ME221 Statics Lecture 2: Laws of Cosines and Sines in Coplanar Forces

1. ME 221 Staticswww.angel.msu.eduSections 2.2 – 2.5 Lecture 2

2. Announcements • ME221 TA’s and Help Sessions • Chad Stimson – stimson1@msu.edu • Homework grading & help room • Tuesdays & Thursdays – 8am to 1pm – 1522EB • Jimmy Issa – jimmy@msu.edu • Quiz & exam grading & help room • Tuesdays & Thursdays – 1pm to 5pm – 2415EB Lecture 2

3. Announcements • HW#1 Due on Friday, May 21 • 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, May 21 Lecture 2

4. Last Lecture • Chapter 1: Basics • Vectors, vectors, vectors • Law of Cosines • Law of Sines • Drawing vector diagrams • Example 1: Addition of Vectors Lecture 2

5. g b a b a c Law of Cosines This will be used often in balancing forces Lecture 2

6. g b a b a c Law of Sines Again, start with the same triangle Lecture 2

7. Example Note: resultant of two forces is the vectorial sum of the two vectors 25o 45o 300 lb 200 lb Lecture 2

8. 200 lb 25o 110o R 300 lb 155o 200 lb  = 90o+25o-a a 25o R 45o 300 lb Lecture 2

9. Line of action stays the same Line of action 0.5 x A A Scalar Multiplication of Vectors Multiplication of a vector by a scalar simply scales the magnitude with the direction unchanged Lecture 2

10. Forces • Review definition • Shear and normal forces • Resultant of coplanar forces Lecture 2

11. Characteristics of a Force • Its magnitude • denoted by |F| • Its direction • Its point of application • important when we discuss moments later Lecture 2

12. P P Cut plane through body P Internal tension Further Categorizing Forces • Internal or external • external forces applied outside body • A section of the body exposes internal body Lecture 2

13. P Intenal shear forces P Shear and Oblique • Shear internal force has line of action contained in cutting plane Lecture 2

14. N P S Oblique Internal Forces • Oblique cutting planes have both normal and shear components Where N + S = P Lecture 2

15. B A 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 Lecture 2

16. 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/s2 = 9.81 m/s2 • English and metric • Weight lb or N • Mass slugs or kg Lecture 2

17. 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 later Lecture 2

18. y A qy qy Ay qx Ax x Perpendicular Vectors y Ax A Ay Ay qx Ax x Ax is the component of vector A in the x-direction Ay is the component of vector A in the y-direction Lecture 2

19. y y Ay A qy qx Ax Ax Ay A= x x Vector Components Vector components are a powerful way to represent vectors in terms of coordinates. where Ax = |A| cos qx Ay = |A| cos qy = |A| sin qx Lecture 2

20. Vector Components (continued) Ax = |A| cos qx cos qx = Ax / |A| = x = y Ay = |A| cos qy cos qy = Ay / |A| = |A| sin qx x and y are called direction cosines x2 +y2 = 1 Note: To apply this rule the two axes must be orthogonal Lecture 2

21. 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 Lecture 2

22. 3-D Vectors; Base Vectors • Rectangular Cartesian coordinates (3-D) • Unit base vectors (2-D and 3-D) • Arbitrary unit vectors • Vector component manipulation Lecture 2

23. y O x z 3-D Rectangular Coordinates • Coordinate axes are defined by Oxyz Coordinates can be rotated any way we like, but ... • Coordinate axes must be a right-handed coordinate system. Lecture 2

24. y y A Ay Az O O Ax x x z z A = Ax + Ay + Az Writing 3-D Components • Component vectors add to give the vector: = Also, Lecture 2

25. y qy A O qx x z qz 3-D Direction Cosines The angle between the vector and coordinate axis measured in the plane of the two Where: lx2+ly2+lz2=1 Lecture 2

26. y O x z Unit Base Vectors Associate with each coordinate, x, y, and z, a unit vector (“hat”). All component calculations use the unit base vectors as grouping vectors. Now write vector as follows: where Ax = |Ax| Ay = |Ay| Az = |Az| Lecture 2

27. Vector Equality in Components • Two vectors are equal if they have equal components when referred to the same reference frame. That is: if Ax = Bx , Ay = By , Az = Bz Lecture 2

28. Additional Vector Operations • To add vectors, simply group base vectors • A scalar times vector A simply scales all the components Lecture 2

29. y b O x z General Unit Vectors • Any vector divided by its magnitude forms a unit vector in the direction of the vector. • Again we use “hats” to designate unit vector Lecture 2

30. y rA O x z rB/A rB Position Vectors in Space • Points A and B in space are referred to in terms of their position vectors. A • Relative position defined by the difference B Lecture 2

31. Vectors in Matrix Form • When using MatLab or setting up a system of equations, we will write vectors in a matrix form: Lecture 2

32. Summary • Write vector components in terms of base vectors • Know how to generate a 3-D unit vector from any given vector Lecture 2

33. Example Problem Lecture 2