3D-Force System

RECTANGULAR COMPONENTS

• Many problems in mechanics require analysis in three dimensions, and for such problems it is often necessary to resolve a force into its three mutually perpendicular components.

x

jFy

F

x

iFx

kFz

yz

z

y

y

xiF x

cos

jF y

cos

yx

F

zzyyxx FFFFFF cos;cos;cos

kFjFiFF

kFjFiFF

zyx

Zyx

)cos()cos()cos(

)( knjmilFF

F

n

ml

z

y

x of cosinesDirection

cos

coscos

222zyx FFFF

1 :Note 222 nml

• Where is unit vector in the direction of force

x

jFy

F

x

iFx

kFz

yz

z

y

)( knjmilFF

)( FnFF

knjmilnF

Fn

Specification of a force vector• (a) Specification by two points on the line of action of the

force.• (b) Specification by two angles which orient the line of action

of the force.

• a) Two points:

• b) Two angles:

)sinsincoscos(cos kjiF

kFjFiFF zyx

Problem-1

x

D

C

O

z

y

A

B

30 mT=24 kN

6m

5m

18 m

The turnbuckle is tightened until the tension in the cable AB equals 24 kN. Express the tension T acting on point A as a vector

A=A(0, 18, 30) B=B(6, 13, 0)

kN 22.2387.364.4

22.2387.364.4)968.0161.0194.0(24

968.0161.0194.0

968.031

300 ;161.031

1813 ;194.031

06m 31)300()1813(0)-(6points obetween tw Distance

; 222

kjiT

kjikjinTT

kjiknjmiln

nml

knjmilnnTT

AB

AB

ABAB

x

D

C

O

z

y

A

B

30 mT=24 kN

6m

5m

18 m

Problem-2

kN 10

y

x

z

045

030

Consider a force as shown in the Figure. The magnitude of this force is 10 kN. Express it as a vector.

kN 00.512.612.6 kjiF

kFjFiFF zyx

kN 0.530sin10

kN 66.830cos100

0

z

xy

F

F

kN 10

y

x

z

045

030

xF

yF

xyF

zF

kN 12.645sin66.845sin

kN 12.645cos66.845cos00

00

xyy

xyx

FF

FF

(Orthogonal) ProjectionF

n

nFnAF direction- in theor BC lineon of ProjectionA

BC

ABCn line ofdirection in ther unit vecto

nnmlknjmiln r unit vecto of cosinesdirection theare ,, Here where,

ABzzn

AByym

ABxxl )( ;)( ;)( 121212

n

),,( 111 zyxA),,( 222 zyxB

212

212

212 )()()( zzyyxxAB

Problem-1

u

v

F = 100 N

O45

15

For the shown force:a. Determine the magnitudes of the projection of the force F = 100 N onto the

u and v axes.b. Determine the magnitudes of the components of force F along the u and v

axes.

• Projections of the force onto u and v axes

• Components of the force along u and v

axes

u

v

100 N

O45

15u

v

100 N

O45

15

N 6.9615cos100

N 7.7045cos100

proj

proj

v

u

F

F

projcomp

projcomp

compcomp

N 6.81

N 9.29120sin

10045sin15sin

vv

uu

vu

FF

FF

FF

Note

Rectangular components of a force along the two chosen perpendicular axes, and projection of the force onto the same axes are the same.

N 1003.646.76:

N 3.6440sin100

N 6.7640cos100

2222

projcomp

projcomp

xx

yy

xx

FFFCheck

FF

FF

Ox

y

100 N

40

Problem-2

y

z

A

O

4 m

3 m

6 m

N 176141106 kjiF

x

. line thealong ) find ..( projection of form vector theDetermine

Figure. in theshown as axes theof Oorigin at the applied is forceA

OAFeiF

x-y-zF

OAOA

384.081.7

)03(

;512.081.7

)04( ;768.081.7

)06(m 81.7)03()04()06(

)()()(222

212

212

212

n

ml

OA

zzyyxxLOA

y

z

A

O4 m

3 m

6 mx

OAF

N 176141106 kjiF

kjin

knjmiln

OA

OA

OA

384.0512.0768.0

:is line thealongr unit vecto Therefore,

N 18.221)384.0512.0768.0).(176141106(. kjikjinFF OAOA

N 93.8424.11387.169

)384.0512.0768.0(18.221).(

kjiF

kjinFnnFF

OA

OAOAOAOAOAOA

y

z

A

O4 m

3 m

6 mx

OAF

N 176141106 kjiF

Problem-3A force with a magnitude of 100 N is applied at the origin O of the axes x-y-z as shown. The line of action of force passes through a point A. Determine the projection Fxy of 100N force on the x-y plane.

x

y

z

A

O

4 m

3 m

6 m

N 100

923.0810.7211.7

)03()04()06(

)00()04()06(cos

222

222

xy

N 3.92)923.0(100cos xyxy FF

(6,4,3)

(6,4,0)

(0,0,0)

x

y

z

A

O

4 m

3 m

6 m

N 100F

xyθB

xyF

Alternative Solution

N ˆ4.38ˆ2.51ˆ8.76810.7

ˆ3ˆ4ˆ6100

)03()04()06(

ˆ)03(ˆ)04(ˆ)06(100)ˆˆˆ(222

kjikjiF

kjiknjmilFF

jikjin

kjiknjmiln

OB

OB

ˆ554.0ˆ832.0211.7

ˆ0ˆ4ˆ6

)00()04()06(

ˆ)00(ˆ)04(ˆ)06(ˆˆˆ222

N 3.92

N 3.9204.38554.02.51832.08.76

xy

OBxy

F

nFF

(6,4,3)

(6,4,0)

(0,0,0)

x

y

z

A

O

4 m

3 m

6 m

N 100F

xyθB

xyF