1. The rod on the power control mechanism for a business jet is subjected to a force of

80 N. Determine the moment of this force about the bearing at A.

60cos15.020sin8060sin15.020cos80 yx FF

AM

mNM A 71.7

(counterclockwise - ccw)

+

Problems (on Moment)

Fx

Fy

2. In order to raise the lamp post from the position

shown, force F is applied to the cable. If F=200 N,

determine the moment produced by F about point A.

q

mNM A 768.471615.23sin200

+

(counterclockwise - ccw)

B

A

Geometry:

6 m

3 m

C3 m

6 mCosine law:

m.BCcosBC 37710563263 222

Sine law:o15.23

sin

3

105sin

37.7 q

q

200sinq

200cosq

Problems (on Moment)

3. Calculate the moment Mo of the 250 N force about the base point O of the robot.

Problems (on Moment)

Fx = 250sin40 NFy = 250cos40 N

Fx

Fy40o40o50o

30o

0.4cos30+0.3cos40

0.5

0.4

sin

30

+0

.3si

n4

0

40sin3.030sin4.05.040sin25040cos3.030cos4.040cos250 oM

mNM o 55.189

+

(counterclockwise - ccw)

A

O

20o

60o

500 mm

400 mm

300 mm

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

moment?

A

B

1.5

m

25 m

O

mkNM o 80204max

cos:/20cos5.1sin25805.1cos425sin4

q

𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑚𝑜𝑚𝑒𝑛𝑡, ABOB

22sec4005.1tan25sec205.1tan25

075.397tan75tan225tan140025.2tan75tan625 222

2252

75.39722547575075.39775225tan

2

2,1

2 mmmm

om 426.561733.1

5067.1tan2,1

o574.33426.5690 q

5. The small crane is mounted along the side of a pickup bed and facilitates the handling of

heavy loads. When the boom elevation angle is q=40o, the force in the hydraulic cylinder BC is

4.5 kN, and this force applied at point C is in the direction from B to C (the cylinder is in

compression). Determine the moment of this 4.5 kN force about the boom pivot point O.

6. 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 a which would

maximize the moment about O; state the value of this maximum moment.

A

O (70+150+70)=290 mm

F = 120 N

(25

+7

0+

70

+2

5)

=1

90

mm

30oIf a=30o, MO=?

190290

2570702530sin120701507030cos120

xy FF

oM

mNmmNM o 5.4141500

+

(clockwise - cw)

Determine the value of a which would maximize the moment about O. MOmax=?

A

O(70+150+70)

=290 mm

F = 120 N

(25

+7

0+

70

+2

5)

=1

90

mm

a

mNmmNM o 6.4185.41603290190120 22

aFor MOmax, F must be perpendicular to OA.

o2.33290

190arctan

a

a

aaaa

aa

o

o

o

FFd

dM

FFM

2.33

29.0

19.0tan19.0cos29.0sin0

19.0sin29.0cos

7. Assume that the hydraulic cylinder AB exerts a force Ԧ𝐹 of constant magnitude 2.5 kN

as the bin is elevated. Determine the moment of Ԧ𝐹 about the point O as a function of qfor the range 0≤ q ≤90o. At what angle q is this moment a maximum and what is themaximum moment?

Problems (on Moment)

8. 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)

9. 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 Ԧ𝐹.

Problems (on Moment)

FxFy

Fz

𝒓𝑪/𝑨

NFxy 28.6930cos80

NFF xyx 53.4440sin28.6940sin

NFF xyy 07.5340cos28.6940cos

NFz 4030sin80

kjiF

4007.5353.44

Fxy

Force in vector form

Position vector from point A to point C

ikijr AC

25.02.03.04.0/

kjir AC

2.04.055.0/

Moment of force 𝑭 about point A

FrM ACA

/

kjikjiM A

4007.5353.442.04.055.0

ijikjkM A

61.1091.81681.172219.29 kjiM A

38.1109.1339.5

𝒓𝑪/𝑨

Magnitude of the moment

FrM ACA

/

mNM A 16.1838.1109.1339.5 222

d: minimum distance

minimum distance from point A to the line of action of force 𝑭

q

q sin// FrFrM ACACA

q sin/ ACrd

F

A FdM

8016.18 d

𝒅 = 𝟎. 𝟐𝟐𝟕𝒎

kjiM A

38.1109.1339.5

10. The pipe assembly is secured on the wall by the two brackets. Determine,

a) the moment of force Ԧ𝐹 about point A,

b) the moment of force F about line AO,

c) the minimum distance from point B to the line AO

Problems (on Moment)

11. 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 Mz of this force about the shaft

axis OA.

𝑭spring𝑭spring=Fspring𝒏C/B

Coordinates : 𝐵 −150𝑠𝑖𝑛30, 150𝑐𝑜𝑠30, 600 𝑩 −𝟕𝟓, 𝟏𝟐𝟗. 𝟗, 𝟔𝟎𝟎𝑪(𝟑𝟓𝟎, 𝟑𝟎𝟎, 𝟎)

kjikji

Fspring

5.3977.11257.281

69.754

6001.170425500

Position vector from point O to point C (It may be written the position vector from point O to point B)

jir OC

3.035.0/

Moment of force 𝑭𝑠𝑝𝑟𝑖𝑛𝑔 about point O springOCO FrM

/

kjikjijiM O

02.4525.13925.1195.3977.11257.2813.035.0

Moment about the shaft axis mNkMnMM OOAOOA 02.45

12. The arms AB and BC of a desk lamp lie in a vertical plane that forms an angle of 30o 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.

30o

20o45o50o

45o

150 mm

AC

D

z

x

y

Direction CD is parallel to the z-axis and

lies in a parallel plane to the horizontal

plane.

mmBCmmAB 325,450

mmBCmmAB 325,450

k.j.i.F

N.CosCosF

N.SinF

N.SinCosF

kFjFiFF

z

y

x

zyx

325665931

32520458

665458

93120458

a)

k.j.i.

k.j.i.k.j.i.M

FrM

k.j.i.r

kSinSin.Sin.jCos.Cos.iCosSin.Sin.r

?M

A

A/CA

A/C

A/C

A

572163192

325665931283501104910

283501104910

3050325045450503250454503050325045450

mmBCmmAB 325,450

k.j.i.k.j.i. 1.81 M

Nm.k.j.i. k.j.i. M

k.j.i.r

rn

k.j.i.r

kSinSin.jCos.iCosSin.r

n M M

?M

AB

AB

AB

ABAB

AB

AB

ABAAB

AB

64028111350710610

811350710610572163192

350710610

159031802750

3045450454503045450

b)

c)

63114833

18350710610325665931

.Cos.

Cosk.j.i.k.j.i.

CosnFnF ABAB

qq

q

q

m..Sin

.

FSin

Md

dFSinM

AB

AB

2490631148

811

q

q

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.

z

13. 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.

y

z

x

B

O

A

30o

30o

0.5 m

𝐴 0.5𝑠𝑖𝑛30, 0.5 + 0.5𝑐𝑜𝑠30, 0𝐵 0, 1𝑐𝑜𝑠30, 1𝑠𝑖𝑛30𝑨 𝟎. 𝟐𝟓, 𝟎. 𝟗𝟑𝟑, 𝟎𝑩(𝟎, 𝟎. 𝟖𝟔𝟔, 𝟎. 𝟓)

kji

kjiF

79.40184.5389.200

5.0067.025.0

05.0933.0866.025.00450

222

𝑭

y

z

x

B

O

A

30o

30o

0.5 m

kjiF

79.40184.5389.200

Position vector from point O to point B (It may be written the position vector from point O to point A)

kjr OB

5.0866.0/

kjikjM O

79.40184.5389.2005.0866.0

FrM OBO

/

kjiM O

97.17945.10086.374

! Obtain the same result using position vector 𝒓𝑨/𝑶

jiijjr OA

933.05.05.030cos5.05.0/

FrM OAO

/

Minimum distance from point O to line AB, d:

𝐹 = 450 𝑁 , 𝑀𝑂 = 𝑀𝑂 = 374.862 + 100.452 + 179.972 = 427.84 𝑁.𝑚

𝑑 =𝑀𝑂

𝐹= 0.951 𝑚

𝑭

14. Concentrated force is acting perpendicular to the crank arm BC at point C. For

the position q=70°, what is Mx , the moment of about the x axis? At this instant,

My =20 N·m and Mz =37.5 N·m.

150 mm

z x

200 mm

Aq

100 mm

B

O

y

P

C

Problems (on Moment)

q=70°, My =20 N·m and Mz =37.5 N·m Mx=?

zx

200 mm

Aq

B

O

y

P

200 mm

z

y

O, A

B, C

P

70o

kPjPP

cossin

kjirr OC

70cos20070sin200250/

kjir

4.6894.187250

kPjPkjiM o

cossin4.6894.187250

PrM o

iPiPjPkPM o

sin4.68cos94.187cos250sin250

C

q=70°, My =20 N·m and Mz =37.5 N·m Mx=?

mNmmNPPM x 294.2525294sin4.68cos94.187

mmNPM y 31020cos250 1

2

2

1

mmNPM z 3105.37sin250

3

3

1020

105.37

cos250

sin250

P

P 875.1tan

93.61

NP 170

iPiPjPkPM o

sin4.68cos94.187cos250sin250

15. Rectangular plate ADCF is held

in equilibrium by string AB which

has a tension of T=14 kN.

Determine,

a) The magnitude of the moment of

about the axis DC,

b) The perpendicular distance

between lines AB and DC.

z

x

y

A (0, 5, 3) m

D (0, 2, 2)

C (4, 6, 0) m

B (6, 3, 0) m

FO

Problems (on Moment)

z

x

y

A (0, 5, 3) m

D (0, 2, 2)

C (4, 6, 0) m

B (6, 3, 0) m

O kjikji

n

kjiT

kjinTT

DC

T

3

1

3

2

3

2

6

244

6412

7

32614

a) The magnitude of the moment of about the axis DC

m kN..- M

kjikjinMM

kji

kjijiTrM

M

DC

DCCDC

CBC

DC

6710339812

3

128

3

212

3

218

3

1

3

2

3

2 281218

281218

641232

?

/

b) The perpendicular distance between lines AB and DC

67.58

52.042

22cos ,cos)1(14

3

22

3

22

3

16

3

24

3

212

cos

3

22 , 6412

q

qq

qDCDC

DC

nTnT

kjinkjiT

The parallel component of a force to a line does not generate moment with respect to that line.

If is resolved into two components parallel and normal to line DC, thenT

mθT

MθTM DC

DC 89.067.58sin14

67.10

sind , dsin

The magnitude of moment MDC is equal to the normal component of to line DC and the

perpendicular distance between and line DC.

T

T

z

x

y

A (0, 5, 3) m

D (0, 2, 2)

C (4, 6, 0) m

B (6, 3, 0) m

O