200 lb

9. ME 221Lecture 39 Scalar Multiplication of Vectors Multiplication of a vector by a scalar simply scales the magnitude with the direction unchanged Line of action A stays the same 0.5 x A

10. ME 221Lecture 310 Forces Review definition Shear and normal forces Resultant of coplanar forces Problems

11. ME 221Lecture 311 Characteristics of a Force Its magnitude –denoted by |F| Its direction Its point of application –important when we discuss moments later

12. ME 221Lecture 312 Further Categorizing Forces Internal or external –external forces applied outside body P P A section of the body exposes internal body P Cut plane through body Internal tension

13. ME 221Lecture 313 Shear and Oblique Shear internal force has line of action contained in cutting plane P P Intenal shear forces

14. ME 221Lecture 314 Oblique Internal Forces Oblique cutting planes have both normal and shear components P S N Where N + S = P

15. ME 221Lecture 315 Transmissibility A force can be replaced by a force of equal magnitude provided it has the same line of action and does not disturb equilibrium B A

16. ME 221Lecture 316 Weight is a Force Weight is the force due to gravity –W = mg where m is mass and g is gravity constant g = 32.2 ft/s 2 = 9.81 m/s 2 English and metric –Weight lb or N –Mass slugs or kg

17. ME 221Lecture 317 Resultant of Coplanar Forces A body’s motion depends on the resultant of all the forces acting on it In 2-D, we can use the Laws of Sines and Cosines to determine the resultant force vector In 3-D, this is not practical and vector components must be utilized more on this in the future

18. ME 221Lecture 318 x y AxAx AyAy xx A yy yy Perpendicular Vectors A x is the component of vector A in the x-direction A y is the component of vector A in the y-direction AxAx xx AyAy x y A AyAy AxAx

19. ME 221Lecture 319 Vector Components Vector components are a powerful way to represent vectors in terms of coordinates. x y xx yy A where A x = |A| cos  x A y = |A| cos  y = |A| sin  x x y AxAx AyAy A= AxAyAxAy

20. ME 221Lecture 320 A x = |A| cos  x A y = |A| cos  y = |A| sin  x cos  x = A x / |A| cos  y = A y / |A| = x = y x and y are called direction cosines x 2 + y 2 = 1 Note: To apply this rule the two axes must be orthogonal Vector Components (continued)

21. ME 221Lecture 321 Summary External forces give rise to –tension and compression internal forces –normal and shear internal forces Forces can translate along their line of action without disturbing equilibrium The resultant force on a particle is the vector sum of the individual applied forces

22. ME 221Lecture 322 Sample Problem