PROBLEMS ON EQUILIBRIUM

OF RIGID BODIES IN TWO DIMENSIONS

1) A 54 kg crate rests on the 27 kg pickup tailgate. Calculate the tension T in each of the two restraining cables, one of which is shown and support force at O. The centers of gravity are at G1 and G2. The crate is located midway between the two cables.

PROBLEMS

m2 = 54 , m1 = 27 kg, calculate the tension T in each of the two restraining cables, one of

which is shown and support force at O.

PROBLEMS

Ox

Oy

2T

54(9.81)=529.74 N

27(9.81)=264.87 N

NT

T

T

T

MO

579

43.098.248

043.041.18557.63

0)31.0(07.44sin2

)35.0(74.529)24.0(87.264

0

07.44,968.0310300

tan

NOOO

NO

O

T

F

NO

OT

F

yx

y

y

y

x

x

x

07.832

79.10

0

07.44sin287.26474.529

0

832

007.44cos2

0

22

Free Body Diagram of crate + tailgate: +

2) Determine the magnitude P of the force required to rotate the release pawl

OB counterclockwise from its locked position. The torsional spring constant is

kT = 3.4 N·m/rad and the pawl end of the spring has been deflected 25°

counterclockwise from the neutral position in the configuration shown. Neglect

any forces at the contact point B. (3/22)

PROBLEMS

P = ? kT = 3.4 N·m/rad, spring deflected 25° ccw. PROBLEMS

mNT

radrad

mNkT T

48.1

436.04.3

NP

PP

PP

MO

89.44

48.100347.00295.0

048.102.010sin03.010cos

0

10cosP

10sinP

+

T

Bx=0

radxradx

rad

436.025

180

By=0

at the instant of release Bx, By will be zero Ox

Oy

3) A 35 N axial force at B is required to open the spring-loaded plunger of the

water nozzle. Determine the required force F applied to the handle at A and the

magnitude of the pin reaction at O. Note that the plunger passes through a

PROBLEMS

vertically elongated

hole in the handle at

B, so that negligible

vertical force is

transmitted there.

PROBLEMS

Ox

15°

NF

FFF

FFMO

98.13

06.4564.742.37630

0)44(10sin)38(10cos)18(35,0

Oy

B = 35 N

B = 35 N (axial force), determine the required force F applied to the handle at A and the magnitude of the pin reaction at O. Negligible vertical force at B.

NO

NO

OF

F

NO

OF

F

y

y

y

x

x

x

83.48

39.2

010sin

0

77.48

03510cos

0

+ Free Body Diagram of OAB:

4) The rack has a mass m = 75 kg.

What moment M must be exerted

on the gear wheel by the motor in

order to lower the rack at a slow

steady speed down the 60°

incline? Neglect all friction. The

fixed motor which drives the gear

wheel via the shaft at O is not

shown. (3/28)

PROBLEMS

radius

mass of rack m = 75 kg. Moment M = ? PROBLEMS

mNM

M

MO

78.47

0)075.0(18.637

0+

N18.637

Free Body Diagram of rack:

NFF

F

18.637,060sin75.735

0

Nmg 75.735)81.9(75

60°

+

Free Body Diagram of gear wheel:

FN

radius

mass of rack m = 75 kg. Moment M = ? PROBLEMS

mNM

M

MO

78.47

0)075.0(60sin75.735

0

18.637

+

or Free Body Diagram of rack + gear wheel

Nmg 75.735)81.9(75

60° Nradius

5) A rocker arm with rollers at A and B is shown in the position when the valve

is open and the valve spring is fully compressed. In this position, the spring

force is 900 N. Determine the force which the rocker arm exerts on the

camshaft C. Also calculate the magnitude of the force supported by the rocker-

arm shaft O.

PROBLEMS

Fspring = 900 N. Determine B and O. PROBLEMS

NB

BMO

66.1381

0)7(28.156)48(33.886)30(0

B

Oy

Ox

Fspring

Fspring x

Fspring y

NFNFNFyx springspringspring 33.88610cos900,28.15610sin900,900

NO

NO

O

F

NO

O

F

x

x

x

y

y

y

38.2273

28.156

028.156

0

2268

033.88666.1381

0

+

Free Body Diagram of AOB:

6) To test the deflection of the uniform 100-kg beam, the 50-

kg boy exerts a pull of 150 N on the rope rigged as shown.

Compute the force supported by the pin at the hinge O. (3/53)

PROBLEMS

NN

N

M

A

A

O

5701

0)4(150)25.3(150)5.2(5.490)2(981)75.0(

,0

PROBLEMS

NA Oy

Ox

Wbeam

Wboy

150 N 150 N

Free Body Diagram of beam + boy:

NW

NW

boy

beam

5.490)81.9(50

981)81.9(100

0,0 xx OF

kNOO

kNNO

OF

y

y

yy

93.3

93.35.3929

01501505.4909815701,0

2 m 0.5 m

mbeam = 100-kg, mboy = 50-kg, T = 150 N, O = ?

7) The uniform rod AB has a mass of 25 kg and a length L = 1 m.

What must be q for equilibrium? Also what are the normal

forces acting from the surfaces to the rod at points A and B?

Assume all surfaces are smooth.

PROBLEMS

30°

45°

25(9.81) = 245.25 N

A

B

Bx

By Ax

Ay

30°

45° 30tan

,30tan

45tan

yx

y

x

x

y

yx

AA

A

A

B

BBB

x

x

xx

xx

yxA

B

B

BB

BB

BBM

625.122tan

cos

sin

sin)625.122(cos

0cossincos625.122

0cos1sin1)cos5.0(25.245,0

qq

q

qq

qqq

qqq

NBNANBB

NB

NA

yx

y

x

95.12653.17977.89

77.8948.15530tan

77.8930tan48.155

NAA

BA

BA

BAF

BAF

yy

yy

yy

xxx

yyy

48.155130tan

25.245,25.245130tan

25.245

030tan

00

025.2450

10.20

37.077.89

77.89625.122tan

q

q

PROBLEMS Free Body Diagram of rod:

8) The uniform bar OC of length L pivots freely about a horizontal axis through O. If the spring of modulus k is unstretched when C is coincident with A, determine the tension T required to hold the bar in the 45° position shown. The diameter of the small pulley at D is negligible.

PROBLEMS

Spring unstretched when C is coincident with A, determine the tension T required to hold the bar in the 45° position shown.

PROBLEMS

PROBLEMS

A

cable

B

C

4

3

36 cm

32 cm

24 cm

9) A cylinder is supported by a bar and cable as shown. The mass

of the cylinder is 150 kg and the mass of the bar is 20 kg. If all

surfaces are smooth, determine the reaction at support C of the

bar and the tension T in the cable.

PROBLEMS

A

cable

B

C

4

3

36 cm

32 cm

24 cm

mcylinder = 150 kg, mbar = 20 kg, determine reaction at C and tension T

in the cable.

10) Pulley A delivers a steady torque of 100 Nm to a pump through its shaft at C. The tension in the lower side of the belt is 600 N. The driving motor B has a mass of 100 kg and rotates clockwise. As a design consideration, determine the magnitude of the force on the supporting pin at O.

PROBLEMS

Pulley A delivers a steady torque of 100 Nm to a pump through its shaft at C. The tension in the lower side of the belt is 600 N. The

driving motor B has a mass of 100 kg and rotates clockwise. As a design consideration, determine the magnitude of the force on the

supporting pin at O.

PROBLEMS

k=50 N/m 300 mm

100 mm

300 mm

30o

11) The toggle switch

consists of a cocking lever

that is pinned to a fixed

frame at A and held in place

by the spring which has an

unstretched length of 200

mm. The cocking lever rests

against a smooth peg at B.

Determine the magnitude of

the support force at A and

the normal force on the peg

at B when the lever is in the

position shown.

PROBLEMS

k=50 N/m 300 mm

100 mm

300 mm 30o

Ax

Ay

300 mm

Fspx

Fspy Fsp

A

D

B

C

150°

Fsp

NB

30°

mmxDC

ACADACADDC

final 64.676

150cos4003002400300

150cos2

22

222

Cosine theorem

NF

mmxxxkxkF

sp

initialinitialfinalsp

85.232.0677.050

200

19.17

,2956.0sin,150sinsin

DCAC

Sine theorem

NAAAA

NAAFNAF

NAAFNAF

NNNABNADFM

yx

yyspyByy

xxspxBxx

BBBspxA

16.28

21.12,19.17cos85.2330sin15.21030sin,0

37.25,19.17sin85.2330cos15.21030cos,0

15.21)100()300(19.17sin85.230,0

22

PROBLEMS Free Body Diagram of lever: