Problems (on Moment)
1. The 1. The rod on the power control mechanism for a business jet is subjected to a force of 80 N. Determ Determine ine the moment moment of this forc force e abou aboutt the the bearing bearing at A. A.
+ M A
0.15 sin 60 80 sin 200.15 cos 60
80 cos 20 F x
M A
F y
7.71 N m
(counter (cou ntercloc clockwis kwise e - ccw)
Problems (on Moment)
2. In 2. In order to raise the lamp post from the position shown, force F F is applied to the cable. If F =200 =200 N, determ determine ine the moment moment produ produced ced by F about about point A. A. Geometry : 200sin
q
6m
B 200cos
6m
2
BC
C
3m
3m
Cosi Co sine ne law law::
3
2
6
2
BC
2 3 6 cos 105
Sine law:
A
7.37
3
sin 105
q 23.15
o
sin q
+ M A
200 sin 23.15 6
471.768 N m
(counter (coun tercloc clockwis kwise e - ccw)
7.37 m
Problems (on Moment)
3. Calculate the moment M o of the 250 N force about the base point O of the robot.
20o F x = 250sin40 N F y = 250cos40 N
400 mm F x
A
o
60
50o o
o
40
30
40o
F y
500 mm 300 mm O 0.4cos30+0.3cos40
+ M o
0 4 n i s 3 . 0 + 0 3 n i s 4 . 0 5 . 0
250 cos 40 0.4 cos 30 0.3 cos 40 M
250 sin 400.5 0.4 sin 30 0.3 sin 40
189 55 N m
(counterclockwise - ccw)
4. The towline exerts a force of P =4 kN at the end of the 20-m-long crane boom. If x =25 m, determine the position of the boom so that this force creates a maximum moment about point O. What is this
B
moment?
O
q
m 5 . 1
, OB
AB
M o
4 cos 1.5 80
4 sin 25
25 tan 1.5 20 sec 625 tan
2
4 20
tan
225m
1.5067
75m 397.75
A
80 kN m
25 tan 1.52 400 sec 2
2
25 sin 1.5 cos 20 / : cos
75 tan 2.25 400 1 tan 2
tan m
m
max
25 m
0
56 426
o
225 tan
2
75 tan 397.75 0
75 75
2
225 397.75 2225
4
m1, 2
q 90 56 426 33 574o
5. The 120 N force is applied as shown to one end of the curved wrench. If a=30o, calculate
the moment of F about the center O of the bolt. Determine the value of
which would
maximize the moment about O; state the value of this maximum moment.
If =30o, M O=?
o F = 120 N 30
A
O
+
M o
120 cos 30
70 150 70 120 sin 3025 70 70 25
F y
M
290
(70+150+70)=290 mm
( 2 5 = + 1 7 9 0 0 + m 7 m 0 + 2 5 )
41500 N mm
F x
41 5 N m
190
(clockwise - cw)
which would maximize the moment about O. M Omax =?
Determine the value of
F = 120 N
A
O M o
( 2 5 = + 1 7 9 0 0 + m 7 m 0 + 2 5 )
(70+150+70) =290 mm 120
190
2
290
2
For M Omax , F must be perpendicular to OA .
190 a arctan 33.2 290
41603.85 N mm
o
41.6 N m
M o F cos a0.29 F sin a0.19 dM o 0 F sin a0.29 F cos a0.19 d a
tan a
0.19 0.29
a 33 2o
Problems (on Moment)
6. The force exerted by the hydraulic cylinder AB on the door is 3 kN (constant) directed from A to B. Determine the moment of this force about point O (M o ) as a fuction of q. Also determine M o for q =50o.
40 cm O
150 cm
A
B
20 cm
Problems (on Moment)
7. The spring-loaded follower A bears
against the circular portion of the cam until the lobe of the cam lifts the plunger. The force required to lift the plunger is proportional to its vertical movement h from its lowest position. For design purposes determine the angle q
for which the moment of the contact
force on the cam about the bearing O is a maximum. In the enlarged view of the contact, neglect the small distance between the actual contact point B and the end C of the lobe.
Problems (on Moment)
8.
The
pipe
assembly
is
subjected to an 80 N force. Determine the moment of this force about point A. Also determine the minimum distance from point A to the line of action of force .
Force in vector form
F xy
80 cos 30
F xy sin 40
F y
F xy cos 40
F z
F x
69.28 sin 40
44.53 N 53.07 N
40 N
F 44.53i
69.28 cos 40
80 sin 30
69.28 N
53.07 j
40k
/
Position vector from point A to point C
r C / A 0.4 j 0.3i 0.2k 0.25i
r C / A
0.55i
0.4 j
0.2
k
F x
Moment of force about point A
M A
0.55i
r C / A F
0.4 j
29 19k 22 j
F z
M
F xy
M A
F y
0.2
k
17 81k
44.53i
16i
53.07 j
8 91 j
40k
10
61i
M
5 39i
13
09 j
11
38k
minimum distance from point A to the line of action of force
d: minimum distance
M A
r C / A F
M A r C / A F r C / A F sin q
/
d r C / A sin q
M A
d F
F
M A
i
5.39
13.09
j 11.38k
Magnitude of the moment
M A
5.39
2
13.09
18.16
2
11.38
2
d 80
= .
18.16
N m
Problems (on Moment)
9. Concrete pump is subjected to force acting on point C as shown in the figure. Arms OA, AB and BC lie in the same plane which makes 35o with xz plane. a) Determine the moment of force about point . b) Determine the moment of force about line .
35o
10. The spring which connects point B of the disk and
point C on the vertical surface is under a tension of 500 N. Write this tension as it acts on point B as a force vector and determine the moment M z of this force about the shaft axis OA.
spring=F springC/B
spring
Coordinates :
−15030, 15030, 600 −, . , (, , )
F spring 500
425i
j 600k
170.1
754.69
281.57i
112.7
Position vector from point O to point C (It may be written the position vector from point O to point B)
j 397.5k
r C / O
Moment of force about point O
M O 0.35i 0.3 j
281.57i
M O
0.35i
0.3 j
r C / O F spring
112.7 j 397.5k 119.25i 139.25 j 45.02k
Moment about the shaft axis
M OA
M O nOA
M O k
45.02 N m
11. Strut AB of the 1 meter diameter hatch door exerts a force of 450 N on point B. Determine
the moment of this force about point O. Line OB of the hatch door lies in the yz plane. z
0.530, 0.5 + 0.530, 0 0, 130, 130 . , . , (,.,.)
z
30o
0.5 m
30o
y
x
F 450
0 0.25i 0.866 0.933 j 0.5 0k
0.25
2
0.067
2
0.5
2
i
200.89
53.84
j 401.79k
z
F 200.89i
53.84
j 401.79k
Position vector from point O to point B (It may be written the position vector from point O to point A)
r B / O
0.866 j
M O
r B / O
0.866 j
F
0.5k
200.89i
53.84
j 401.79k
! Obtain the same result using position vector /
r A / O
0.5 j
0.5 cos 30 j
0.5i
0.5i
30o
x
0.5 m
M O 374.86i 100.45 j 179.97k
y
30o
M O
0.5k
0.933 j
M O
r A / O
F
Minimum distance from point O to line AB, d:
= 450 ,
= =
= = 0.951
374.86 + 100.45 + 179.97 = 427.84 .
Problems (on Moment)
12. Concentrated force is acting perpendicular to the crank arm BC at point C . For
the position q=70°, what is M x , the moment of about the x axis? At this instant, M y =20 N·m and M z =37.5 N·m.
P 100 mm
y
B C
O
200 mm
A z
x 150 mm
q=70°,
P
M y =20 N·m and M z =37.5 N·m M x=?
P
P P sin j P cos k
y
B
B, C
C y
200 mm
z
200 mm
O
70o
O, A
z
A
r C / O r 250i 200 sin 70 j 200 cos 70k
r 250i
187.94
M o
M o
M
j 68.4k
r P
250i
187.94 j 68.4k P sin j P cos k
250 P sin k
250 P cos j
187 94 P cos i 68 4 P sin i
x
q=70°,
M y =20 N·m and M z =37.5 N·m M x=?
M o 250 P sin k 250 P cos j 187.94 P cos i 68.4 P sin i M y 250 P cos 20 10 N mm 3
M z
2 1
250 P sin 37.5 103
250 P sin 250 P cos
N mm
37.5 103 20 10
3
1 2
tan 1.875
61.93
P
M x
187.94 P cos 68.4 P sin 25294
170 N
N mm 25.294 N m
Problems (on Moment)
z
13. Rectangular plate ADCF is held
A (0, 5, 3) m
in equilibrium by string AB which D (0, 2, 2)
y
has
a
tension
of T =14
kN.
Determine, F
O
a) The magnitude of the moment of C (4, 6, 0) m
about the axis DC , b) The perpendicular distance
B (6, 3, 0) m
x
between lines AB and DC .
z
6i 2 j 3k T T nT 14 7 T 12i 4 j 6k 4i 4 j 2k 2 2 1 n DC i j k 6 3 3 3
A (0, 5, 3) m D (0, 2, 2)
y
O C (4, 6, 0) m
a) The magnitude of the moment of about the axis DC
B (6, 3, 0) m
M DC ? M C r B / C T 2i 3 j 12i 4 j 6k
x
18i 12 j 28k 2 2 1 M DC M C n DC 18i 12 j 28k i j k 3 3 3 2 2 1 18 12 28 3 3 3 M DC 12 8-9.33 10.67 kN m
z
b) The perpendicular distance between lines AB and DC
A (0, 5, 3) m
T 12i 4 j 6k ,
D (0, 2, 2)
n DC
2i 2 j k 3
T n DC T n DC cosq
y
2 2 1 22 12 4 6 3 3 3 3
O C (4, 6, 0) m
22 3
B (6, 3, 0) m
14(1) cos q , q
cos q
58.67
22 42
0.52
x The parallel component of a force to a line does not generate moment with respect to that line.
If T is resolved into two components parallel and normal to line DC, then
M DC
T
sin θ d
,
d
M DC
T sin θ
10.67 14 sin 58.67
0.89 m
The magnitude of moment M DC is equal to the normal component of T to line DC and the
perpendicular distance between T and line DC.
14. The arms AB and BC of a desk lamp lie in a vertical plane that forms an angle of 30 o with xy
plane. To reposition the light, a force of magnitude 8 N is applied as shown. Determine, a) the moment of the force about point A, b) the moment of the force about the axis of arm AB, c) the angle between the line of action of force and line AB, d) the perpendicular distance between the line AB and line of action of force. Direction CD is parallel to the z -axis and lies in a parallel plane to the horizontal plane.
45o y
AB
450 mm , BC
D
50o
45o
325 mm C
A
150 mm
x 30o
z
20o
F F x i F y j F z k
F x 8Cos 45Sin20 1.93 N
F y 8Sin45 5.66 N F 1.93i 5.66 j 5.32k F z 8Cos 45Cos 20 5.32 N
a) M A
?
r C / A
0.45Sin45 0.325Sin50Cos30i 0.45Cos 45 0.325Cos50 j 0.45Sin45 0.325Sin50Sin30k
r C / A
0.491i
r C / A F
0.491i
M A
0.11 j
0.2835k
M A
2.19i
3
0.11 j
0.2835k
.93i
1
.66 j 5.32k
5
.16 j 2.57k
AB
450 mm , BC 325 mm
b) M AB
?
M AB M A n AB
r AB
0.45Sin 45Cos30i
r AB
0.275i
r AB
0.45Cos 45 j
0.45Sin 45Sin30k
0.318 j
0.159k
n AB
r AB
2.19i
c) F n AB
.93i
1
0.35k
.16 j 2.57k 0.61i
3
1.81 0.61i
0.71 j
M AB
AB
M AB
0.61i
0.71 j
0.35k
0.71 j
.1i
1
0.35k
450 mm , BC
325 mm
.81 Nm
1
.28 j 0.64k
1
F n AB Cosq
0.61i
.66 j 5.32k
5
0.71 j
0.35k
81Cosq
q 114.63
.3 8Cosq
3
d) If the force is resolved into two components parallel and normal to line AB, only normal component generate a moment. The parallel component of a force to a line does not generate moment with respect to that line.
M AB d
FSinq d
M AB
1.81
0.249
m
