the force acting on a body has two effects...the force acting on a body has two effects: the first...

The force acting on a body has two effects:

the first one is the tendency to push or pull the body in the direction

of the force, and the second one is to rotate the body about any fixed

axis which does not intersect nor is parallel to the line of the force.

This dual effect can more easily be represented by replacing the

given force by an equal parallel force and a couple to compensate for

the change in the moment of the force.

F

Let us consider for acting at point A in a rigid body. It is possible

to slide force along its line of action, but it is not possible to

directly move it to point B without changing the external effect on the

rigid body.

F

F

In order to do this, two equal and opposite forces and are added to

point B without introducing any net external effects on the body. It is

seen that, the original force at A and and the equal and opposite one at

B constitute the couple M=Fd, which is counterclockwise for this case.

F

F

Therefore, we have replaced the original force at A by the same

force acting at a different point B and a couple, without altering

the external effects of the original force on the body. Since is

a free vector, its location is of no concern. The combination of

the force and couple is referred to as a force-couple system.

M

By reversing this process, we can combine a given couple and a

force which lies in the plane of the couple (normal to the couple

vector) to produce a single, equivalent force. Force can be moved

to a point by constructing a moment equal in magnitude and

opposite in direction . The magnitude and direction of

remains the same, but its new line of action will be distance

away from point B.

M

F

F

Md

1. The tie-rod AB exerts the 250 N force on the steering knuckle

AO as shown. Replace this force by an equivalent force-couple

system at O.

If two force systems are creating the same external effect

on the rigid body they are exerted on, they are said to be

“ ”.

The resultant of a force system is the simplest

combination that they can be reduced without altering the

external effects they produce on the body.

Coplanar Force Systems

If the resultant of all forces lying in a single plane

such as xy is , this resultant is calculated by the vector sum of these

forces.

nFFFFR

...321

nFFFF

...,,,, 321

R

x

y

yx

yyxx

R

R

FFR

FRFR

1

22

tan

The location of the line of action of the resultant force to an

arbitrary point (such as point O is the origin of the xy coordinate

system) can be determined by using the Varignon’s theorem. The

moment of about point O will be equal the sum of the couple

moments constructed by moving its components to point O.

R

R

FdMMo

FR

R

Md o

2. The device shown is part of an automobile seat-back-release

mechanism. The part is subjected to the 4 N force exerted at A and a

300 Nmm restoring moment exerted by a hidden torsional spring.

Determine the y-intercept of the line of action of the single equivalent

force.

3. Under nonuniform and slippery road conditions, the four forces shown are exerted

on the four drive wheels of the all-wheel-drive vehicle. Determine the resultant of

this system and the x- and y-intercepts of its line of action. Note that the front and

rear tracks are equal (i. e., ). CDAB

Three Dimensional Force Systems

The same principles can be enlarged to three dimensional force

systems. The resultant of forces acting on a

body can be obtained by moving them to a desired point. In this

way, the given force system will be converted to

1) Three dimensional, concurrent forces comprising the same

magnitudes and directions as the original forces,

2) Three dimensional couples.

nFFFF

...,,,, 321

By calculating the resultants of these forces and couples, a single

resultant force and a single couple can be obtained.

The resultant force,

222

321 ...

zyx

zzyyxx

n

FFFR

FRFRFR

FFFFFR

The resultant couple moment,

The selection of point O is arbitrary, but the magnitude and direction

of will depend on this point; whereas, the magnitude and

direction of are the same no matter which point is selected.

CFrM

M

R

C C

As a special case, if the resultant couple is perpendicular to

the resultant force , these two vectors can further be simplified to

obtain a single resultant force . The force can be slided a

distance d to form a moment , which is equal in magnitude

and opposite in direction , so that they will cancel each other

out. The distance d will be equal to d=SM/R.

R

R

R

M

M

M

4. The special-purpose milling cutter is subjected to the force of

1200 N and a couple of 240 N.m as shown. Replace the given force

system with an equivalent force-couple system at O.

5. Represent the resultant of the force system acting on the pipe

assembly by a single force at A and a couple .

When the resultant couple vector is parallel to the resultant

force , the resultant is called a “wrench”.

The wrench is the simplest form in which the resultant of a general

force system may be expressed.

By definition, a wrench is positive if the couple and force vectors

point in the same direction, and negative if they point in opposite

directions.

R

M

Wrench Resultants

A common example of a wrench is found with the application of a

screw driver.

All force systems can be reduced to a wrench acting at a particular

line of action.

M R

//RR

RR//R

MMMM

nnMMM

2

1

Equivalent force-couple system at point O M is resolved into components M1 along the

direction of R and M2 normal to R.

Positive wrench

R

Md 2

M

R

M R

6. In tightening a bolt whose center is at point O, a person exerts a 180 N force on the

ratchet handle with his right hand. In addition, with his left hand he exerts a 90-N force as

shown in order to secure the socket onto the bolt head. Determine the equivalent force-

couple system at O. The find the point in the x-y plane through which the line of action of

the resultant force of the wrench passes.

7. The force-couple system acting at O is equivalent to the wrench acting at A.

If and , determine . NkjiR

7001400600 mNM //R 1200

oM

8. a) Reduce the general three-dimensional force system to a force-couple system at O.

b) Replace the force-couple system obtained by a wrench and determine the coordinates of the

point in the yz plane through which the line of the wrench passes.

Dimensions in meters.

9. Two forces and a couple (applied to portion AB) are applied to the car’ s exhaust

system as shown. Replace the given force system with an equivalent force-couple

system at D.

25 N.m

115 N

100 N

10. The direction cosines of robot arm AB are cos x=0.6, cos y (y <90°) and cos z=0. For

arm BC the direction cosines are, cos x=7/9, cos y=4/9 and cos z=4/9. A force of

magnitude F=250 N and a couple of magnitude C=27 Nm along the axis BC are applied to

the end C of arm BC. Determine the moment about line AD. Replace the force and couple

acting on the robot assembly with an equivalent force-couple at point A.

* Line MN lies in a plane parallel to the horizontal plane

* Line AD lies in the xz plane and makes a 37° angle with the

x axis.

11. The threading dye is screwed onto the end of the fixed pipe which is bent

through an angle of 20°. Replace the two forces by an equivalent force at O and a

couple . Find and calculate the magnitude M of the moment which tends to screw

the pipe into the fixed block about its angled axis through O.

Nm.k.i.kiMMM

k.i.kcosisinn

kiM

ji.k.kcos.isin.

ji.k.kcos.isin.M

jjjFR

oOC

OC

o

o

68859403408517

9403402020

8517

150250202015020150

200250202015020150

50200150

S

S

S

S

y

x

500 N

1700 N 3400 N

3

4

tan

30 cm 50 cm

34 cm

50 cm

50 cm z

800 N.m

tan 15

8

12. The pulleys and the gear are subjected to the loads shown. For these forces,

determine the equivalent force-couple system at point A.

15

8

jiR

jiijiFR

jijiF

iF

jijiF

35201040

800150050027202040

800150017

81700

17

151700

500

272020405

43400

5

33400

3

2

1

S

kjiCFrM

kC

kjijikjFr

kijFr

kjiFr

jikjijijikFr

CFrM

A

A

6.11391770960

800

75075040080015005.05.0

1505003.0

6.48910201360

272020405.016.03.02720204017

8

17

1534.05.0

33

22

11

11

SS

SS

C3 = 80 Nm (in plane ABCD) y > 90o

F1 = 30 N

F2 = 75 N

F3 = 40 N

C1 = 60 Nm

C2 = 100 Nm (in yz plane)

O (0, 0, 0) m A (12, 0, 0) m B (in xz plane) C (12, 8, 0) m E (6, 10, -3) m G (10, 4, 4) m

13. Replace the system comprising two forces, two couples and a positive wrench with an

equivalent force-couple acting at point O. Then, reduce the system further into a wrench and

determine the coordinates of point P, of which the line of action of the wrench cuts through

the yz plane.

A

B

C

O

D

E

G

37°

X Y

Z

45°

60° y

53°

30°

6 m

4 m

1C

1F

2C

3F

3C

2F

k1ji.kcos60jcos6icos45F

cos6cos45cos oyy

222

5152121030

6010

1

Force:


F1 = 30 N

F2 = 75 N

F3 = 40 N

C1 = 60 Nm


A

B

C

O

D

E

G

37°

X Y

Z

45°

60° y

53°

30°

6 m

4 m

1C

1F

2C

3F

3C

2F

Force:


F1 = 30 N

F2 = 75 N

F3 = 40 N

C1 = 60 Nm


A

B

C

O

D

E

G

37°

X Y

Z

45°

60° y

53°

30°

6 m

4 m

1C

1F

2C

3F

3C

2F

k.j.i.

k)(j)(iF

884688461535

884

8008181275

2222


Force:


F1 = 30 N

F2 = 75 N

F3 = 40 N

C1 = 60 Nm


A

B

C

O

D

E

G

37°

X Y

Z

45°

60° y

53°

30°

6 m

4 m

1C

1F

2C

3F

3C

2F


k.j.i.

...

k.j.i.

R

Rn

k.j.i.FR

k.jikcosjsincosicoscosF

.

222R

7806300150

52968877951

52968877951

52968877951

643416123040536040536040

04124

3

S

kj.2i.k.jikji6Fr

k.j562.56i.k.j.i.ji12Fr

k.j.k1ji.k4ji10Fr

CFrMo

248443439464341612310

84843043758846884615358

1665166551521214

33

22

11

SSMoment:


F1 = 30 N

F2 = 75 N

F3 = 40 N

C1 = 60 Nm


A

B

C

O

D

E

G

37°

X Y

Z

45°

60° y

53°

30°

6 m

4 m

1C

1F

2C

3F

3C

2F



F1 = 30 N

F2 = 75 N

F3 = 40 N

C1 = 60 Nm


A

B

C

O

D

E

G

37°

X Y

Z

45°

60° y

53°

30°

6 m

4 m

1C

1F

2C

3F

3C

2F


k3i

2432

k24iC

k24ijki6BDBA

iC

kji.4kcosjcosicos45C

223

2

1

26432

80

3248

100

3030422606060

40

kj.i.M

k3iikji.4

kj.2i.k.j562.56i.k.j.M

o

o

8675684186847

2641003030422

2484434394848430437516651665

Moment:

k86756.841i86.847M

k52.9677.88i95.1

o

j

jR

Equivalent force-couple system at point O

k86756.841i86.847M

k52.9677.88i95.1

o

j

jR

Equivalent force-couple system at point O

Reduction to a wrench in yz plane

k.j.ik.j.i..nMM

Nm.k.j.i.kj.i47.86nMM

R////

Ro//

0210402842780630015035133

351337806300150867568418

Positive wrench

k02.10402.48i2M

k52.9677.88i95.1

//

j

jR

O

x y

z

k02.10402.48i2M//

j

k52.9677.88i95.1

jR

oM

M

Positive wrench

The coordinates of point P, of which the line of action of the wrench cuts through the yz plane:

myyk

mzzj

jizjziyky

jj

j

jj

Rr

27.39198.76295.1

66.47458.92595.1

k98.76258.259i49.8688.7745.152.9695.1

k98.76258.259i49.868k52.968.77i95.1kzjy

k98.76258.259i49.868M

k02.10402.48i2k867841.56i47.868MMM

M

//o

O x y

z

k02.10402.48i2M//

j

k52.9677.88i95.1

jR

Positive wrench

P(0;391.27;474.66) r

the force acting on a body has two effects...the force acting on a body has two effects: the first...

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