Kuala Perlis

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2. Engineering MechanicsSTATIC FORCE VECTORS

3. Chapter Outline • Scalars and Vectors • Vector Operations • Vector Addition of Forces • Addition of a System of Coplanar Forces • Cartesian Vectors • Addition and Subtraction of Cartesian Vectors • Position Vectors • Force Vector Directed along a Line • Dot Product

4. 2.1 Scalars and Vectors • Scalar – A quantity characterized by a positive or negative number – Indicated by letters in italic such as A e.g. Mass, volume and length

5. 2.1 Scalars and Vectors • Vector – A quantity that has magnitude and direction e.g.force and moment – Represent by a letter with an arrow over it, – Magnitude is designated as – In this subject, vector is presented as A and its magnitude (positive quantity) as A

6. 2.2 Vector Operations • Multiplication and Division of a Vector by a Scalar - Product of vector A and scalar a = aA - Magnitude = - Law of multiplication applies e.g. A/a = ( 1/a )A, a≠0

7. 2.2 Vector Operations • Vector Addition - Addition of two vectors A and B gives a resultant vector R by the parallelogram law - Result R can be found by triangle construction - Communicative e.g. R = A + B = B + A - Special case: Vectors A and B are collinear (both have the same line of action)

8. 2.2 Vector Operations • Vector Subtraction - Special case of addition e.g. R’ = A–B = A + ( - B ) - Rules of Vector Addition Applies

9. 2.3 Vector Addition of Forces Finding a Resultant Force • Parallelogram law is carried out to find the resultant force • Resultant, FR = ( F1 + F2 )

10. 2.3 Vector Addition of Forces Procedure for Analysis • Parallelogram Law • Make a sketch using the parallelogram law • 2 components forces add to form the resultant force • Resultant force is shown by the diagonal of the parallelogram • The components is shown by the sides of the parallelogram

11. 2.3 Vector Addition of Forces Procedure for Analysis • Trigonometry • Redraw half portion of the parallelogram • Magnitude of the resultant force can be determined by the law of cosines • Direction if the resultant force can be determined by the law of sine • Magnitude of the two components can be determined by the law of sine

12. Example 2.1 The screw eye is subjected to two forces, F1 and F2. Determine the magnitude and direction of the resultant force.

13. Solution Parallelogram Law Unknown: magnitude of FR and angle θ

14. Solution Trigonometry Law of Cosines Law of Sines

15. Solution Trigonometry Direction Φ of FR measured from the horizontal

16. Exercise 1: • Determine magnitude of the resultant force acting on the screw eye and its direction measured clockwise from the x- axis

17. Exercise 2: • Two forces act on the hook. Determine the magnitude of the resultant force.

18. 2.4 Addition of a System of Coplanar Forces • When a force resolved into two components along the x and y axes, the components are called rectangular components. • Can represent in scalar notation or cartesan vector notation.

19. 2.4 Addition of a System of Coplanar Forces • Scalar Notation • x and y axes are designated positive and negative • Components of forces expressed as algebraic scalars

20. 2.4 Addition of a System of Coplanar Forces • Cartesian Vector Notation • Cartesian unit vectors i and j are used to designate the x and y directions • Unit vectors i and j have dimensionless magnitude of unity ( = 1 ) • Magnitude is always a positive quantity, represented by scalars Fx and Fy

21. 2.4 Addition of a System of Coplanar Forces • Coplanar Force Resultants To determine resultant of several coplanar forces: • Resolve force into x and y components • Addition of the respective components using scalar algebra • Resultant force is found using the parallelogram law • Cartesian vector notation:

22. 2.4 Addition of a System of Coplanar Forces • Coplanar Force Resultants • Vector resultant is therefore • If scalar notation are used

23. 2.4 Addition of a System of Coplanar Forces • Coplanar Force Resultants • In all cases we have • Magnitude of FRcan be found by Pythagorean Theorem * Take note of sign conventions

24. Example 2.5 Determine x and y components of F1 and F2 acting on the boom. Express each force as a Cartesian vector.

25. Solution Scalar Notation Hence, from the slope triangle, we have

26. Solution By similar triangles we have Scalar Notation: Cartesian Vector Notation:

27. Solution Scalar Notation Hence, from the slope triangle, we have: Cartesian Vector Notation

28. Example 2.6 The link is subjected to two forces F1 and F2. Determine the magnitude and orientation of the resultant force.

29. Solution I Scalar Notation:

30. Solution I Resultant Force From vector addition, direction angle θ is

31. Solution II Cartesian Vector Notation F1 = { 600cos30°i + 600sin30°j } N F2 = { -400sin45°i + 400cos45°j } N Thus, FR = F1 + F2 = (600cos30ºN - 400sin45ºN)i + (600sin30ºN + 400cos45ºN)j = {236.8i + 582.8j}N The magnitude and direction of FRare determined in the same manner as before.

32. Exercise 1: • Determine the magnitude and direction of the resultant force.

33. Exercise 2: • Determine the magnitude of the resultant force and its direction θmeasured counterclockwise from the positive x-axis.

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