Mechanics: Scalars and Vectors

Transcription

1 Mechanics: Scalars and Vectors Scalar Onl magnitude is associated with it Vector e.g., time, volume, densit, speed, energ, mass etc. Possess direction as well as magnitude Parallelogram law of addition (and the triangle law) e.g., displacement, velocit, acceleration etc. Tensor e.g., stress (33 components) 1

2 Mechanics: Scalars and Vectors A Vector V can be written as: V = Vn V = magnitude of V n = unit vector whose magnitude is one and whose direction coincides with that of V Unit vector can be formed b dividing an vector, such as the geometric position vector, b its length or magnitude Vectors represented b Bold and Non-Italic letters (V) Magnitude of vectors represented b Non-Bold, Italic letters (V) x j i z k 2

3 Tpes of Vectors: Fixed Vector Fixed Vector Constant magnitude and direction Unique point of application e.g., force on a deformable bod F F Local depression e.g., force on a given particle 3

4 Tpes of Vectors: Sliding Vector Sliding Vector Constant magnitude and direction Unique line of action Slide along the line of action No unique point of application Force on coach F Force on coach F 4

5 Tpes of Vectors: Sliding Vector Sliding Vector Principle of Transmissibilit Application of force at an point along a particular line of action No change in resultant external effects of the force 5

6 Tpes of Vectors: Free Vector Free Vector Freel movable in space No unique line of action No unique point of application e.g., moment of a couple 6

7 Vectors: Rules of addition Parallelogram Law Equivalent vector represented b the diagonal of a parallelogram V = V 1 + V 2 (Vector Sum) V V 1 + V 2 (Scalar sum) 7

8 Vectors: Parallelogram law of addition Addition of two parallel vectors F 1 + F 2 = R -F F R 2 F 2 F 1 R 1 R 2 R 1 R 8

9 Vectors: Parallelogram law of addition Addition of 3 vectors F 1 + F 2 + F 3 = R 9

10 Vectors: Rules of addition Trigonometric Rule Law of Sines Law of Cosine A B C 10

11 Force Sstems Cable Tension

12 Force Sstems Cable Tension P Force: Represented b vector Magnitude, direction, point of application P: fixed vector (or sliding vector??) External Effect Applied force; Forces exerted b bracket, bolts, Foundation (reactive force) 12

13 Force Sstems Rigid Bodies External effects onl Line of action of force is important Not its point of application Force as sliding vector 13

14 Force Sstems Concurrent forces Lines of action intersect at a point A F 2 R Plane F 1 A F 2 F 2 F 1 R F 1 R = F 1 +F 2 R A F 1 F 2 Concurrent Forces F 1 and F 2 Principle of Transmissibilit R = F 1 + F 2 14

15 Components and Projections of a Force Components and Projections Equal when axes are orthogonal F 1 and F 2 are components of R R = F 1 + F 2 :F a and F b are perpendicular projections on axes a and b : R F a + F b unless a and b are perpendicular to each other 15

16 Components of a Force Examples 16

17 Components of a Force Examples 17

18 Components of a Force Example 1: Determine the x and scalar components of F 1, F 2, and F 3 acting at point A of the bracket 18

19 Components of Force Solution: 19

20 Components of Force Alternative Solution: Scalar components of F 3 can be obtained b writing F 3 as a magnitude times a unit vector n AB in the direction of the line segment AB. Unit vector can be formed b dividing an vector, such as the geometric position vector b its length or magnitude. 20

21 Components of Force Example 2: The two forces act on a bolt at A. Determine their resultant. Graphical solution Construct a parallelogram with sides in the same direction as P and Q and lengths in proportion. Graphicall evaluate the resultant which is equivalent in direction and proportional in magnitude to the diagonal. Trigonometric solution Use the law of cosines and law of sines to find the resultant. 21

22 Components of Force Solution: Graphical solution - A parallelogram with sides equal to P and Q is drawn to scale. The magnitude and direction of the resultant or of the diagonal to the parallelogram are measured, R 98 N 35 Graphical solution - A triangle is drawn with P and Q head-to-tail and to scale. The magnitude and direction of the resultant or of the third side of the triangle are measured, R 98 N 35 22

23 Components of Force Trigonometric Solution: R 2 P 2 Q 2 2PQ cos B N 60N 240N60Ncos155 R 97.73N sin A Q sin A sin B R Q sin B R sin155 A A N 97.73N 23

24 Components of Force Example 3: Tension in cable BC is 725 N; determine the resultant of the three forces exerted at point B of beam AB. Solution: Resolve each force into rectangular components. Determine the components of the resultant b adding the corresponding force components. Calculate the magnitude and direction of the resultant. 24

25 Components of Force Solution Resolve each force into rectangular components. Calculate the magnitude and direction. 25

26 Rectangular Components in Space The vector F is Resolve into contained in the horizontal and vertical plane OBAC. components. F F h F F cos F sin Resolve F h into rectangular components F F x z F F h h cos F sin cos sin F sin sin 26

27 Rectangular Components in Space With the angles between and the axes, Fx F cos x F F cos Fz F cos z F Fxi F j Fzk Fcos xi cos j cos zk F cos xi cos j cos zk is a unit vector along the line of action of and cos x, cos, and are the direction cosines for F cos z F 27 F

28 Rectangular Components in Space Direction of the force is defined b the location of two points: ,, and,, z x N z x M d Fd F d Fd F d Fd F k d j d i d d F F z z d d x x d k d j d i d N M d z z x x z x z x z x 1 and vector joining