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FORCE SYSTEMS
PROBLEMS on
1. The guy cables AB and AC are attached to the top of the transmission tower. The tension in cable AB is 8 kN. Determine the required tension T in cable AC such that
PROBLEMS
the net effect of the two cable tensions is a downward force at point A. Determine the magnitude R of this downward force. (2/12)
TAB =8 kN, TAC =? Net effect of two cable tensions is a downward force at point A. R =?
PROBLEMS R
ABT
ACTABT
ACT
A
R
kNjiT
ji
r
rnTT
mC
mBA
AB
AB
ABABABAB
25.65
5040
504088
)30,50(
)50,40()0,0(
03.64
22/
/
jiTTji
Tr
rTnTT ACACAC
AC
ACACACACAC
514.0857.0,3050
3050
31.58
22/
/
x
y
TAB =8 kN, TAC =? Net effect of two cable tensions is a downward force at point A. R =?
PROBLEMS R
ABT
ACTABT
ACT
A
R
kNRRTj
kNTTi
jiTjijR
jRTTR
jiTT
jiT
AC
ACAC
AC
ACAB
ACAC
AB
25.9,514.025.6
83.5,0857.050
514.0857.025.65
514.0857.0
25.65
2. Power is to be transferred from the pinion A to the output gear C inside a mechanical drive. Because of out put motion requirements and space limitations, an idler gear B is introduced as shown. A force analysis has determined that the total contact force between each pair of meshing teeth has a magnitude Fn = 5500 N, and these forces are shown acting on idler gear B. Determine the magnitude of the resultant of the two contact forces acting on the idler gear. (2/28)
PROBLEMS
R
Fn = 5500 N, determine the magnitude of the resultant .
PROBLEMS R
3. In the design of the robot to insert the small cylindrical part into a close-fitting circular hole, the robot arm must exert a 90 N force P on the part parallel to the axis of the hole as shown. Determine the components of the force which the part exerts on the robot along axes (a) parallel and perpendicular to the arm AB, and (b) parallel and perpendicular to the arm BC. (2/28)
PROBLEMS
Determine the components of P along axes (a) parallel and perpendicular to the arm AB, and (b) parallel and perpendicular to the arm BC.
PROBLEMS
4. The unstretched
length of the spring is r.
When pin P is in an
arbitrary position q,
determine the x- and y-
components of the
force which the spring
exerts on the pin. Fspring=kd
Evaluate your answer for r = 400 mm, k = 1.4 kN/m and q = 40°.
Also when pin P is in the position q = 20°, determine the n- and
t-components of the force which the spring of modulus k = 1.4
kN/m exerts on the pin. Take r = 400 mm. (2/29)
PROBLEMS
l0=r. For q, determine x- and y-components of the force spring exerts on the pin. Find force for r = 400 mm, k = 1.4 kN/m and q = 40°. For q = 20°, determine the n- and t-components of the F.
PROBLEMS
sin
cos
springyspring
springxspring
FF
FF
springFyspringF
xspringF
Sine theorem
AP
r
APr
q
qsin2
sin
sinsin2
AP
r
AP
r
AP
r
qqqq
sin2sin90
sin2sin90
90sin2
sinsinsin
11
11
q r
2r
A
P
O
springF
l0=r. For q, determine x- and y-components of the force spring exerts on the pin. Find force for r = 400 mm, k = 1.4 kN/m and q = 40°. For q = 20°, determine the n- and t-components of the F.
PROBLEMS
AP
rFF
AP
rFF
springspring
springspring
y
x
sin2sin90sin
sin2sin90cos
1
1
Cosine theorem
NF
NF
NllkkF
mAP
valuesgiventheFor
rAP
rrrrrrAP
y
x
spring
spring
spring
4.1945.62sin219
2.1015.62cos219
219)4.0557.0(1400)(
5.62557.0
40sin)4.0(2sin9040
557.040cos454.0
cos45
cos45cos)2)((2)2(
0
1
22222
d
q
springFyspringF
xspringF
2r
q r
2r
A
P
O
springF
PROBLEMS
5. The circular cam has an offset e = 50 mm and a radius r = 100 mm. For the position where q = 30°, the smooth undersurface of the plunger exerts a downward force of 1600 N on the cam normal to the contacting surfaces. In designing the cam bearing it is necessary to calculate the rectangular component F' of this force along the line joining the contact point and the center of the shaft which turns the cam. Find F'.
PROBLEMS e = 50 mm, r = 100 mm. For q = 30°, smooth plunger exerts a downward force of 1600 N on the cam normal to the contacting surfaces. Calculate the rectangular component F' of this force along the line joining the contact point and the center of the shaft which turns the cam.
6. The tension in the supporting cable AB is T = 425 N. Write this tension as a vector (a) as it acts on point A and (b) as it acts on point B. Assume a value of q = 30°.
PROBLEMS
T = 425 N. Write T as a vector (a) as it acts on
point A and (b) as it acts on point B. q = 30°.
PROBLEMS
until it produces a tension T in CD of 1.2 kN. Express as a vector in terms of its unit vectors , and . Determine the direction angles qx, qy and qz which makes with the positive x-, y-, and z-axes. Does it make any difference in the result which coordinate system is used? (2/107)
5. The rigid pole and cross arm assembly is supported by the three cables shown. A turnbuckle at D is tightened
i
T
j
k
T
PROBLEMS
PROBLEMS T =1.2 kN. = ? Determine the direction angles qx, qy and qz which makes with the positive x-, y-, and z-axes. Does it make any difference in the result which coordinate system is used?
T
T
kNkjiT
kjiT
kjin
kji
r
rn
mDmC
nTT
CD
CD
CDCD
CD
962.0641.0321.0
802.0535.0267.02.1
802.0535.0267.0
5.435.1
5.435.1
)0,3,0()5.4,0,5.1(
612.5
222/
/
CDr /
CDn
ji
k
PROBLEMS T =1.2 kN. = ? Determine the direction angles qx, qy and qz which makes with the positive x-, y-, and z-axes. Does it make any difference in the result which coordinate system is used?
T
T
32.143,802.0cos
66.57,535.0cos
51.74
267.02.1
321.0cos,321.0cos
coscoscos
zz
z
y
y
y
x
xxxx
n
zyxzyx
T
TT
T
T
TTT
kjiTkTjTiTT
CD
q
qqq
It does not matter which coordinate system is used. x-y-z or x'-y'-z' systems can be used. Both systems have the same unit vectors.
xq yq
zq
kji
802.0535.0267.0
6. Three forces are acting on the eye bolt. If the
resultant force has a magnitude and direction as
shown,
determine the
magnitude and
direction angles
of .
RF
3F
PROBLEMS
PROBLEMS Given determine the magnitude and direction angles of .
RF
3F
PROBLEMS A
35° 60°
25° 40°
5 km
y
x
z
2 km B
O
7. At a certain
instant, the locations
of a plane at A and a
train at B are
monitored by a radar
antenna at O.
Determine the
distance d between A
and B for this instant.
PROBLEMS A
35° 60°
25° 40°
5 km
y
x
z
2 km B
O
Determine the distance d between A and B
kmrd
kjir
kjir
kmB
B
kmA
A
BA
BA
BA
72.618.582.222.3
18.582.222.3
)85.0(33.439.143.117.105.2
)85.0,39.1,17.1(
)25sin2,40cos25cos2,40sin25cos2(
)33.4,43.1,05.2(
)60sin5,35sin60cos5,35cos60cos5(
222
/
/
/
BAr /
8. Two cables exert
forces on the pipe.
Determine the
magnitude of the
projected
component of
along the line of
action of . Also
determine the angle
q between the two
cables.
1F
2F q
PROBLEMS
q
PROBLEMS Determine the magnitude of the projected component of along the line of action of and the angle q between the two cables.
1F
2F
9. The access door
is held in the 30°
open position by
the chain AB. If
the tension in the
chain is 100 N,
determine the
projection of the
tension force onto
the diagonal axis
CD of the door.
(2/116)
PROBLEMS
PROBLEMS TAB =100 N, determine the projection of TAB onto the diagonal axis CD of the door
NkjiT
kjinTT
NT
kjikjinTT
kjin
kji
r
rn
NkjiT
kjikji
n
nnTT
mB
mA
A
CD
DCCDCD
CD
DCABCD
DC
CD
CDDC
AB
AB
ABABABAB
8.1392.238.36
3.052.08.046
46)3.0)(30()52.0)(52()8.0)(80(
3.052.08.0305280
3.052.08.0
50.1
45.078.02.1
305280
3.052.08.050.1
45.078.02.1
100
)9.0,0,0(
)45.0,78.0,2.1(
)30sin9.0,30cos9.0,2.1(
/
/
BAT /
CDr /
x y
z
mD
mC
45.0,78.0,0
0,0,2.1
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 qxy between the cable and the vertical x-y plane. (2/115)
PROBLEMS
PROBLEMS boxcar begins to move when the x-component of the cable tension reaches 3 kN, calculate tension T in the cable. Determine the angle qxy between the cable and the vertical x-y plane.
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 qA and qB 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.
PROBLEMS
PROBLEMS k = 2.6 kN/m, spring is unstretched when qA and qB are both zero. Determine a vector expression for the force the spring exerts at point A.
