PROBLEMS ON

RESULTANTS

PROBLEMS

1) Replace the three horizontal forces and applied couple with an

equivalent force-couple system at O by specifying the resultant

and couple MO.

R

70°

70°

70°

PROBLEMS Replace the three horizontal forces and applied couple with an equivalent force-couple system at O by specifying the resultant and couple MO.

)(10.22

15)4(70cos10)5.2(70cos6)3.0(70sin6)3.0(70sin4)1(70cos4

)(74.252.7

)70cos670sin6(

)70cos1070sin10()70cos470sin4(321

ccwmmkNM

M

kNjiR

ji

jijiFFFFR

O

O

R

+

R

OM

x

PROBLEMS

2) Where does the

resultant of the two

forces act?

PROBLEMS Where does the resultant act?

)(198)3.0(6601 ccwmNM

mx

cwmNxM

9.920

198

)(198)(202

660 N

20 N

1M

2M

x

20 N

Resultant (20 N) acts 9.9+0.3+0.5= 10.7 m left

of point A

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

PROBLEMS

Determine the resultant of the system and the x- and y-intercepts of its line of action.

PROBLEMS

CDAB

PROBLEMS

4) The pedal-chain wheel unit of a bicycle is shown in the figure. The left foot of the rider exerts the 160 N force, while the use of toe-clips allows the right foot to exert the nearly upward 80 N force. Determine the equivalent force couple system at point O. Also, determine the equation of the line of action of the system resultant treated as a single force R. Treat the problem as two dimensional.

PROBLEMS Determine the equivalent force couple system at point O and the equation of the line of action of the system resultant treated as a single force R.

5) The forces acting on the belts on a pulley system are equal with a magnitude of 800 N. The pulley is secured to the steel column by means of two bolts at A and B. Replace the two forces with a force-couple system, in which the equivalent force will be at the midpoint of the bolts. Then, determine the force each bolt will sustain by distributing this force and couple to forces acting at points A and B.

160 mm

380 mm

150 mm

320 mm

C

B

A

800 N

800 N

Replace with a force-couple system, in which the equivalent force will be at the midpoint of the bolts. Determine the force each bolt will sustain by distributing this force and couple to forces acting at points A and B.

160 mm

380 mm

150 mm

320 mm

C

B

A

800 N

800 N

6) 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 N·mm restoring moment exerted by a hidden torsional spring.

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

force.

mmP

yxP

mmyy

)41.40,0(

),(

41.40,156)(86.3

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

single equivalent force.

)(156300144

300)10(15sin4)40(15cos4

04.186.315sin415cos4

04.186.3

ccwmmN

M

NjijiF

+

F

F

NF 4mmNM 156

F

F

y

x

{ y

mmNM 156

P

NF 4

P

7) Determine the

force-couple system

at O which is

equivalent to the

two forces applied

to the shaft AOB.

Is perpendicular

to ?

R

OM

PROBLEMS

Determine the force-couple system at O which is equivalent to the two forces applied to the shaft AOB. Is perpendicular to ? R

OM

)(31.108569.265

)45cos80045sin800()30cos60030sin600(

69.56569.56562.519300 21

NkjR

kjkjR

kjFkjF

)(49.11492.48

49.9049.9057.4124

69.56569.56516.0

62.51930008.02211

mNkjM

jkjkM

kji

kjiM

FrFrM

O

O

O

O

othereachtolarperpendicunotarethey

MR

MRthenMRif

O

OO

0111260

49.11431.108592.4869.265

0,

PROBLEMS

8) The two forces and one

couple act on the

elements of a drill press

as shown. Determine the

equivalent force-couple

system at point O.

Determine the equivalent force-

couple system at point O.

)(30020 NkiFR

)(964.1014.1130

1286.3036.13015

1220193.00518.0

3001.005.0

2211

mNkjiM

kjkijM

kikj

kjiM

CFrFrM

O

O

O

O

mmB

mmzAO A

)15cos200,15sin200,0(

,),100,50(,)0,0,0(

19.19376.51

9) Express and

identify the resultant

of the two forces and

one couple shown

acting on the shaft

angled in the x-z

plane.

600 N.m

1.5 kN

1.5 kN

450 mm

150 mm

600 N.m

1.5 kN

1.5 kN

450 mm

150 mm

B

C z

CBr /

ABn

?RM

0FR

21 CCM R

mzC

mzzB

),0,0(

)304.0,0,423.0()20sin45.015.0,0,20cos45.0(

)(71.83982.107

)(21.20582.563

)20sin20(cos600

)(4565.634

1500)304.0423.0(

2

22

1

/1

mNkiM

mNkiC

kinCC

mNikC

jkiFrC

R

AB

CB

z

x

A

B

20ABn

10) The floor exerts the four indicated forces on the

wheels of an engine hoist.

Determine the location

in the x-y plane

at which the

resultant of

the forces act.

Determine the location in the x-y plane at which the resultant of the forces act.

)(2280

)8401080120240(

NkR

kFR

)(2253612108025.08409.1475.0

1204.05.02404.06.0

21605401596399

486096144

44332211

mmNjikjikji

kjikjiM

FrFrFrFrM

ijij

ijij

O

O

mDmC

mBmAO

)0,2,5.0(,)0,9.1,475.0(

)0,4.0,5.0(,)0,4.0,6.0(,)0,0,0(

Determine the location in the x-y plane at which the resultant of the forces act.

)(2253612,)(2280 mNjiMNkR O

mP

mxxj

myyi

mNjiM

iyjxkjyixM

RrMorRrM

O

O

OPOOPO

)58.1,0987.0(

0987.0,2252280

58.1,36122280

)(2253612

228022802280)(

//

R

OM

R

R

P (x, y)

OPr /

OM

11) The weight of the robot arm

AB is 200 N and its direction

cosines are cos qx=0.6, cos qy (qy

<90°) and cos qz=0. For arm BC the

weight is 160 N with cos qx=7/9, cos

qy=4/9 and cos qz=-4/9. The mass

centers of both arms are at their

midpoints. 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.

Replace the three forces and one

couple acting on the robot assembly

with an equivalent force-couple at

point A.

PROBLEMS

C

C=27 N.m

//z M

N

37o

F=250 N

x z

y

53o

//x

//y

500 mm

540 mm

A

B

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

200 N

160 N

WAB = 200 N, cos qx=0.6, cos qy (qy <90°) and cos qz=0. WBC = 160 N, cos

qx=7/9, cos qy=4/9 and cos qz=-4/9. F=250 N and C=27 Nm act to the end C of arm BC. Replace the three forces and one couple acting

PROBLEMS

C

C=27 N.m

//z M

N

37o

F=250 N

x z

y

53o

//x

//y

500 mm

540 mm

A

B

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

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

200 N

160 N

PROBLEMS

12) The force-couple system acting at O is equivalent to

the wrench acting at A. If and

, determine .

PROBLEMS

OMR

z

x y 2 m 3 m

A

//RM

R

z

x y

)(7001400600 NkjiR

)(1200// mNM R

OM

PROBLEMS The force-couple sytem acting at O is equivalent to the wrench acting

at A. If and

determine .

PROBLEMS

OMR

z

x y 2 m 3 m

A

//RM

R

z

x y

)(7001400600 NkjiR

)(1200// mNM R

OM

PROBLEMS 13) A portion of the flue for a furnace is attached to the ceiling at A. While supporting the free end of the flue at F, a worker applies the forces indicated at E and F and the couple M (applied to portion CB) to align end E with the furnace. Knowing that the 95 N force at F lies in a plane parallel to the yz plane, determine (a) the angle a the force at F should form with the horizontal if duct AB is not to tend to rotate about the vertical axis, (b) the equivalent force-couple system at A of the given system, (c) further

Line EG lies in the xz plane and forms a 45° angle with the // x axis. The line of action of the 75 N force forms an angle of 53° with line EG.

reduce the system to a wrench and determine the coordinates of the point the line of action of the wrench passes in the yz plane.

PROBLEMS Determine (a) the angle a the force at F should form with the horizontal if duct AB is not to tend to rotate about the vertical axis, (b) the equivalent force-couple system at A of the given system, (c) further reduce the system to a wrench and determine the coordinates of the point the line of action of the wrench passes in the yz plane.

95 N force at F lies in a plane parallel to the yz plane, line EG lies in the xz plane and forms a 45° angle with the // x axis. The line of action of the 75 N force forms an angle of 53° with line EG.

C3 = 80 Nm (in plane ABCD)

qy < 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

14) 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° qy

53°

30°

6 m

4 m

1C

1F

2C

3F

3C

2F

k78.063.0i015.052.9688.7795.1

k52.9688.77i95.1

R

Rn

k52.9688.77i95.1FR

k64.3416i12k30cos4053sin60cos40i53cos60cos40

k88.4688.46i15.35884

k)80()08(i181275

k5115i21.21kcos600cos6icos4530

601cos06cos45cos

04.124

222R

3

2222

1

222

jj

j

jjF

jj

F

jjF

o

yy qq

Force:

k86756.841i86.847M

k23i64i100k3030i42.24

k2484.432i4.394k84.843562.56i04.375k16.6516.65M

k23i642432

k24i3280C

k24i324k8i6BDBA

i100C

k3030i42.24k60cos60cosicos4560C

k2484.432i4.394k64.3416i12k310i6

k84.843562.56i04.375k88.4688.46i15.358i12

k16.6516.65k5115i21.21k44i10

O

O

40

223

2

1

33

22

11

j

j

jjj

j

jj

jjjFr

jjjFr

jjjFr

CFrM o

Moment:

k86756.841i86.847M

k52.9677.88i95.1

O

j

jR

Equivalent force-couple system at point

O

Reduction to a wrench in yz plane

k02.10402.84i2k78.063.0i015.035.133MM

35.133k78.063.0i015.0k86756.841i47.868M

////

//

jjn

NmjjnM

R

RO

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