PROBLEMS ON

EQUILIBRIUM OF

PARTICLES

1. Find the angle of tilt q with the horizontal so that the

contact force at B will be one-half that at A for the

smooth cylinder. (3/15)

PROBLEMS

q =?, contact force at B will be one-half that at A

tangent

tangent

AN

AB NN21

=

q

mg

q

45

q45

q45

q45

q

45

q45

q45

q45

q45

AN

AB NN21

=

mg

q45

q45

PROBLEMS

q =?, contact force at B will be one-half that at A

43.18

57.2645

212

1

)45tan(

0

=

=

==

=

q

q

qA

A

N

N

F

AN

AB NN21

=

mg q45

43.1857.2645

21

)45tan()45cos(

)45sin(

)45cos(21

)45sin(

0

==

==

=

=

qq

qq

q

qq AA

x

NN

F

or

AN

AB NN21

=

mg

q45

q45

x

y

PROBLEMS

2. The 40-kg block rests on the rough surface. Length of

the spring is 180 mm in the position shown. Unstretched

length of the spring is 200 mm. Determine the

coefficient of friction required for the equilibrium.

PROBLEMS

m =40-kg, l = 180 mm, l0 = 200 mm, m =?

N NmgW 4.392)81.9(40 ===

springF

fF

PROBLEMS

20° 20°

NmmN

F

kF

spring

spring

5018.02.02500 =

=

=

020sin4.39250020sin0

74.36820cos4.392020cos0

===

====

ffspringx

y

FWFFF

NNWNF

5.074.36821.184

,21.184 =====N

FNFNF

f

ff mm

3. ABC is a cable passing

over a frictionless pulley

at B where a force F is

applied. Let h1 = 12 cm and

h2 = 24 cm. The length of

the cable ABC is 50 cm.

Determine the magnitude

and the direction of force

F such that the tension in

the cable is 65 N.

PROBLEMS

h1 = 12 cm, h2 = 24 cm. The length of the cable lABC = 50 cm,

TABC = 65 N determine F.

94.54,82.0

68.205024

sin

45.35,58.068.20

12sin

68.20,2068100

5761002500144

245012

32.29

22

22222

==

=

===

==

=

==

qq

cmxx

xxx

xxa

q

N6512 cm

24 cm

x

50-x

N65

x

y

a

NF

FF

FFF

F

y

x

61.91,75.9

17.0tan,51.15sincos

29.90

094.54sin6545.35sin65sin0cos

29.90,29.90cos,094.54cos6545.35cos65cos

0

==

==

==

===

=

PROBLEMS

4. A 4 kg sphere rests on the smooth parabolic surface. Determine the normal force it exerts on the surface and the mass mB of block B needed to hold it in the equilibrium position shown.

PROBLEMS

m =4 kg, determine the normal force it exerts on the surface and the mass mB of block B.

N

tangent T

mg

60

q

dy

dxq

43.63

2)4.0(5tan

)5.2(2tan

5.2

tan

2

=

==

==

=

=

q

q

q

q

xdx

dy

xydx

dy

NTNN

NNTF

NTNT

F

N

y

x

12.3562.19

24.392,0)81.9(443.63cos60sin0

79.1,043.63sin60cos

0

24.3979.1

==

===

==

=

(opposite direction to that shown)

PROBLEMS

m =4 kg, determine the normal force it exerts on the surface and the mass mB of block B.

NT 12.35=

NTgmB 12.35==

kgmB 58.381.912.35

==

21 TT =

B

pulley

PROBLEMS

PROBLEMS

5. The cable and pulley

system shown is used to

hoist a weight W. If cables

AB and AC have breaking

strengths of 900 N and

cable DAE has a breaking

strength of 450 N,

determine the largest

weight W that may be lifted.

PROBLEMS cables AB and AC have breaking strengths of 900 N, cable DAE has a breaking strength of 450 N, determine the largest weight W that may be lifted

r=200 mm

h=120 mm

A

B

h r

C

C

6. The cylinder of mass 1 kg having a very small diameter is held against a semi-cylinder with a much larger diameter by two identical springs, which are fixed to points C and C on the ground. The springs are unstretched when at point A. Knowing that the small cylinder is in equilibrium at point B, what is the spring constant?

PROBLEMS

r=200 mm

h=120 mm

A

B

h r

C

C

PROBLEMS m= 1 kg, springs are unstretched when at point A, small cylinder is in equilibrium at point B, what is the spring constant?

PROBLEMS

7. Three cables are

used to hold a balloon

as shown. Knowing that

the balloon exerts an

800 N vertical force at

A, determine the

tension in each cable.

PROBLEMS balloon exerts an 800 N vertical force at A, determine the tension in each cable

ABT

NjF

kjTnTT

kjiTnTT

jiTnTT

FTTT

ADADADAD

ACACACAC

ABABABAB

ADACAB

800

5.6

3.36.54.7

2.46.54.27

6.52.4

0

=

==

==

==

=

ACT

ADT

F

mDmCmBmA )3.3,0,0()2.4,0,4.2()0,0,2.4()0,6.5,0(

PROBLEMS balloon exerts an 800 N vertical force at A, determine the tension in each cable

ABT

ACT

ADT

F

NTNTNT

TTTT

TTTF

TTTT

TTF

TTTT

TTF

ADABAC

AC

T

ADAC

T

AB

ADACABy

ACADADAC

ADACz

ACABACAB

ACABx

ACAC

04.41659.20046.371

80015.2,8005.66.5

4.76.5

76.5

08005.66.5

4.76.5

76.5

0

12.1,5.63.3

4.72.4

05.63.3

4.72.4

0

54.0,4.74.2

72.4

04.74.2

72.4

0

12.154.0

===

==

==

==

==

==

=

=

8. A small peg P rests on a spring that is contained inside the

smooth pipe. When the spring is compressed so that s = 0.15

m, the spring exerts an upward force of 60 N on the peg.

B

A

x

0.3 m

y x

z

y

s

C

P

0.2 m 0.4 m

Determine the point of

attachment A (x, y, 0) of

cord PA so that the

tension in cords PB and

PC equals 30 N and 50 N,

respectively.

PROBLEMS

PROBLEMS s = 0.15 m, Fspring = 60 N (↑), A (x, y, 0) = ? TPB = 30 N TPC = 50 N

B

A

x

0.3 m

y

x

z

y

s

C

P

0.2 m 0.4 m

PCT

PBT

PAT

springFNkFTTTF springPAPCPBspring

60,0 ==

myxAmCmBmP )0,,()0,2.0,3.0()0,4.0,0()15.0,0,0(

a

PAPAPAPA

PCPCPC

PBPBPB

yx

kjyixTnTT

kji

kjinTT

kj

kjnTT

222 15.0

15.0

23.1964.2546.3839.0

15.02.03.050

54.1010.28427.0

15.04.030

==

=

==

=

==

PROBLEMS s = 0.15 m, Fspring = 60 N (↑), A (x, y, 0) = ? TPB = 30 N TPC = 50 N

NTTa

F

NTTa

yF

NTTax

F

z

zPA

y

yPA

x

xPA

PA

T

PAz

PA

T

PAy

PA

T

PAx

23.30,015.0

23.1954.10600

46.2,064.2510.280

46.38,046.380

===

===

===

PCT

PBT

PAT

springF

P

C

A

B

mA

myTa

yTyfrom

mxTax

Txfrom

aTa

Tzfrom

NT

NkjiT

PAPA

PAPA

PAPA

PA

PA

y

x

z

0,012.0,19.0

012.0,46.2

19.0,46.38

243.0,15.0

23.30

98.4823.3046.246.38

23.3046.246.38222

===

===

===

==

=

9. Cables AB and AC can sustain a maximum tension of 500 N and the pole can sustain a maximum compression of 300 N.

Determine the maximum weight of the lamp that can be supported in the position shown. The force in the pole acts along the axis of the pole.

PROBLEMS

PROBLEMS Cables AB and AC can sustain a maximum tension of 500 N and the pole can sustain a maximum compression of 300 N. Determine the maximum weight of the lamp that can be supported in the position shown.

10. The crate weighing 580 N is held on the incline by the wire AB and by the horizontal force which is directed parallel to the z axis. Since the crate is mounted on casters, the force exerted by the

P

incline on the crate is perpendicular to the incline. Determine the magnitude of and the tension in wire AB.

E B

A

2.2 m

3 m

1.5 m 3 m

x z

y

O

P

C

4 m

P

PROBLEMS

E B

A

2.2 m

3 m

1.5 m 3 m

x z

y

O

P

C

4 m

PROBLEMS W = 580 N, force exerted by the incline on crate perpendicular to the incline, determine the magnitude of and the tension in wire AB

P

P

T

W

N

kPP

NjW

NTPWF

=

=

==

580

0

mEmDmC

mBmA

mA

5.1,3,0,0,0,4,0,3,0

0,2.5,0,5.1,2.1,4.2

5.1,13.53cos33,13.53sin3

x

y

4 m

3 m

53.13°

A

xA

yA

D

E B

A

2.2 m

3 m

1.5 m 3 m

x z

y

O

P

C

4 m

PROBLEMS W = 580 N, force exerted by the incline on crate perpendicular to the incline, determine the magnitude of and the tension in wire AB

P

P

T

W

N

jiNN

jijin

or

jiji

n

ijkjirr

eithernfor

nNN

kjiT

r

rTnTT

N

N

CECD

N

N

AB

ABT

8.06.0

8.06.013.53sin13.53cos

8.06.05.7

65.4

5.465.134

9.4

5.144.2

//

/

/

=

==

=

=

==

=

===

x

y

4 m

3 m

53.13°

A

Nn

53.13°

CDr /

D

CEr /

mEmDmC

mBmA

mA

5.1,3,0,0,0,4,0,3,0

0,2.5,0,5.1,2.1,4.2

5.1,13.53cos33,13.53sin3

E B

A

2.2 m

3 m

1.5 m 3 m

x z

y

O

P

C

4 m

PROBLEMS W = 580 N, force exerted by the incline on crate perpendicular to the incline, determine the magnitude of and the tension in wire AB

P

P

T

W

N

NNNPTPk

NTTTNj

TNTNi

kTPjTNiTN

T

322120,9.45.1

395,58047.1,9.4

45808.0

94.24.2

,9.44.2

6.0

09.45.1

9.44

5808.09.44.2

6.0

94.24.2

===

==

==

=

CDr /

D

CEr /

0=F

11. Straight bar AB is fixed in space. Spring CD has a stiffness of 3 N/mm and its unstretched length is 200 mm. If there is no friction between collar C and bar AB,

determine the weight W of the collar that produces the equilibrium condition shown and the reaction between the collar and bar AB.

PROBLEMS

PROBLEMS k =3 N/mm, l0=200 mm, neglecting friction determine the weight W of the collar for equilibrium and the reaction between the collar and bar AB

W

A

B

12. If

WA=WB=1400

N, determine

the force P,

TAB and the

reactions

between the

collars and

bars.

PROBLEMS

PROBLEMS WA=WB=1400 N, determine the force P, TAB and the reactions between the collars and bars

13. Smooth collars A, B and C, each weighing 360 N, are connected by the wires AB and BC and may slide freely on the smooth rod having the shape shown. Determine the magnitude of the horizontal force which must be applied to the collar A to maintain equilibrium. DEFG portion of the rod is parallel to xy-plane.

P

PROBLEMS

P

NAy

NAx

TAB

W

x z

y

x z

y

W

TBA

TBC

NBz

NBxy

x z

y

TCB

W

NCx NCz

FBD of collar A

FBD of collar B FBD of collar C

Four unknowns

Four unknowns

Three unknowns

x z

y

TCB

W

NCx NCz

FBD of collar C

A (0;9;3)

B (2;5;7)

C (6;1;0)

E

B

F

6 m

1.5 m 4.5 m xB

65.4

5.1 Bx= NT

TF

kNiNN

jW

kjiTT

NWTF

BC

BCy

CCC

BCBC

CBC

zx

810

03609

40

360

9

744

00

=

==

=

=

=

==

NP

PF

kPP

jNiNN

jW

kjiTT

PNWTF

z

yAxAA

BAAB

AAB

720

07200

360

720720360

00

=

==

=

=

=

==

==

P

NAy

NAx

TAB

W

x z

y

FBD of collar A

14. Smooth collars A and B are connected by the spring. Spring has a constant of 120 N/cm and its unstretched length is 30 cm. Determine the magnitude of the force P which must be applied to the collar A to maintain equilibrium and the reaction between the collar and bar. Neglect the weight of the collars. Take A (40;0;40) and B (0;20;80).

y x

z

30 cm

40 cm 40 cm

20 cm 80 cm

80 cm

P

Q

A

B

PROBLEMS

FBD of collar A

P

NAxz NAy

Fspring

y x

z

y x

z

30 cm

40 cm

40 cm

20 cm

80 cm 80 cm

P

Q

A

B

NPNN

NPF

NNNF

NPF

jNkNiNNkPiPP

cmABkjiABkjiF

kjiABkji

F

Axz

Axzz

AyAyy

Axzx

AyAxzAxzA

spring

spring

87.3483387

05

3

5

424000

1200012000

05

4

5

324000

5

3

5

4

5

4

5

3

60402040240012002400

)4080()020()400(60

402040)3060(120

==

==

===

==

==

===

=

=

x

A

z

30 cm

NAxz

q

q

00 == Aspring NPFF

Correct sense