𝐹Ԧ1 𝐹Ԧ2 magnitude and direction of the resultant...
Transcript of 𝐹Ԧ1 𝐹Ԧ2 magnitude and direction of the resultant...
P-1: The screw eye is subjected to two forces, Ԧ𝐹1 and Ԧ𝐹2. Determine themagnitude and direction of the resultant force.
P-2: Resolve the horizontal 600-N force in the figure into componentsacting along the u and v axes and determine the magnitudes of thesecomponents. 600-N
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P-3: The vector 𝑉 lies in the plane defined by the intersecting lines LA and LB. Its magnitude is 400
units. Suppose that you want to resolve 𝑉 into vector components parallel to LA and LB. Determine the
magnitudes of the vector components.
80°
60°
V
LA
LB
P-4: Determine the projections Pa and Pb of 𝑉 onto the lines LA and LB.
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P-5: Express forces as the vector form in terms of the unit vectors Ԧ𝑖 and Ԧ𝑗 .
(a) (b) (c)
Fx
Fy
jiF
jiF
jFiFF yx
39.32102.383
40sin50040cos500
Fx
Fy
Fx
Fy
40o
30o
jiF
jiF
jFiFF yx
2004.346
30sin40030cos400
jiF
F
F
jFiFF
y
x
yx
28.4
512
52.5
512
122.5
13
22
13
22
P-6: An elastic rubber band is attached to points A and B.Determine its length and its direction measured from A toward B.
P-8: The force acts on the hook. Express it as aCartesian vector.
P-7: Express the force as a Cartesian vector.
4
P-9: Express the forces shown in the figures as a Cartesian vector.
5
F=50 N
P-10: Determine the magnitude of the resultant 𝑉=𝑉1+𝑉2 and the angle q which 𝑉 makes with the
positive x–axis.
V1 = 9 units
V2 = 7 units
30°
y
x
4
3
6
7
P-11: The frame shown is subjected to a horizontal force Ԧ𝐹 = 300Ԧ𝑗. Determine the magnitude of thecomponents of this force parallel and perpendicular to member AB.
Ԧ𝐹 = 300Ԧ𝑗
P-12: Ԧ𝐴=3Ԧ𝑖 +2Ԧ𝑗 5𝑘 and 𝐵 =2Ԧ𝑖 7Ԧ𝑗 6𝑘. Determine;
?AB
?BA
?AB
?BA
?BA
?BA
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Consider the straight lines OA and OB.
a) Determine the components of a unit vector that is
perpendicular to both OA and OB.
b) What is the minimum distance from point A to the line OB?
A (10, 2, 3) m
y
xq
z
B (6, 6, 3) m
O
P-13:
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a) Determine the components of a unit vector that is perpendicular to both OA and OB.
A (10, 2, 3) m
q
B (6, 6, 3) m
O
b) What is the minimum distance from point A to the line OB?
OBr /
OAr /
kjirkjir OAOB
3210366 //
md
drrrr
rd
OBOA
r
OBOA
OA
71.9
36636.87sin
sin
9
222
////
/
q
q
Vector normal to both OA and OB is
kjir
ijikjkr
kjikjirrr OBOA
724812
18186123060
3663210//
Unit vector normal to both OA and OB is
kjinkji
r
rn
824.0549.0137.0724812
724812
36.87
222
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Consider the triangle ABC.
a) What is the surface area of ABC?
b) Determine the unit vector of the outer normal of surface ABC.
c) What is the angle between AC and AB?
A (12, 0, 0) m
y
x
z
B (0, 16, 0) m
C (0, 0, 5) m
P-14:
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a)What is the surface area of ABC?
A (12, 0, 0) m
y
x
z
B (0, 16, 0) m
C (0, 0, 5) mB
A
C
c) What is the angle between AC and AB?
kiACjiAB
5121612
AC
AB
ikjACAB
kijiACAB
8019260
5121612
2222
24.1082
1926080mABCs
b) Determine the unit vector of the outer normal of surface ABC.
kjikji
ACAB
ACABn
887.0277.0369.0
48.216
1926080
368.56cos1320144
cos51216125121612
cos
2222
kiji
ACABACAB
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P-15: Determine the magnitude and direction angles of theresultant force acting at A .
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P-16: The door is held opened by two chains. Forces in AB and CD are FAB=300 N and FCD=250 N, respectively.
a) Write these forces in vector form.
b) Determine the resultant of these forces and the direction angles of the resultant force.
c) Determine the projection of force FAB on line CD.
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P-17: Cables extend from A to B and from A to C. The cable
AC exerts a 1000 N force vector Ԧ𝐹 at A.
a) What is the angle between the cables AB and AC?
b) Determine the vector component of Ԧ𝐹 parallel to the
cable AB and write the vector expression of this
component. Also determine the normal component of Ԧ𝐹
to the cable AB.
F
A (0, 7, 0) m
y
x
zB (0, 0, 10) m
C (14, 0, 14) m
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P-18: Observers on Earth at points A and B measure direction cosines of position vectors to the
space shuttle at C as
For Ԧ𝑟𝐶/𝐴: cos qx = 0.360, cos qy = 0.480 and cos qz <90°.
For Ԧ𝑟𝐶/𝐵: cos qx = –0.515, cos qy = – 0.606 and cos qz <90°.
Determine the x, y, z coordinates of the space shuttle located at point C and also the shortest distance
from point C to direction AB.
A
B
C
x
y
z
(520, 640, 0) km 16
P-19: Consider force Ԧ𝐹 = 12Ԧ𝑖 + 24Ԧ𝑗 + 24𝑘 𝑁. Resolve Ԧ𝐹
into two components, one parallel to line AB and one
perpendicular to line AB.
A(10, 2, 14) Position vector
F
kjir
kjir
AB
AB
884
146261014
/
/
B(14, 6, 6)
Unit vector
n
kjikji
r
rn
AB
AB
3
2
3
2
3
1
884
884
12
222/
/
Parallel component
kjikjinFF
NkjikjinFF
3
8
3
8
3
4
3
2
3
2
3
14
43
224
3
224
3
112
3
2
3
2
3
1242412
////
//
(in vector form)
Normal component
kjikjikjiFFF
67.2633.2167.103
8
3
8
3
4242412//
//F
F
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P-20: A house with 98000 kg mass is built on a steep slope defined by points A, B, and C.
To help assess the possibility of slope failure (mud slide), it is necessary to
a) Determine the components of the weight in directions normal and parallel (tangent) to
the slope.
b) Determine the component vectors of the weight in directions normal and parallel
(tangent) to the slope.
c) Determine the smallest distance from point O to the slope.
(g=9.81 m/s2)
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a-b) Determine the components of the weight in directions normal and parallel (tangent) to the slope.
W
Wn
Wt
W: Weight of house
Wn: Normal component of the weight to slope
Wt : Parallel component of the weight to slope
kjiN
jikjACABN
21600720010800
18012060180
jiAC
kjAB
180120
60180
kjikji
N
Nn
7
6
7
2
7
3
21600720010800
21600720010800
25200
222
kkW
96138081.998000
AB
AC
B
A
C
NkjiknWWn 8240407
6
7
2
7
3961380
kjikjinWW nn
706320235440353160
7
6
7
2
7
3824040
(Vector normal to the slope)
(Unit vector normal to the slope)
(its magnitude)
(in vector form)
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c) Determine the smallest distance from point O to the slope.
W
Wn
Wt
kjiW
kjikWWW
t
nt
255060235440353160
706320235440353160961380
(its magnitude)
(in vector form)
NW
W
t
t
4.495186
255060235440353160 222
O
B
C
n
mkjiinOCd 43.517
6
7
2
7
3120
mkjijnOAd
mkjiknOBd
43.517
6
7
2
7
3180
43.517
6
7
2
7
360
CThe smallest distance can also be determined as follows
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