7. The turnbuckle T is tightened until the tension in cable OA is 5 kN.
Express the force acting on point O as a vector. Determine the projection
of onto the y-axis and onto line OB. Note that OB and OC lies in the x-y
plane.
F
F
8. Determine the magnitude and
direction angles of the resultant force
acting on the bracket.
kjiF
xyxyFF
45sin45030cos45cos45030sin45cos450
11
1
21 FFFR
kjiF
2.31857.2751.1591
Resultant
kjiF
120cos60060cos60045cos6002
kjiF
30030026.4242
Direction angles for 𝑭 𝟐 906045 zyx
1coscoscos 222 zyx
1222 nml
Direction cosines
1cos60cos45cos 222 z
5.0cos25.0cos2 zz
1205.0cos90 zzz
kjiF
30030026.4242
Direction angles for 𝑹
029.097.633
2.18cos907.0
97.633
57.575cos418.0
97.633
16.265cos
coscoscos
zyx
zz
y
yx
xR
R
R
R
R
R
Direction Cosines of Resultant Force
3.88)029.0arccos(9.24)907.0arccos(3.65)418.0arccos( zyx
kjiF
2.31857.2751.1591
Resultant 21 FFFR
kjiR
3002.31830057.27526.4241.159
kjiR
2.1857.57516.265
Magnitude of Resultant Force
NRR 97.6332.1857.57516.265 222
9. Determine the parallel and normal components of force 𝐹 in vector form with
respect to a line passing through points A and B.
35
50°x
y
z
A
B
(3, 2, 5) m
(6, 4, 8) m
F=75 kN
F
kNFxy 311.6435
575
22
kNFF xyx 34.4150cos
kNFF xyy 265.4950sin
kNFz 59.3835
375
22
kjiF
59.38265.4934.41
Cartesian components of 𝑭
35
50°x
y
z
A
B
(3, 2, 5) m
(6, 4, 8) m
F=75 kN
F
kjiF
59.38265.4934.41
Unit vector of line AB
kjikji
nAB
639.0426.0639.0
323
323
222
Parallel component of 𝑭 to line AB (its scalar value) :
kNF
kjikjiF
nFF AB
14.72639.059.38426.0265.49639.034.41
639.0426.0639.059.38265.4934.41
//
//
//
Parallel component of 𝑭 to line AB (in vector form):
kjikjinFF AB
14.4676.3014.46639.0426.0639.014.72////
Normal component of 𝑭 to line AB (in vector form):
kjiFFF
55.7505.188.4//
ABn
10. An overhead crane is used to reposition the boxcar within a
railroad car-repair shop. If the boxcar begins to move along the rails
when the x-component of the cable tension reaches 3 kN, calculate
the necessary tension T in the cable. Determine the angle xy between
the cable and the vertical x-y plane.
Tx=3 kN, calculate tension T, the angle xy between the cable and the vertical x-y plane.
x-component of 𝑻
kjin
kjin
T
T
154.0617.077.0
145
45
222
kjiTT
154.0617.077.0
kNTT 896.3377.0 (magnitude of 𝑻)
kNTkNT zy 6.0896.3154.04.2896.3617.0
y and z-components of 𝑻
kjiT
6.04.23
Unit vector of 𝑻
𝑻 in vector form
kNTTT yxxy 84.34.23 2222
598.9896.089.3
84.3cos xy
xy
xyT
T
11. The spring of constant k = 2.6 kN/m is attached to the disk at point A and
to the end fitting at point B as shown. The spring is unstretched when A and
B are both zero. If the disk is rotated 15° clockwise and the end fitting is
rotated 30 ° counterclockwise, determine a vector expression for the force
which the spring exerts at point A.
12. The rectangular plate is supported by hinges along its side BC and
by the cable AE. If the cable tension is 300 N, determine the projection
onto line BC of the force exerted on the plate by the cable. Note that E is
the midpoint of the horizontal upper edge of the structural support.
If T=300 N, determine the projection onto line BC of the force exerted on the plate by the cable.
x
y
z
𝑪𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔 𝒐𝒇 𝒑𝒐𝒊𝒏𝒕𝒔 𝑨,𝑩, 𝑪 𝒂𝒏𝒅 𝑬 𝒘𝒊𝒕𝒉 𝒓𝒆𝒔𝒑𝒆𝒄𝒕 𝒕𝒐 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆 𝒔𝒚𝒔𝒕𝒆𝒎
𝑨 400, 0, 0 B (0, 0, 0)
𝑪 0, 1200𝑠𝑖𝑛25,−1200𝑐𝑜𝑠25C (0, 507.14, 1087.57)
𝑬 0, 1200𝑠𝑖𝑛25,−600𝑐𝑜𝑠25E (0, 507.14, 543.78)
kjiT
kjinTT T
21.19319.18012.142
33.844
78.54314.507400300
kjkjinTT BCBC
906.0423.021.19319.18012.142
Unit vector of 𝒍𝒊𝒏𝒆 𝑩𝑪
z
y
25o25o
BCn
kjjknBC
906.0423.025sin25cos
Projection of T onto line BC
NTBC 26.251
13. The y and z scalar components of a force are 100 N and 200 N,
respectively. If the direction cosine l=cosx of the line of action of the
force is 0.5, write 𝐹 as a vector.
5.0200100 lNFNF zy
75.05.011 22222222 nmnmnml
yzzy FNFF 61.22322
FnFFFmFF zzyy coscos
NFF 2.25861.22375.0
NFF xx 1.129cos
kjiF
2001001.129
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