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e14 - applied mechanics: statics

2textbook. russell c hibbeler: statics.

e14 - applied mechanics: statics

textbook.russell c. hibbelerprentice hall, 12th editionengineering mechanics staticsstatics study pack for engineering mechanics

3syllabus

e14 - applied mechanics: statics

first homework

42. force vectors

• to show how to add 3dforces and resolve forcesinto components using theparallelogram law• to express 3d force andposition in Cartesian vectorform and explain how to de-termine the vector’s magni-tude and direction• to introduce the 3d dotproduct in order to deter-mine the angle between twovectors or the projection ofone vector onto another

today‘s objectives

52.2 vector operations

vector addition and subtractionvector addition R = A + B = B + A

vector subtraction R = A - B = -B + A

62.3 vector addition and forces

I. find the resultant force

FR = F1 + F2

parallelogram vs triangle

72.3 vector addition and forces

II. find the components of a force

parallelogram vs triangle

FR = Fu + Fv

82.3 vector addition and forces

Ia. find the resultant of several forces

FR = F1 + F2 + F3

double parallelogram

92.4 addition of forces (2d)

force addition in cartesian coordinates• we can easily add and subtractforces using a Cartesian coordinatesystem

• we can determine the magnitudeof a force using Pythagoras (2d)

• we can determine the direction of a force using trigonometry

FRx = "FxFRy = "Fy

FRxFRy

FR =

FR = # [ FRx2 + FRy

2 ]

! = tan-1[ FRy / FRx ]

Fx = |F| cos !Fy = |F| sin !

FxFy

F =

10

• we can easily add and subtractforces using a Cartesian coordinatesystem

• we can determine the magnitudeof a force using Pythagoras (3d)

• we can determine the direction ofa force using trigonometry

2.5 addition of forces (3d)

force addition in cartesian coordinates

FRx = "FxFRy = "FyFRz = "Fz

FxFyFz

F =

FR = # [ FRx2 + FRy

2 + FRz2 ]

" = cos-1[ FRx / F ]

Fx = |F| cos "Fy = |F| cos !Fz = |F| cos "

FRxFRyFRz

FR =F

Fy y

Fz z

Fx x

112.5 addition of cartesian vectors (3d)

force addition in cartesian coordinates

the resultant force acting on the bow of the ship can bedetermined by first representing each rope force as acartesian vector and then summing the components Fx , Fy , Fz

12

express force as Cartesian vector• calculate the angle "

2.6 addition of cartesian vectors (3d)

example 2.8

• calculate the components Fx , Fy , Fz

• control F

13

express force as Cartesian vector• calculate the angle "

2.6 addition of cartesian vectors (3d)

example 2.8

FxFyFz

F =

cos2" + cos2!+ cos2" = 1cos2" + cos2600+ cos2450 = 1cos" = # [ 1 - 0.52 - 0.7072 ] = 0.5" = 600

F cos "F cos #F cos $

= =100 N100 N141 N

• calculate the components Fx , Fy , Fz

• control FF = #! 1002 + 1002 + 1412 ]N = 200 N

142.7 position vectors (3d)

positon vectors define the direction

establish a cartesian coordinate system and determine thecoordinates of points A and B. determine the position vector r= rB - rA along the cable. its magnitude r determines the lengthof the cable, its unit vector u = r / r defines the direction.

152.8 forces along a line

force projection onto a given direction

the force F along the chain can be represented as a cartesianvector by calculating the position vector r along the chain.then we determine the unit vector u = r / r and the magnitudeF, such that F = F u.

162.9 dot product

I. angle between two force vectors

we can determine the angle ! between the rope and the beamby formulating the unit vetors along the beam ub and the ropeur and then using the dot product cos ! = ub # ur .

172.9 dot product

II. projection of force along a direction

we can determine the projection of the cable force along thebeam Fb by first finding the unit vector ub along the beam andthen taking the dot product Fb = Fb # ub.

182.9 dot product

example 2.17

the frame is subjected to a horizontal force F = [ 0, 300, 0 ] N.determine the magnitude of this force along the member AB.