1. The uniform I-beam has a mass of 60 kg per meter of its length.

Determine the tension in the two supporting cables and the reaction

at D.

PROBLEMS

m = 60 kg/m, determine the tension in the two supporting cables and the reaction at D.

PROBLEMS

mGmDmCmBmA )0,0,3(,)0,0,0(,)3,2,5(,)3,2,5(,)0,0,5(

ABT

ACT

W

DN

x

y

z

G

kj

ACAC

kj

ABAB

kjTT

kjTT

83.055.0

83.055.0

606.3

32

606.3

32

)(8.470881.98602

Nkks

mm

m

kgW

kNN DD

NN

jNjkNiki

NrWrM

D

DD

DADAGA

52.1883

056.941758.47082

00 //

m = 60 kg/m, determine the tension in the two supporting cables and the reaction at D.

PROBLEMS

mGmDmCmBmA )0,0,3(,)0,0,0(,)3,2,5(,)3,2,5(,)0,0,5(

ABT

ACT

W

DN

x

y

z

G

NTT

TF

TTF

kjTT

kjTT

ACAB

ABz

ACABy

ACAC

ABAB

98.1701

052.18838.4708)83.0(20

0

83.055.0

83.055.0

PROBLEMS

2. A 36 kg sheet of plywood rests on two small wooden blocks as

shown. It is allowed to lean 20° from the vertical under the action

of a force P which is perpendicular to the sheet. Friction at all

surfaces of

blocks A and B is

sufficient to

prevent slipping.

Determine the

magnitude P and

the vertical

reaction forces

at A and B.

PROBLEMS m = 36 kg. Friction at all surfaces of blocks A and B is sufficient to

prevent slipping. Determine the magnitude P and the vertical

reaction forces at A and B.

PROBLEMS

3. The uniform 7 m steel shaft

has a mass of 200 kg and end

A is supported by the ball-and-

socket connection in the

horizontal x-y plane. The ball

end B rests against the smooth

vertical walls. One of the

vertical walls supporting end B

of the shaft is turned through a 30° angle as shown. Calculate the

magnitudes of the forces and exerted on the ball end B of the

shaft by the vertical walls C and D, respectively.

P

R

PROBLEMS m = 200 kg, ball-and-socket connection at A, end B rests against the smooth vertical walls.. Calculate magnitudes of forces and exerted on B by the vertical walls C and D. P

R

P

xA

yA

zA

W

G

RTop view

x

y

30

P

RyR

xR

30

jRiRjRiRRN

jPPN

NkkW

kAjAiAA

yxD

C

zyx

30sin30cos

)(1962)81.9(200

PROBLEMS m = 200 kg, ball-and-socket connection at A, end B rests against the

smooth vertical walls.. Calculate magnitudes of forces and exerted

on B by the vertical walls C and D. P

R

P

xA

yA

zA

W

G

R

mGmBmA )5.1,3,1(,)3,0,0(,)0,6,2(

058861962

30sin330cos330cos630sin232

019625.1330sin30cos362

0

0

//

ij

iRjRkRkRiPkP

kkjijRiRjPkji

WrRPr

M

AGAB

A

G

m5.3

m5.3

m325.626 22

m3

m5.1

m3m1jRiRR

jPPNkW

30sin30cos

,)(1962

PROBLEMS m = 200 kg, ball-and-socket connection at A, end B rests against the

smooth vertical walls.. Calculate magnitudes of forces and exerted

on B by the vertical walls C and D. P

R

P

xA

yA

zA

W

G

R

030cos630sin22

196230cos3

588630sin33

0

kRRP

jR

iRP

M A

jRiRRjPPNkWkAjAiAA zyx

30sin30cos,,)(1962,

j

NR

R

17.755

196230cos3

i

NP

RP

42.1584

588630sin3317.755

4. 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.

PROBLEMS

NjPNkkW

kBjBiBBjAiAA

kjFnFF

kjkj

n

zyxyx

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) m

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 E

G

W

Ax Ay

Bx

By

Bz FC

D

H

O

P PROBLEMS

kFjAFiAFM

jAiAkjki

kjFkikkiM

M

CxCyCB

yx

CB

B

2406.058.1766.18.03606.148.0

6.12008.12.1

8.06.08.03.2948.06.0

0

NA

NAFAj

NAAFi

NFFk

xy

xCx

yyC

CC

06.138

64.89,58.1768.06.1

105,6.136048.0

400,2406.0

NB

NBFBAF

NABBAF

xy

yCyyy

xxxxx

47.170

14506.02000

64.8900

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

E

G

W

Ax Ay

Bx

By

Bz FC

D

H

O

P

PROBLEMS

PROBLEMS

5. 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 on

bearings A and B, and rotation

is prevented by link CD.

Determine the forces

exerted on the shaft by the

bearings A and B while the 30

N·m couple is applied to the

handle as shown. Would these

forces change if the couple were applied to the shaft AB rather than to the

handle?

PROBLEMS Determine the forces exerted on the shaft by the bearings A and B while the 30 N·m couple is applied to the handle as shown. Would these forces change if the couple were applied to the shaft AB rather than to the handle?

Ax

Ay

Bx

By

W

F

C

PROBLEMS 6. Under the action of the 40 N·m

torque applied to the vertical shaft,

the restraining cable AC limits the

rotation of the arm OA and

attached shaft to an angle of 60°

measured from the y axis. The

collar D fastened to the shaft

prevents downward motion of the

shaft in its bearing. Calculate the

bending moment M, the compression

P and the shear force V in the

shaft at section B. (Bending

moment, expressed as a vector, is

normal to the shaft axis and shear

force is also normal to the shaft

axis.)

PROBLEMS collar D prevents downward motion of the shaft in its bearing, calculate the bending moment M, the compression P and the shear force V in the shaft at section B.

(Bending moment, expressed as a vector, is normal to the shaft axis and shear force is also normal to the shaft axis.)

xB

T

yB

zB

xM

yM

kjiTkji

TT

76.038.053.0263.0

2.01.01386.0

mCmB

mA

)2.0,18.0,0(,)11.0,0,0(

)0,08.0,1386.0()0,60cos16.0,60sin16.0(

040

76.038.053.018.009.0

0400 /

jMiMk

kjiTjk

jMiMkTrM

yx

yxBCB

040095.0047.0

106.0

kTjMT

iMTM

y

xB

PROBLEMS collar D prevents downward motion of the shaft in its bearing, calculate the bending moment M, the compression P and the shear force V in the shaft at section B.

(Bending moment, expressed as a vector, is normal to the shaft axis and shear force is also normal to the shaft axis.)

xB

T

yB

zB

xM

yM

kjiTT

76.038.053.0

mNMi

mNMj

NTk

x

y

63.44

79.19

05.421

mNMMM yx 82.4822

NPBTBF

NBV

NBTBF

NBTBF

zzz

xy

yyy

xxx

19.320,076.00

62.273

19.160,038.00

89.221,053.00

7. 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.

PROBLEMS

R = 20 N

W = 29.43 N

Cz Cy My

Mx

F = 0.6R=12 N

Mz

Bx

Bz

m =3 kg, normal force R =20 N due entirely to force component Bx (i.e., Cx = 0), the friction force F acting on the disk 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.

PROBLEMS

mmG

mmC

mmB

mmA

)0,0,40(

)300,0,0(

)0,200,40(

)100,0,120(

01220120400

43.294030072.142020040300

0

0 ///

kMjMiMjiik

kikkijik

kMjMiM

jFiRrWrkBiBrM

zyx

zyx

CACGzxCBC

NW

CBCB

F

CNB

zzzz

z

xx

715.142

,43.29

0

0,20

R = 20 N

W = 29.43 N

Cz Cy My

Mx

F = 0.6R=12 N

Mz

Bx

Bz

m =3 kg, normal force R =20 N due entirely to force component Bx (i.e., Cx = 0), the friction force F acting on the disk 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.

PROBLEMS

mmNM

mmNM

mmNM

z

y

x

2560

6.1411

1856

014404000

8002.11778.5886000

48002944

kM

jM

iMM

z

y

xC

jCy

12

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.

PROBLEMS

Tz

Ty

Tyz

y

z

20

°

x

W=10(9.81)=98.1 N = 0.0981 kN

W

Ox

Oy Oz

MOy

MOx

MOz

kNTTT

kNTTT

kNTT

kNTT

yzz

yzy

yz

x

89.820sin30cos20sin

41.2420cos30cos20cos

98.2530cos

1530sin

A

PROBLEMS Determine all of the reactions at point O, neglect the weight of the fixed cylinder

O (0, 0, 0)

A (30sin60, 30cos60, 800) = (26, 15, 800)

B (0, 30, 1100) mm

B

A

Tx

y

x T

30°

60° 60° 30 mm

z B

Top view

Front view

Determine all of the reactions at point O, neglect the weight of the fixed cylinder

0

89.841.24158001526

0981.0110030

00

?

?

//

kMjMiM

kjikjj

ikjM

MTrWrM

kMjMiMM

kOjOiOO

OzOyOx

O

OOAOBO

OzOyOxO

zyx

mkNMmkNkjiM

mkNmmkNM

mkNmmkNM

mkNmmkNM

kM

jMiMM

OO

Oz

Oy

Ox

Oz

OyOxO

88.22,86.01.124.19

86.072.856

1.1234.12123

4.1965.19394

072.856

34.1212365.19394

kNOTOF

kNOTOF

kNOTOF

zzzz

yyyy

xxxx

89.8,00

41.24,00

902.14,0098.00

kNO

kNkjiO

95.29

89.841.249.14

W

Ox

Oy Oz

MOy

MOx

MOz

PROBLEMS

B

O (0, 0, 0)

A (26, 15, 800)

B (0, 30, 1100) mm

PROBLEMS

9. The two bars AC and CD are homogeneous and each weigh 200 N/m.

Joints A, C and D are ball-and-socket joints and a cable is connected

between B and E. Determine all reactions that act on bar AC.

PROBLEMS AC and CD are homogeneous and each weigh 200 N/m. Joints A, C and D are ball-and-socket joints and a cable is connected between B and E. Determine all reactions that act on bar AC.

10. The structure is subjected to the loadings shown. Member AB is

supported by a ball and socket joint at A and a smooth collar at B. Member

CD is supported by a pin at C which acts like a thrust bearing. Determine the

reactions at A and C.

PROBLEMS

Ball and socket joint at A, smooth collar at B, thrust bearing at C. Determine reactions at A and C.

PROBLEMS

G

G

D

11. A circular plate with a weight of 900 N acting at its center point G is

supported by cord DE and a thrust bearing at B. Shaft AC is fixed and

frictionless. Determine the cable tension and the reactions acting on the

circular plate from the bearing at B.

PROBLEMS

W = 900 N, circular plate supported by cord DE and a thrust bearing at B. Shaft AC is fixed and frictionless. Determine the cable tension and the reactions at bearing B.

PROBLEMS

G

G

D