Have both a magnitude and direction Examples: Position, force, moment

Transcription

1 Force Vectors

2 Vectors Vector Quantities Have both a magnitude and direction Examples: Position, force, moment Vector Notation Vectors are given a variable, such as A or B Handwritten notation usually includes an arrow, such as A or B

3 Illustrating Vectors Vectors are represented by arrows. Include magnitude, direction, and sense. Magnitude: The length of the line segment Magnitude = X

4 Illustrating Vectors Vectors are represented by arrows. Include magnitude, direction, and sense Direction: The angle between a reference axis and the arrow s line of action. Direction = 30 counterclockwise from the positive X axis. 30 +X

5 Illustrating Vectors Vectors are represented by arrows Include magnitude, direction, and sense Sense: Indicated by the direction of the tip of the arrow. Sense = Upward and to the right. 30 +X

6 Sense +y (up) +y (up) -x (left) -y (down) (0,0) +x (right) -y (down) -x (left) +x (right)

7 Trigonometry Review Right Triangle A triangle with a 90 angle. Sum of all interior angles = 180 Pythagorean Theorem: A 2 + B 2 = C 2 Opposite Side (opp) 90 Adjacent Side (adj)

8 Trigonometry Review Trigonometric Functions soh cah toa sin θ = opp / hyp cos θ = adj / hyp tan θ = opp / adj Opposite Side (opp) 90 Adjacent Side (adj)

9 Trigonometry Application The hypotenuse is the Magnitude of the Force, F. The adjacent side is the x-component, F x. The opposite side is the y-component, F y. Opposite Side F y 90 Adjacent Side F x

10 Trigonometry Application sin θ = F y / F cos θ = F x / F F y = F sin θ F x = F cos θ tan θ = F y / F x Opposite Side F y 90 Adjacent Side F x

11 Vector X and Y Components Vector A Magnitude = 75.0 Direction = 35.0 from the horizontal +y Sense = right, up A 75.0 opp = F Ay x +x -y adj = F Ax

12 Vector X and Y Components Solve for F AX cos adj hyp cos FAX A cos35.0 F AX y FAX 75.0 cos35.0 A 75.0 FAX 61.4 opp = F Ay x +x -y adj = F Ax

13 Vector X and Y Components Solve for F AY sin opp hyp sin FAY A sin35.0 F AY y FAY 75.0 sin35.0 A 75.0 FAY 43.0 opp = F Ay x +x -y adj = F Ax

14 Vector X and Y Components Your Turn +y Vector B Magnitude = 75.0 Direction = 35.0 from the horizontal Sense = right, down -x adj = F Bx x opp = F By -y B 75.0

15 Vector X and Y Components Your Turn Solve for F BX cos adj hyp cos FBX B cos35.0 F BX y FBX 75.0 cos35.0 -x adj = F Bx x FBX 61.4 opp = F By -y B 75.0

16 Vector X and Y Components Your Turn Solve for F BY sin opp hyp sin B F By sin35.0 F By y FBy 75.0 sin35.0 -x adj = F Bx x FBy 43.0 opp = F By -y B 75.0

17 Resultant Force Two people are pulling a boat to shore. They are pulling with the same magnitude. A B 75.0

18 Resultant Force F Ay = A 75 F Ax = 61.4 F Bx = 61.4 List the forces according to sense. Label right and up forces as positive, and label left and down forces as negative. F x F Ax = F Bx = F y F Ay = F By = B 75 F By = -43.0

19 Resultant Force F x F Ax = F Bx = F y F Ay = F By = Sum (S) the forces SF x = F Ax + F Bx SF x = SF x = (right) SF y = F Ay + F By SF y = ( ) = 0 Magnitude is Direction is 0 from the x axis Sense is right.

20 Resultant Force Draw the resultant force (F R ) Magnitude is 123 Direction is 0 from the x axis F Ay = 43.0 Sense is right F Ax = 61.4 F Bx = 61.4 F R = F By = -43.0

21 Resultant Force C Determine the sense, magnitude, and direction for the resultant force. 30. D 400.

22 Resultant Force Find the x and y components of vector C. F Cx = 300. cos60. C 300. F Cx = 150 F Cy 60. F Cx F Cy = 300. sin60. F Cy = 260

23 Resultant Force Find the x and y components of vector D. F Dx 30. F Dx = 400 cos30. F Dx = 350 F Dy D 400. F Dy = -400 sin30. F Dy = -200

24 Resultant Force List the forces according to sense. F Cy = F Dy = C F Cx = F Dx = D 400 Label right and up forces as positive, and label left and down forces as negative. F x F Cx = F Dx = Fy F Cy = F Dy =

25 Resultant Force Sum (S) the forces F x F Cx = F Dx = SF x = F Cx + F Dx SF x = = (right) F Y F Cy = F Dy = SF y = F Cy + F Dy SF y = ( ) = 59.8 (up) Sense is right and up.

26 Resultant Force Draw the x and y components of the resultant force. SF x = (right) SF y = 59.8 (up) Two ways to draw the X and Y components F R F R

27 Resultant Force Solve for magnitude. F a 2 + b 2 = c 2 R 59.8 (59.8 ) 2 +(496.4 ) 2 =F 2 R (59.8 ) (496.4 ) FR F R = 500 or 5.0x10 2 Magnitude is 5.0x10 2 (500 s)

28 Resultant Force Solve for direction. tan opp adj tan tan Direction is 7 counterclockwise from the positive X axis.

29 Resultant Force Draw the resultant force (F R ) Magnitude is 500. Direction is 7 counterclockwise from the positive x axis. Sense is right and up