BAB 3
SISTIM RESULTAN GAYA
Sasaran bab ini adalah : Mahasiswa Mampu untuk
1. Menjelaskan konsep moment satu gaya dan menghitungnya dalam 2 dan 3 dimensi.
2. Menetapkan momen suatu gaya di suatu garis/sumbu. 3. Mendefine the moment of a couple.
4. Menghitung Resultan sistim gaya nonconcurrent/tidak bersamaan.
5. Mengindikasikan bagaimana mengurangi beban yang terdistribusi ke dalam sisitim resultan gaya pada suatu lokasi tertentu.
Definisi Moment Gaya
The moment of a force about a point or an axis provides a measure of the tendency of the force to cause a body to rotate about the point or axis
Momen di sumbu z akibat gaya pada sumbu x tegak lurus sumbu y
Fx - horizontal force
dy - distance from point O to force Mo - moment of force about point O (Mo)z -moment of force about axis z
Momen di sumbu x akibat gaya pada sumbu z tegak lurus sumbu y
Fz - horizontal force
NO moment
Resultant Moment of a System of Coplanar Forces
Counterclockwise is positive by scalar sign convention
CONTOH : For each case, find the moment of the force about the point O
M Fd
Magnitude of the moment
=
Direction of the moment Right Hand Rule
O
R
M Fd
+ =
∑
(
)( )
O
CROSS PRODUCT
Another method of vector multiplication
Read as C equals A cross B
Magnitude:
Direction: Right Hand Rule
1. Cross Product
Not Commutative.
2. Scalar Multiplication
3. Distributive Law:
Unit Vektor
Right Hand Rule
Bentuk Cartesiannya :
Operasi proses :
Formula ekivalen :
Moment of a Force -Vector Formulation
PRINSIP TRANSMISIBLE
r vector can be taken to any point on line of action of F
For Element
Bentuk Cartesiannya dan Formulasi vektor cartesian :
MOMENT
RESULTAN MOMEN DARI SUATU SISTIM GAYA
Contoh soal : Cari moment di titik A
2. Force vector is 60 N times a unit vector in direction of
3. Moment
Vektor posisi :
Vektor Gaya :
Vektor momen
Contoh soal : Determine the resultant moment at O and the coordinate direction angles for the moment.
Vektor posisi (lihat gambar 04.17.b)
Vektor gayanya :
Vektor momennya :
Sudut Arahnya :
Principle of Moments
The moment of a force about a point is equal to the sum of the moments of the force’s components about the point.
A
[5(20) 40(0)]i [0]j [0(40) 60(5)]k
ˆ ˆ ˆ
0.3841i 0.5121j 0.7682k
= − + ⋅
ˆu 0.3841i 0.5121j 0.7682k
cos 0.3841 67.4
cos 0.5121 121
cos 0.7682 39.8
= − +
α = α =
β = − β =
Contoh Soal : Determine the moment of the force about A
Moment of a Force About a Specified Axis
1. Sometimes need the component of a moment about a particular axis
2. Scalar analysis: ma=Fda where da is the ⊥ or shortest distance from the force line of action to the axis of interest
3. Vector analysis: ma= ua • (r x F) 4. This called the triple scalar product
(
)(
)
(
)
o
A
A
CB d 100cos 45 70.71mm 0.07071m
M Fd 200N 0.07071m 14.1N m
ˆ
M 14.1k N m
= = = =
= = = ⋅
= ⋅
1. Ma is a scalar with positive or negative sign.
2. Positive sign indicates the sense is the same as unit vector ua.
3. Negative sign indicates the sense is opposite the sense of unit vector ua.
O
F, O
a ' a '
M F O a ' a '
In general case, to find
moment of at point ,
about axis , we project
of about onto .
x,y,z components of unit vector directed along the axis.
r r r
x,y,z components of position vector directed from point O on the axis to any
point along the line of action of the force.
F F F
⇒
⇒
⇒
