Rectangular componentspioneer.netserv.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation... ·...
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Rectangular components y x z F r i F x ˆ j F y ˆ k F z ˆ k ˆ j ˆ i ˆ z θ x θ y θ 2 2 2 z y x F F F F + + = ) cos( x x F F θ = ) cos( y y F F θ = ) cos( z z F F θ = k F j F i F F z y x ˆ ˆ ˆ + + = r ) ˆ cos ˆ cos ˆ (cos k j i F F z y x θ θ θ + + = r ) ˆ ˆ ˆ ( k n j m i l F F + + = r F n F F ˆ = r x l θ cos = y m θ cos = z n θ cos = Direction cosine 1 2 2 2 = + + n m l
Transcript of Rectangular componentspioneer.netserv.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation... ·...
Rectangular components
y
x
z
Fr
iFxˆ
jFyˆ
kFzˆ
k j
i
zθ
xθyθ 222
zyx FFFF ++=
)cos( xx FF θ=)cos( yy FF θ=
)cos( zz FF θ=
kFjFiFF zyxˆˆˆ ++=
r
)ˆcosˆcosˆ(cos kjiFF zyx θθθ ++=r
)ˆˆˆ( knjmilFF ++=r
FnFF ˆ=r
xl θcos= ym θcos= zn θcos=
Direction cosine
1222 =++ nml
Writing a vector in 3D (1)1 Specification by two points on the line of action
xy
z
A (x1, y1, z1)
Fr B (x2, y2, z2)
ABAB rrr rrr−=
kzjyixrAˆˆˆ
111 ++=r
kzjyixrBˆˆˆ
222 ++=r
||ˆ
AB
ABF r
rFnFF r
rr==
kzzjyyixxrABˆ)(ˆ)(ˆ)( 121212 −+−+−=
r
212
212
212
121212
)()()(
ˆ)(ˆ)(ˆ)(zzyyxx
kzzjyyixxFF−+−+−
−+−+−=
r
Writing a vector in 3D (2)2 Specification by two angles which orient the line of action
y
x
z
Fr
xFr
yFrzF
rk j
i
θ
φ
xyFr
Fxy = F cos(φ)
Fz = F sin(φ)
Fx = Fxy cos(θ) = F cos(φ) cos(θ)
Fy = Fxy sin(θ) = F cos(φ) sin(θ)
Dot product(scalar)P
r
Qr
α Pcos(α)Projection of in the direction of
Pr
Qr . Q
)cos(αPQQP =⋅rr
Fr
nrnFFn ˆ⋅=
rnnFFn ˆˆ⋅=
rr
If is a unit vector , dot product expresses the projection of vector in the unit vector direction
Qr
)ˆ(n
)ˆˆˆ()ˆˆˆ(ˆ kjiknjmilFnFFn γβα ++⋅++=⋅=r
)( γβα nmlF ++=
1ˆˆˆˆˆˆ =⋅=⋅=⋅ kkjjii
0ˆˆˆˆˆˆˆˆˆˆˆˆ =⋅=⋅=⋅=⋅=⋅=⋅ jkkjikkiijji
Angle between two vectors)cos(θPQQP =⋅
rrFrom relation of dot product
The angle between vectors and isPr
Qr
PQQPrr
⋅= −1cosθ
The angle between vectors and isFr
n
FnF ˆ
cos 1 ⋅= −
r
θ
0ˆ =⋅nFr
nF ˆ⊥r
Sample 1A force F with a magnitude of 100 N is applied at the origin O of the axes x-y-zas shown. The line of action of F passes through a point A whose coordinates are 3m, 4m and 5m. Determine (a) the x, yand z scalar components of F, (b) the projection Fxy of F on the x-y plane, and (c) the projection FOB of F along the line OB.
Sample 2The cable BC carries a tension of 750 N. Write this tension as a force T acting on point B in terms of the unit vector i, j and k. The elbow at A forms a right angle.
Sample 3The tension in supporting cable BC is 3200 N. Write the force which this cable exerts on the boom OAB as a vector T. Determine the angles θx, θy and θz which the line of action of Tforms with the positive x-, y- and z-axes.
