Bölüm 3
RİJİT CİSİMLER: Eşdeğer Kuvvet Sistemleri
F
F’
Bir rijit cisme etki eden
kuvvetler iki gruba
ayrılabilir: (1) dış
kuvvetlet(anılan rijit
cismin üzerine diğer rijit
cisimlerin uyguladığı
kuvvetleri temsil eder)
ve (2) iç kuvvetler (rijit cismi meydana getiren parçacıkları
bir arada tutan kuvvetleri temsil eder)
F
F’
kuvvetin kaydırılabilme ilkesine göre, bir rijit cisim üzerine
etkiyen bir dış kuvvetin etykisi, kuvvet etki çizgisi üzerinde
hareket ettirilirse, değişmez. Bir rijit cisme iki farklı noktada
etki eden iki kuvvetin, eğer şiddetleri, yönleri ve etki çizgileri
aynı ise, etkileri aynıdır. Bu, iki kuvvet sisteminin eşdeğer
olduğu şeklinde ifade edilir.
V =PxQ
İki vektörün vektörel çarpımı
şu şekilde tarif edilir.
Q
q
V = Px Q
P
Elde edilen vektör V, P ve
Q vektörlerinin oluşturduğu
düzleme diktir, ve şiddeti ise,
V = PQ sin q ile verilir
V vöktörünün ucundaki bir kişi için, P den Q ya doğru saat
yönünün tersine doğru bir dönme gözlenir. P, Q ve V vektörleri
Sağ el kuralına göre tarif edilmiş bir sistem oluştururlar.
Buradan,
Q x P = - (P x Q) yazılabilir.
j
k
i
It follows from the definition of the vector
product of two vectors that the vector
products of unit vectors i, j, and k are
ixi=jxj=kxk=0
ixj=k , jxk=i, kxi=j , ixk=-j , jxi=-k, kxj=-i
The rectangular components of the vector product V of two
vectors P and Q are determined as follows: Given
P = Px i + Py j + Pz k
Q = Qx i + Q y j + Qz k
The determinant containing each component of P and Q is
expanded to define the vector V, as well as its scalar
components
P = Px i + Py j + Pz k
i
V = P x Q = Px
Qx
Q = Qx i + Qy j + Qz k
j
k
Py Pz = Vx i + Vy j + Vz k
Qy Q z
Öyleki,
Vx = Py Qz - Pz Qy
Vy = Pz Qx - Px Qz
Vz = Px Qy - Py Qx
Mo
The moment of force F
about point O is defined
as the vector product
F
MO = r x F
where r is the position
r
O
q
vector drawn from point O
d
to the point of application
A
of the force F. The angle
between the lines of action
of r and F is q.
The magnitude of the moment of F about O can be expressed as
MO = rF sin q = Fd
where d is the perpendicular distance from O to the line of action
of F.
y
Fy j
A (x , y, z )
yj
r
O
Fz k
xi
zk
z
Bir F kuvvetinin
Fx i oluşturduğu Mo
momentinin kartezyen
bileşenleri, r x F
ifadesinin
x
determinantının açılımı
ile belirlenir. .
i j k
Mo = r x F = x y z = Mx i + My j + Mzk
Fx Fy Fz
Yani,
Mx = y Fz - z Fy
Mz = x Fy - y Fx
My = zFx - x Fz
y
A (x A, yA, z A)
Fy j
B (x B, yB, z B)
Fx i
r
Fz k
O
x
z
En genel halde,
herhangi bir A
noktasına uygulanan bir
F kuvvetinin, herhangi
bir B noktasına göre
momenti şunu
yazabiliriz:
i
j
Öyleki,
MB = rA/B x F = xA/B yA/B
Fx Fy
rA/B = xA/B i + yA/B j + zA/B k
ve
xA/B = xA- xB
yA/B = yA- yB
k
zA/B
Fz
zA/B = zA- zB
y
F
Fy j
rA/B
(yA - yB ) j
A
Fx i
B
(xA - xB ) i
O
z
MB = M B k
x
In the case of problems
involving only two
dimensions, the force F
can be assumed to lie
in the xy plane. Its
moment about point B
is perpendicular to that
plane. It can be
completely defined by
the scalar
MB = (xA- xB )Fy - (yA- yB ) Fx
The right-hand rule is useful for defining the direction of the
moment as either into or out of the plane (positive or negative
k direction).
Q
q
The scalar product of two
vectors P and Q is denoted
as P Q ,and is defined as
P Q = PQ cos q
P
where q is the angle between
the two vectors
The scalar product of P and Q is expressed in terms of the
rectangular components of the two vectors as
P Q = Px Qx + Py Qy + Pz Qz
y
L
qy
l
qx P
O
z
A
x
qz
The projection of a vector
P on an axis OL can be
obtained by forming the
scalar product of P and the
unit vector l along OL.
POL = P l
Using rectangular components,
POL = Px cos qx + Py cos qy + Pz cos qz
The mixed triple product of three vectors S, P, and Q is
S (P x Q ) =
Sx Sy S z
Px Py Pz
Q x Qy Q z
The elements of the determinant are the rectangular
components of the three vectors.
y
L
MO
l
F
C
r
O
A (x, y, z)
x
z
MOL = l
MO = l (r x F) =
The moment of a
force F about an axis
OL is the projection
OC on OL of the
moment MO of the
force F. This can be
written as a mixed
triple product.
lx ly lz
x y z
Fx Fy Fz
lx, ly , lz = direction cosines of axis OL
x, y , z = components of r
Fx, Fy , Fz = components of F
M
-F
d
F
Two forces F and -F having the same magnitude,
parallel lines of action, and opposite sense are said
to form a couple.
The moment of a couple is independent of the point
about which it is computed; it is a vector M perpendicular
to the plane of the couple and equal in magnitude to the
product Fd.
M
y
y
-F
y
My
(M = Fd)
d
O
z
F
x
Mx
O
O
z
M
x
z
Mz
Two couples having the same moment M are equivalent
(they have the same effect on a given rigid body).
x
F
r
O
A
F
MO
A
O
Any force F acting at a point A of a rigid body can be replaced
by a force-couple system at an arbitrary point O, consisting
of the force F applied at O and a couple of moment MO equal to
the moment about point O of the force F in its original position.
The force vector F and the couple vector MO are always
perpendicular to each other.
F3
F1
A3
r3
M1
r1 A1
r2 A
2
O
F2
F3
F1
O
M3
R
M2
O
F2
R
MO
Any system of forces can be reduced to a force-couple
system at a given point O. First, each of the forces of the
system is replaced by an equivalent force-couple system at O.
Then all of the forces are added to obtain a resultant force R,
and all of couples are added to obtain a resultant couple vector
R
R
MO. In general, the resultant force R and the couple vector MO
will not be perpendicular to each other.
F3
F1
R
A3
r3
r1 A1
r2 A
2
O
F2
O
R
MO
Rijit cisimler dikkate alındığı sürece, iki kuvvet sistemi,
F1, F2, F3 . . . , ve F’1, F’2, F’3 . . . , sadece ve sadece aşağıdaki
şartlar altında eşdeğerdirler:
S F = S F’
ve
S Mo = S Mo’
F3
F1
R
A3
r1 A1
r2 A
2
O
F2
O
R
MO
If the resultant force R
and the resultant
R
couple vector MO are
perpendicular to each
other, the force-couple
system at O can be
further reduced to a
single resultant force.
This is the case for systems consisting of either
(a) concurrent forces,
(b) aynı düzlemdeki kuvvetler, veya
(c) paralel kuvvetler.
If the resultant force and couple are directed along the same
line, the force-couple system is termed a wrench.