Force Vectors
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
Illustrating Vectors Vectors are represented by arrows. Include magnitude, direction, and sense. Magnitude: The length of the line segment Magnitude = 3 30 +X
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
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
Sense +y (up) +y (up) -x (left) -y (down) (0,0) +x (right) -y (down) -x (left) +x (right)
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)
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)
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
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
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 35.0 -x +x -y adj = F Ax
Vector X and Y Components Solve for F AX cos adj hyp cos FAX A cos35.0 F AX 75.0 +y FAX 75.0 cos35.0 A 75.0 FAX 61.4 opp = F Ay 35.0 -x +x -y adj = F Ax
Vector X and Y Components Solve for F AY sin opp hyp sin FAY A sin35.0 F AY 75.0 +y FAY 75.0 sin35.0 A 75.0 FAY 43.0 opp = F Ay 35.0 -x +x -y adj = F Ax
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 35.0 +x opp = F By -y B 75.0
Vector X and Y Components Your Turn Solve for F BX cos adj hyp cos FBX B cos35.0 F BX 75.0 +y FBX 75.0 cos35.0 -x adj = F Bx 35.0 +x FBX 61.4 opp = F By -y B 75.0
Vector X and Y Components Your Turn Solve for F BY sin opp hyp sin B F By sin35.0 F By 75.0 +y FBy 75.0 sin35.0 -x adj = F Bx 35.0 +x FBy 43.0 opp = F By -y B 75.0
Resultant Force Two people are pulling a boat to shore. They are pulling with the same magnitude. A 75.0 35.0 35.0 B 75.0
Resultant Force F Ay = 43.0 35 35 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 = +61.4 F Bx = +61.4 F y F Ay = +43.0 F By = -43.0 B 75 F By = -43.0
Resultant Force F x F Ax = +61.4 F Bx = +61.4 F y F Ay = +43.0 F By = -43.0 Sum (S) the forces SF x = F Ax + F Bx SF x = 61.436 + 61.436 SF x = 122.9 (right) SF y = F Ay + F By SF y = 43.018 + (-43.018 ) = 0 Magnitude is 122.9. Direction is 0 from the x axis Sense is right.
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 = 122.9 F By = -43.0
Resultant Force C 300. 60. Determine the sense, magnitude, and direction for the resultant force. 30. D 400.
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
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
Resultant Force List the forces according to sense. F Cy = 259.8 F Dy = -200.0 C 300 60 F Cx = 150.0 30 F Dx = 346.4 D 400 Label right and up forces as positive, and label left and down forces as negative. F x F Cx = +150.0 F Dx = +346.4 Fy F Cy = +259.8 F Dy = -200.0
Resultant Force Sum (S) the forces F x F Cx = +150.0 F Dx = +346.4 SF x = F Cx + F Dx SF x = 150.0 + 346.4 = 496.4 (right) F Y F Cy = +259.8 F Dy = -200.0 SF y = F Cy + F Dy SF y = 259.8 + (-200.0 ) = 59.8 (up) Sense is right and up.
Resultant Force Draw the x and y components of the resultant force. SF x = 496.4 (right) SF y = 59.8 (up) Two ways to draw the X and Y components F R 496.4 59.8 59.8 496.4 F R
Resultant Force Solve for magnitude. F a 2 + b 2 = c 2 R 59.8 (59.8 ) 2 +(496.4 ) 2 =F 2 R 496.4 2 2 (59.8 ) (496.4 ) FR F R = 500 or 5.0x10 2 Magnitude is 5.0x10 2 (500 s)
Resultant Force Solve for direction. tan opp adj 59.8 500 tan 59.8 496.4 1 59.8 tan 496.4 496.4 7 Direction is 7 counterclockwise from the positive X axis.
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. 7 500