Problems (Equilibrium of Particles) 1. -...
Transcript of Problems (Equilibrium of Particles) 1. -...
1. The 40-kg block rests on the rough surface. Length of the spring is 180 mm in the position
shown. Unstretched length of the spring is 200 mm. Determine the coefficient of friction
required for the equilibrium. ( m = ? )
Problems (Equilibrium of Particles)
FBD of Block
20o
x
yW=mg
For spring: l=180 mm, lo=200 mm (spring is compressed)
NlkFspring 5018.02.02500
NNNmgFy 74.36820cos81.940020cos:0
NFmgFFF frictionfrictionspringx 21.18420sin81.94050020sin:0
5.0N
FNF
friction
friction mm
Equations of equilibrium
2. 4-kg sphere rests on the smooth parabolic surface. Determine the reaction of the
surface on the sphere and the mass mB required to hold it in the equilibrium position.
A
mB
x
y
0.4 m
60o
y=2.5x2
Problems (Equilibrium of Particles)
A
x
y
0.4 m
60o
y=2.5x2
FBD of Sphere
q=63.43o
N
W=mg T
Tangent to parabola
o
.x
.xdx
dytan 436325
40
N.T
.sinNcosT
Fx
791
0436360
0
Equations of equilibrium
N.TN.N
..cosNsinT
F
N.
y
12356219
08194436360
0
791
FBD of Block B
T
mBg
kg.m
.mT
F
B
B
.
y
583
0819
0
1235
3. Determine the stretch in springs AC and AB for equilibrium of the 2 kg block. The springs
are shown in the equilibrium position.
Problems (Equilibrium of Particles)
SOLUTION:
FBD of A
From equilibrium of A :
45o
FAC
AFAB
36.87o
x
y
W=mg=2(9.81) N
045cos87.36cos:0 ACABx FFF
2
1
081.9245sin87.36sin:0 ACABy FFF
mxNF ACAC 793.020
86.1586.15
mxNF ABAB 467.030
01.1401.14
4. A continuous cable of total length 4 m is wrapped
around the small pulleys at A, B, C and D. If each spring is
stretched 300 mm, determine the mass m of each block
for equilibrium. Neglect the weight of the pulleys and
cords. The springs are unstretched when d=2 m.
Problems (Equilibrium of Particles)
FBD of Pulley B
Tqq
C
T
Fspring
NlkFspring 1503.0500
sin
750sin2150
0
TT
Fy
md 290 qC'
A'
A
C
Bq
1 m0.7 m
NT 1.1074.441
7.0arcsin
q
q
T
Tq
Bmg
FBD of Pulley C
kgm
gmT
Fx
6.15
0cos2
0
81.9
q
5. The pail and its contents have a mass of 10 kg. If the length of cable ABC is 7 m, determine
the horizontal distance x for equilibrium. Also find the tension in the cable. The mass of the
small pulley at B is small and can be neglected.
1 m
4 mx
A
B C
Problems (Equilibrium of Particles)
L=7 m, x = ? (for equilibrium). T=?
1 m
4 mx
A
B C
x
yA
B
C
yy 1
q q'
FBD of B
T
W=mg
q q'
T From equilibrium of B :
qqqq 0coscos:0 TTFx
0sin2:0 WTFy q
From geometry: xaaxaaxxa
x
a
x
4
747
7
4cos
q
15.557
4
4
7cos qq
x
x
a
xxy
x
y44.144.1tan q
mxx
x
x
x
x
y
x
y347.2
4
144.144.1
4
1
(*)
(*) NTT 817.59081.91015.55sin2
B
x4x
6. The cylinder of mass 1 kg having a
very small diameter is held against a
semi-cylinder with a much larger
diameter by two identical springs,
which are fixed to points C and C on
the ground. The springs are
unstretched when at point A. Knowing
that the small cylinder is in
equilibrium at point B, what is the
spring constant?
r=200 mm
h=120 mm
A
B
h r
C
C
C,C
Problems (Equilibrium of Particles)
7. Two bodies weighing 150 N and 200 N, respectively, rest on a cylinder and are
connected by a rope as shown. Find the reactions of the cylinder on the bodies, the
tension in the rope and the angle q. Assume all surfaces to be smooth.
ropeW1=200 N
W2=150 N
q90q
Problems (Equilibrium of Particles)
W1=200 Nq
90q
qN1
T
q
Problems (Equilibrium of Particles)
FBD of Particle 1
y
x
q
qqq
qqq
q
q
qqq
sinT
sinsinTcosT
sinTcossin
cosT
sinTcosN
F
sin
cosTNcosTsinN
F
y
x
200
200
0200
0200
0
0
0
22
1
11
SOLUTION:
Problems (Equilibrium of Particles)
8. A force of F=100 N holds the 400-N crate in equilibrium. Determine the coordinates(0, y, z) of point A if the tension in cords AC and AB is 700 N each.
Problems (Equilibrium of Particles)
5 m
5 m
4 m
SOLUTION:FBD of A
FAC=700 NA
FAB=700 N
W=400 N
F=100 N
Coordinates:
𝐴 0, 𝑦, 𝑧𝐵 −5, 0,4𝐶 5, 0,4
222 45
45700
zy
kzjyiFAC
222 45
45700
zy
kzjyiFAB
jF
100 kW
400
Problems (Equilibrium of Particles)
222 45
45700
zy
kzjyiFAC
222 45
45700
zy
kzjyiFAB
jF
100 kW
400
0
45
3500
45
35000
222222
zyzyFx
0100
45
700
45
7000
222222
zy
y
zy
yFy
0400
45
4700
45
47000
222222
zy
z
zy
zFz
1
2
3
22 4251001400 zyy 2'
22 42540041400 zyz 3'
2'
3' zy
z
y
44
4
1
4
2' 22222 16251416251001400 yyyyyy
mzmy 51.23737.0
9. The shear leg derrick is used to haul the 200 kg net of fish onto the dock. Determine the
compressive force along each of the legs AB and CB and the tension in the winch cable DB.
Assume the force in each leg acts along its axis.
Problems (Equilibrium of Particles)
SOLUTION: FBD of B
𝑭BD
B
W=200(9.81) N𝑭AB
𝑭CB
6
442 kjiFF ABAB
kW
1962
Coordinates:
𝐴 2,0,0 𝐵 0,4,4𝐶 −2,0,0 𝐷 0,−5.6,0
6
442 kjiFF CBCB
4.10
46.9 kjFF BDBD
Problems (Equilibrium of Particles)
6
442 kjiFF ABAB
kW
1962
6
442 kjiFF CBCB
4.10
46.9 kjFF BDBD
00 WFFFF BDCBAB
Equations of Equilibrium
There are three unknowns.
03
1
3
10 CBABx FFF1
04.10
6.9
3
2
3
20 BDCBABy FFFF2
019624.10
4
3
2
3
20 BDCBABz FFFF3
kNF
kNF
kNF
BD
CB
AB
64.3
52.2
52.2
10. A small peg P rests on a spring that is contained inside the smooth pipe. The
spring exerts an upward force of 284 N on the peg. Determine the point of
attachment A (x, y, 0) of cord PA so that the tension in cords PB and PC equals
130 N and 84 N, respectively.
B
A
x
0.3 m
yx
z
y
0.6 m
C
P
0.2 m0.4 m
Problems (Equilibrium of Particles)
Fspring = 284 N, TB = 130 N , TC = 84 N; determine coordinates of point A (x, y, 0).
B
Ax
0.3 m
y
x
z
y
0.6 m
C
P
0.2 m0.4 m
FBD of peg P
AT
BT
CT
springF
Coordinates
𝐴 𝑥, 𝑦, 0 𝐵 0,−0.4, 0
𝐶 −0.3,0.2,0 𝑃 0, 0, 0.6
a
AA
yx
kjyixTT
222 6.0
6.0
kjT
kjTT
B
BB
33.10822.72
72.0
6.04.0
130
kjiT
kjiTT
C
CC
722436
7.0
6.02.03.0
84
kFspring
284
From equilibrium of peg P 0F
360360 a
xT
a
xTF AAx
22.4802422.720 a
yT
a
yTF AAy
67.1036.0
07233.1086.0
2840 a
Ta
TF AAz
1
2
31
3mx
aT
a
xT
A
A
208.067.103
36
6.0
2
3my
aT
a
yT
A
A
279.067.103
22.48
6.0
TA = 119.845 N
11. Crate A weighing 580 N rests on the inclined surface by the cable AB and force P which is
parallel to the z-axis. Determine the tension in the cable AB and force P for equilibrium.
Since the crate is mounted on casters, the force exerted by the incline on the crate is
perpendicular to the incline.
P
A
1.5 m
4 m
3 m
2.2 m
x
y
zDO
C
B
E
3 m
Problems (Equilibrium of Particles)
𝑷𝑾
𝑻
𝑵
4 m
3 m
x
3 m
yFBD of Crate A
𝐴 (2.4; 1.2;−1.5)
𝐴 (3𝑐𝑜𝑠37; 3 − 3𝑠𝑖𝑛37;−1.5)
a
a
a 37o
A
𝐵 (0; 5.2; 0)jW
580
Forces acting on the crate
94
51442
94
5102125420
.
k.ji.TT
.
k.j..i.TT
kPP
Reaction (normal force) exerted by incline on crate
4 m
3 m
y
a
A
N
xa
jN.iN.N
isinNjcosNN
8060
3737
(Reaction force (normal force) must be perpendicular to the incline.)
SOLUTION:
jW
580
Forces acting on the crate
94
51442
.
k.ji.TT
kPP
jN.iN.N
8060
00 NTPWF
094
42600 T
.
.N.Fx
094
4805800 T
.N.Fy
094
510 T
.
.PFz
1
2
3
1 2 3From , and ; T=395 N, N=322 N, P=120 N
Equations of Equilibrium
𝑷𝑾
𝑻
𝑵
155 mm 200 mm
460 mm
Coordinates: 𝐴 0,155,0 𝐵 200,0,460
𝑾
AxAz
𝑷
𝑻
𝑷
FBD of collar A
Reactions exerted by the vertical bar on collar A : kAiAN zxA
525
460155200 kjiTT
jP
351
jW
10
00 WTPNF A
Equations of Equilibrium
12. Collars A and B are connected by wire AB and canslide freely on frictionless rods. If a force P=351 N isapplied to collar A, determine (a) the tension in thewire, (b) the magnitude of the force Q required tomaintain the equilibrium of the system and reactionson collar A by the vertical rod. WA=WB=10 N
Forces acting on collar A
Problems (Equilibrium of Particles)
SOLUTION:
kAiAN zxA
525
460155200 kjiTT
jP
351 jW
10
NT
TFy
1155
010351525
1550
1
NATAF xxx 4400525
2000 2
(correct sense)
3 NATAF zzz 10120525
4600
FBD of collar B
Ax
𝑻′
𝑸
𝑾
Bx
By
𝑾
AxAz
𝑷
𝑻
Reactions exerted by the horizontal on collar B : jBiBN yxB
525
4601552001155
kjiTT
jW
10 kQQ
NQQFz 101201155525
4600
NBBF xxx 44001155525
2000
NBBF yyy 33101155525
155100
13. If WA=WB=1400 N, determine the force P, TAB and the reactions between the
collars and bars.
Problems (Equilibrium of Particles)
14. The 100-kg collar A rests on the smooth straight fixed bar CD by the cable AB.
Determine the tension in the cable and the reaction between the collar and bar CD.
C
Problems (Equilibrium of Particles)
C
FBD of collar A
xz
y
C
D
Length of fixed bar CD: mCD 9447 222
Coordinates of point A: m.zx AA 6729
64 m.yA 332
9
37 𝐴 (2.67; 2.33; 2.67)
𝐵 (0; 7; 4)
jj.W
981819100
Forces acting on the collar
kjiTkji
TT
24.084.048.054.5
33.167.467.2
kNjNiNN zyx
(Reaction exerted by bar on the collar)
15. The 100-kg collar A rests on the smooth straight fixed bar CD by the cable AB. Determine the tension in
the cable and the reaction between the collar and bar CD.
𝑻
𝑵
𝑾
C
jW
981Forces acting on the collar
kjiTT
24.084.048.0 kNjNiNN zyx
(There are four unknowns: T, Nx, Ny, Nz)
(But, there are there equations of equilibrium.)
SOLUTION METHOD:Reaction force between the collar and bar CD is
perpendicular to the axis of the bar. ( 𝑪𝑫𝑵 or
𝒏𝑪𝑫𝑵 ). So, scalar product of these vectors must be equal to zero.
0 CDnN
Additionally, since 0F
0 CDnF
00
0
CDCDCDABCDAB nWnNnTnWNT
𝐶 (4; 0; 4)
𝐷 (0; 7; 0)
9
474 kjinCD
076311066021009
474981240840480
T.T.T.
kjijk.j.i.T
k.j.i.TNT AB
0824428854324881017
k.j.i.NWTNF AB
0824472126324880
3.4 m
r =3.4 m
4.5 m
158
16. Collar A weighing 170 N, which can slide freely
on the quarter circle, is held in equilibrium by cable
AB.
a) Determine the tension in the cable.
b) Determine the magnitude and the components of
the contact force acting on the collar from the circle.
Problems (Equilibrium of Particles)
xz
y
FBD of collar A
W
ABT
N
Coordinates of points A and B:
0
17
843
17
1543 ;.;.A 𝐴 (3; 1.6; 0) 𝐵 (0; 3.4; 4.5)
3.4 m
r =3.4 m
4.5 m
158
x
y
z
y
W
ABT
N
Forces acting on the collar
75
54813
.
k.j.iTTAB
jW
170
xyN
kNjNiNN zxyxy
17
8
17
15
xy plane yz plane
zN
(Reaction exerted by bar on the collar)
00 NTWF AB
017
15
75
30 xyx NT
.F
017
8
75
811700 xyy NT
.
.F
075
540 T
.
.NF zz
1
2
3
Equation of Equilibrium
1 2 3From , and ; T=285 N, Nxy=170 N, Nz=225 N
17. Smooth collars A, B and C, each weighing 360 N, are connected by the
wires AB and BC and may slide freely on the smooth rod having the shape
shown. Determine the magnitude of the horizontal force P which must be
applied to the collar A to maintain equilibrium. DEFG portion of the rod is
parallel to xy-plane.
Problems (Equilibrium of Particles)
P
NAy
NAx
TAB
W
xz
y
xz
y
W
TBA
TBC
NBz
NBxy
xz
y
TCB
W
NCxNCz
FBD of collar A
FBD of collar BFBD of collar C
Four unknowns
Four unknowns
Three unknowns
xz
y
TCB
W
NCxNCz
FBD of collar C
A (0;9;3)
B (2;5;7)
C (6;1;0)
E
B
F
6 m
1.5 m4.5 m xB
65.4
5.1 Bx NT
TF
kNiNN
jW
kjiTT
NWTF
BC
BCy
CCC
BCBC
CBC
zx
810
03609
40
360
9
744
00
6.0cos
6.0sin
87.36
q
q
q o
NT
TNF
TNF
kNjNiNN
jW
kjiTT
kjiTT
NWTTF
BA
BAxyBy
BAxyBx
BxyBxyBB
BABA
CBBC
BABBC
z
1080
03606
43608.00
06
23606.00
8.06.0
360
6
442
630360360
00
FBD of collar B
xz
y
W
TBA
TBC
NBzNBxy
F
E
E
B
F
6 m
4.5 mNBxy qq
NP
PF
kPP
jNiNN
jW
kjiTT
PNWTF
z
yAxAA
BAAB
AAB
720
07200
360
720720360
00
P
NAy
NAx
TAB
W
xz
y
FBD of collar A
18. Smooth collars A and B are connected by the spring. Spring has a
constant of 120 N/cm and its unstretched length is 30 cm. Determine the
magnitude of the force P which must be applied to the collar A to maintain
equilibrium and the reaction between the collar and bar. Neglect the weight of
the collars. Take A (40;0;40) and B (0;20;80).
yx
z
30 cm
40 cm40 cm
20 cm80 cm
80 cm
P
Q
A
B
Problems (Equilibrium of Particles)
FBD of collar A
P
NAxz
NAy
Fspring
yx
z
yx
z
30 cm
40 cm
40 cm
20 cm
80 cm80 cm
P
Q
A
B
NPNN
NPF
NNNF
NPF
jNkNiNNkPiPP
cmABkjiABkjiF
kjiABkji
F
Axz
Axzz
AyAyy
Axzx
AyAxzAxzA
spring
spring
87.3483387
05
3
5
424000
1200012000
05
4
5
324000
5
3
5
4
5
4
5
3
60402040240012002400
)4080()020()400(60
402040)3060(120
40 cm
x
A
z
30 cm
NAxz
q
q
00 Aspring NPFF
Correct sense