1

1. Determine the tension in cables BD and CD and the x, y, z components of reaction at theball-and-socket joint at A.

“Ball-and-socket joint”

Reactions at A:

SOLUTION

kAjAiAR zyxA

FBD of Plate

Az

Ay

Ax

TBD

TCD

kjiTT BDBD

8571.04286.02857.0

kjiTT CDCD

8571.04286.02857.0

kF

3002

kAjAiAR zyxA

FBD of Plate

Az

Ay

Ax

TBD

TCD

kjiTT BDBD

8571.04286.02857.0

kjiTT CDCD

8571.04286.02857.0

kF

300

0300

8571.04286.02857.05.15.1

8571.04286.02857.05.15.1

0

ki

kjiTji

kjiTji

M

BD

CD

A

0300

2856.14286.02856.16429.0

2856.14286.02856.16429.0

j

iTkTjTkT

iTkTjTkT

BDBDBDBD

CDCDCDCD

04286.06429.04286.06429.0

03002856.12856.1

02856.12856.1

BDBDCDCD

BDCD

BDCD

TTTTk

TTj

TTi

NTT BDCD 677.116

NATTAF xCDBDxx 68.6602857.02857.00

004286.04286.00 yCDBDyy ATTAF

NATTAF zCDBDzz 10003008571.08571.00 3

2. The uniform 48 N plate is welded to the vertical shaft AB of negligible

weight. The assembly is supported by a radial bearing at A and a thrust

bearing at B. Determine the force cable CD and the reactions at A and B.

4

FBD of Plate

Ay

radial bearing

thrust bearing

Ax

By

Bz

Bz

𝑾

𝑻

Reactions at A: jAiAR yxA

Reactions at B: kBjBiBR zyxB

C (0, 4.8, 1.8)D (3.6, 0, 0)

kjiTT

kjiTT

288.0767.0575.0

8.18.46.3

8.18.46.3

222

0288.0767.0575.06.3

608.43.5

484.255.3

1.6

0

kjiTi

ijk

kjk

jAiAk

M

yx

B

076.2288

00368.13181.6

02.1151.6

00368.176.22883182.1151.61.6

Tk

TAj

Ai

jTkTkjiiAjA

x

y

yx

NT

NA

NA

y

x

348.104

88.18

395.34

5

Reactions at A: jAiAR yxA

Reactions at B: kBjBiBR zyxB

kjiTkji

TT

288.0767.0575.08.18.46.3

8.18.46.3

222

0288.0480

0767.00

0575.0600

TBF

TBAF

TBAF

zz

yyy

xxx

NT

NA

NA

y

x

348.104

88.18

395.34

NB

NB

NB

z

y

x

05.78

915.98

395.34

6

3. Structure ABCD is supported by a collar at D that can rotate and slide along bar EF which is

fixed and is frictionless. Structure ABCD makes contact with smooth surfaces at A and C

where normal direction to the surface at A lies in a plane that is parallel to the xy plane.

Force P is parallel to the y axis. If P=10 kN, determine the reactions at A, C and D. Note that

collar D acts like a wide radial bearing.

n

7

Dy

Dz

NC

NA

My

Mz

FBD of ABCD

8

4. The lever AB is welded to the bent rod BCD which is supported by bearing E and cable DG. Assuming

that the bearing can exert an axial thrust and couples about axes parallel to the x and z axes, determine the

tension in cable DG and the reaction at E under the action of the 220 N force. The mass of ABCD is

neglected.

220 N

240 mm

60 mm

250 mm

225 mm

160 mm

120 mm

GD

C

E

B

y

z

x

A

Reactions at E

(Single thrust bearing) :

kEjEiER zyxE

kjTT

kjTT

iF

88.047.0

225120

225120

220

22

kMiMM EzExE

kEjEiER zyxE

kjTTiF

88.047.0220 kMiMM EzExE

088.047.019016022024060 kMiMkjTjiikjM EzExE

065.167 ExMTi

018.14152800 Tj

029.7513200 EzMTk

T=374 NMEx=62701.1 N.mmMEz=14958.46 N.mm

065.16718.14129.755280013200 kMiMiTjTkTjkM EzExE

2

NE

EF

NE

EF

NE

EF

z

T

zz

y

T

yy

x

xx

12.329

037488.00

78.175

037447.00

220

02200

1

2

220 N

240 mm

60 mm

250 mm

225 mm

160 mm

120 mm

GD

C

E

B

y

z

x

A

5. Turnbuckle T1 is tightened to a tension of 750 N and turnbuckle T2 is tightened to

500 N. Determine the components of the corresponding force and moment reactions

at the built-in support at O. Neglect the weight of the structure.

11

Ox

Mx OyMy

Oz

Mz

𝑻𝟐

𝑻𝟏

mNMmNMmNM

kMjMiM

kijikjji

M

zyx

zyx

O

38.17499.31218.310

0

316.447658.2236.045.0857.642714.3859.08.0

0

kiik

TT

kjjk

TT

316.447658.223006.1

45.09.0

857.642714.38575.1

9.05.1

500

22

750

11

FBD of Pole

12

Ox

Mx OyMy

Oz

Mz

𝑻𝟐

𝑻𝟏

kiT

kjT

316.447658.223

857.642714.385

2

1

0316.447857.6420

0714.3850

0658.2230

zz

yy

xx

OF

OF

OF

NO

NO

NO

z

y

x

173.1090

7145.385

658.223

13

6. Determine the magnitudes of the force R and couple M exerted by the nut and bolt

on the loaded bracket at O to maintain equilibrium.

OxMx

OyMy

Oz

Mz

FBD of Bracket

Answer:

mkNMmkNMmkNM zyx 245.0359.01172.0

kNOkNOkNO zyx 2.1853.0028.1 14

7. A vertical force P on the foot pedal of the bell crank is required to produce a

tension T of 400 N in the vertical control rod. Determine the corresponding bearing

reactions at A and B.

15

Az

AyBz

By

0569.41642.01.02.02.0

040030sin12.030cos12.016.0

2.01.0

2.0

0

ijiPjPjBkB

kkji

kPji

kBjBi

M

zy

zy

A

002.0

9225.42302.0641.0

845.2070569.412.0

yy

zz

BBk

NBBPj

NPPi

NAABF

ABAF

zzzz

yyyy

9225.1830845.2074000

000

9225.423

16

8. Because of friction a very large force T is required to raise the 10 kg body when the cable

is wound around a fixed rough cylinder as shown. For the static equilibrium values shown,

determine all of the reactions at point O. The 0.3 m dimension refers to the point A where

the cable loses contact with the cylinder. Neglect the weight of the fixed cylinder.

17

Ox

Mx

OyMy Oz

Mz

W=10(9.81) N=0.098 kN18

Ox

Tyz

OyMy Oz

Mz

W=10(9.81) N=0.098 kN

Reactions at O:

y

20o

z

y

x Txy

30o

30o

A

Mx

A

kMjMiMM

kOjOiOR

zyxo

zyxo

[kN]098.0[N]1.9881.910

89.841.2415

20sin30cos20cos30cos30sin

iiiW

kjiT

kTjTiTT

19

Ox

OyMy

Oz

Mz

W=10(9.81) N=0.098 kN

y

x Txy

30o

30o

A

30cos3026 mm

30sin3015mm

Mx

mmkN.MmmkN.MmmkN.M

kMjMiM

k.j.ikjii.jk

M

zyx

zyx

O

7285634121236519394

0

89841241580015260980301100

0

kNOTOF

kNOTOF

kNOTOF

zzzz

yyyy

xxxx

89.800

41.2400

902.140098.00

20

9. The uniform panel door has a mass of 30 kg

and is prevented from opening by the strut C,

which is a light two-force member whose

upper end is secured under the door knob and

whose lower end is attached to a rubber cup

which does not slip on the floor. Of the door

hinges A and B, only B can support force in the

vertical z-direction. Calculate the compression

C in the strut and the horizontal components

of the forces supported by hinges A and B

when a horizontal force P = 200 N is applied

normal to the plane of the door as shown.

21

NjP

NkkW

kBjBiBB

jAiAA

kjFnFF

kjkj

n

zyx

yx

CDHCC

DH

200

3.294)81.9(30

8.06.0

8.06.025.1

75.0

O (0, 0, 0) A (1.2, 0, 1.8) B (1.2, 0, 0.2)

G (0.6, 0, 1) E (0, 0, 2) D (0.2, 0, 1)

H (0.2, 0.75, 0)

jAiAkjki

kjFkikkiM

M

yx

CB

B

6.12008.12.1

8.06.08.03.2948.06.0

0

E

G

W

AxAy

Bx

By

BzFC

D

H

O

P

22

NAAA

NAAFj

NAAFi

NFFk

yxxy

xxC

yyC

CC

06.138

64.89058.1766.18.0

10503606.148.0

40002406.0

22

NBBB

NBFBAF

NBBAF

yxxy

yCyyy

xxxx

47.170

14506.02000

64.8900

22

E

G

W

AxAy

Bx

By

BzFC

D

H

O

P

02406.058.1766.18.03606.148.0 kFjAFiAFM CxCyCB

23

10. The control surface of an aircraft is supported by a thrust bearing at point C and is

actuated by a bar connected to point A. The 1 kN force acts in the negative z direction, and

the line connecting points A and B is parallel to the z axis. Determine the value of force F

needed for equilibrium and all support reactions.

24

F Cx

Cy

Cz

Mx

Mz

F

1 kN

kF,kFjFiFF D

1

15

5

15

2

15

14

kMiMM,kCjCiCR zxCzyxC

Reactions at C:

06.03.015

5

15

2

15

141.02.0

0

kMiMkjikjiFkj

M

zx

C

kNMMFk

mkNMkNFFj

MFFi

kMiMijiFjFiFkF

zz

x

x

zx

6.0015

8.2

343.021.303.015

4.1

06.015

2.0

15

1

06.03.015

2.0

15

4.1

15

1

15

8.2

25

F Cx

Cy

Cz

Mx

Mz

F

1 kN

kF,kFjFiFF D

1

15

5

15

2

15

14

kMiMM,kCjCiCR zxCzyxC

Reactions at C:

kN.CCFF

kN.CCFF

kN.CCFF

zzz

yyy

xxx

0720115

50

4280015

20

9962015

140

26

11. The shaft, lever and handle are welded together and constitute a single rigid

body. Their combined mass is 28 kg with mass center at G. The assembly is

mounted in bearings A and B, and rotation is prevented by link CD. Determine the

forces exerted on the shaft by bearings A and B while the 30 N.m couple is applied

to the handle as shown.

27

Ax

Ay

Bx

By

W

F

C

F

x

y 600 mm

45

0 m

m

jF.iF.F

8060

06.06.03043.604.8236.008.006.0

06.03081.92822.03.08.06.01.06.0

0

iBjBkkjkFiFjF

jBiBkkijkjiFkj

M

yx

yx

A

NBBFj

NBBFi

NFFk

xx

yy

2.11206.04.8206.0

49.3306.008.0

2.251043.9036.0

28

Ax

Ay

Bx

By

W

F

C

F

x

y 600 mm

45

0 m

m

jF.iF.F

8060

NABAFF

NAAFF

yyyy

xxx

46.16708.00

78.1107.2742.1126.00

NB

NB

NF

x

y

2.112

49.33

2.251

29

12. The electric sander has a mass of 3 kg

with mass center at G and is held in a slightly

tilted position (z-axis vertical) so that the

sanding disk makes contact at its top with the

surface being sanded. The sander is gripped

by its handles at B and C. If the normal force

R against the disk is maintained at 20 N and

is due entirely to the force component Bx

(i.e., Cx = 0), and if the friction force F acting

on the disk is 60 percent of R, determine the

components of the couple M which must be

applied to the handle at C to hold the sander

in position. Assume that half of the weight is

supported at C.

30

Cy

Cz

29.43 N

My Mx

Mz

Nm.MNm.MNm.M

kMjMiM

k.k.i.

k.ik.j.i.

jik.i.

M

zyx

zyx

C

56241118571

0

432930040

71514203020040

122040120

0

N

WCC

NFNR

zx 715.142

81.93

20

12100

602020

NCB

CBWF

NC

CF

NB

BF

zz

zzz

y

yy

x

xx

715.14

00

12

0120

20

0200

31