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• Ömer faruk Sil

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PROBLEMS ON EQUILIBRIUM OF PARTICLES

PROBLEMS 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

45q 45q

NB = 1 N A 2

45q

45q

NA 45q 45q

45q

mg

45q

45q N = 1 N B A

45

q

NA

q

45

2

q

mg

PROBLEMS

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

y

NB = 1 N A 2

NA

NB = 1 N A 2

45q 45q

x

mg

45q

NA

mg or

F =0 x

N A sin( 45 q ) = 1 N A cos( 45 q ) 2 sin( 45 q ) = tan( 45 q ) = 1 cos( 45 q ) 2 45 q = 26.57   q =18.43

 F = 0 1N A 2 tan( 45 q ) = =1 NA 2 45 q = 26.57  q =18.43

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 =?

Fspring Ff 20°

Fspring = k

N

20°

W = mg = 40(9.81) = 392.4 N

Fspring = 2500  N 0.2  0.18m= 50 N  m  N W cos 20 = 0 N = 392.4cos 20 = 368.74 N Fy = 0 Fspring  Ff W sin 20 = 0 50  Ff  392.4sin 20 = 0 Fx = 0 F f 184.21 F f =184.21 N , F f = mN  m = = = 0.5 N 368.74

PROBLEMS 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.

a 2 = x 2 12 2 = 50  x   24 2 x 2 144 = 2500 100 x  x 2  576 100 x = 2068 , x = 20.68 cm

x

2

sin q = 12 = 0.58, q = 35.45 20.68 24 sin  = = 0.82,  = 54.94 50  20 . 68 

12 cm

y 65 N

a

q

x

 65 N

24 cm

50-x

29.32

F = 0 x

F cos  65cos35.45  65cos54.94 = 0

F

,

F cos = 90.29

= 0 F sin   65sin 35.45  65sin 54.94 = 0 90.29 sin  =15.51 , tan = 0.17 cos  = 9.75 , F = 91.61 N y

,

F = 90.29 cos

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.

dy T dx tangent 2 y = 2 .5 x 60 dy = tanq = 2( 2.5) x dx mg q N tanq = 5(0.4) = 2 q = 63.43 dy q dx tanq =

F = 0 x

T cos 60  N sin 63.43 = 0 , T =1.79 N

F

y

=0

N =19.62 N

T sin 60  N cos 63.43 4(9.81) = 0 

1.79 N

39.24

(opposite direction to that shown)

,

2 N = 39.24

T = 35.12 N

PROBLEMS

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

pulley T1 =T2 T = 35.12 N

B mB g =T = 35.12 N

mB = 35.12 = 3.58 kg 9.81

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 determine

of the

450

N,

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

PROBLEMS A B h

r

6. The cylinder of mass 1 kg having a very small diameter is held against a semi-cylinder r=200 mm with a much larger diameter by two h=120 mm identical springs, which are fixed to points C and C on the ground. The springs are unstretched when at C point A. Knowing that the small cylinder is in equilibrium at point B, what is the spring C constant?

PROBLEMS

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

A r=200 mm

B h

h=120 mm

r

C

C

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 A (0, 5.6, 0) m

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

B (4.2, 0, 0) m

D (0, 0,  3.3) m

C (2.4, 0, 4.2) m

    TAB TAC TAD  F = 0      4.2i  5.6 j TAB = TAB n AB = TAB  7    2.4i  5.6 j  4.2k TAC = TAC n AC = TAC  7.4     5.6 j  3.3k TAD = TAD n AD = TAD 6.5   F = 800 j N

 F

 TAB

 TAD  TAC

PROBLEMS

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

 4.2 T  2.4 T = 0 7 AB 7.4 AC 4.2 T = 2.4 T , TAB = 0.54TAC AB AC 7 7.4

 Fx = 0

 F

 TAB

 Fz = 0

4.2 T  3.3 T = 0 7.4 AC 6.5 AD 4.2 T = 3.3 T , TAD =1.12TAC AC AD 7. 4 6.5

 Fy = 0

 5.6 TAB  5.6 TAC  5.6 TAD  800 = 0 7 7.4 6.5

5.6 T  5.6 T  5.6 T = 800 , 2.15T = 800 AB AD AC 7  7.4 AC 6.5  0.54TAC

1.12TAC

TAC = 371.46 N TAB = 200.59 N TAD = 416.04 N

 TAD  TAC

PROBLEMS 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. Determine the point of z attachment A (x, y, 0) of P cord PA so that the tension in cords PB and PC equals 30 N and 50 N, s respectively. B

0.2 m

0.4 m

C

0.3 m y

x

x A

y

PROBLEMS

TPB = 30 N TPC = 50 N

s = 0.15 m, Fspring = 60 N (↑), A (x, y, 0) = ?

P (0, 0, 0.15) m

B (0,  0.4, 0) m

    Fspring TPB TPC TPA = 0

C (0.3, 0.2, 0) m

  Fspring = 60k N z  Fspring

,

     0.4 j  0.15k TPB = TPB nPB = 30 0.427  = 28.10 j 10.54k

P

      0.3i  0.2 j  0.15k TPC = TPC nPC = 50   0.39  = 38.46i  25.64 j 19.23k   TPA = TPAnPA = TPA

B

   xi  yj  0.15k x 2  y 2  0.152  a

A ( x, y, 0) m

s

 TPB

 TPC

0.2 m

0.4 m

 TPA

y x A

C 0.3 m

x

y

PROBLEMS

 Fx = 0

TPB = 30 N TPC = 50 N

s = 0.15 m, Fspring = 60 N (↑), A (x, y, 0) = ?

 38.46  x TPA = 0 a 

,

TPAx = 38.46 N

TPAx

 Fy = 0

y  28.10  25.64  TPA = 0 a 

TPAy = 2.46 N

,

TPAy

F = 0 z

60 10.54 19.23 0.15 TPA = 0 a   

, TPAz = 30.23 N

 Fspring P

TPAz

    TPA = 38.46i  2.46 j  30.23k N

TPA = 38.46  2.46  30.23 = 48.98 from z TPAz =30.23 = 0.15 TPA a from x TPAx =38.46 = x TPA a y from y TPAy = 2.46 = TPA a A 0.19, 0.012, 0 m 2

2

2

 B TPB

N ,

 TPC C

a = 0.243

,

x = 0.19 m

,

y = 0.012 m

A

 TPA

PROBLEMS 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

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.

PROBLEMS 9. The crate weighing 580 N is held on the incline  by the wire AB and by the horizontal force P which is directed parallel to the z axis. Since the crate is mounted on casters, the force exerted by the incline on the y crate is 1.5 m 3m perpendicular to the incline. B E Determine  P themagnitude 2.2 m C of P and the A tension in 3m wire AB. O 4m z

x

PROBLEMS

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

A 3sin 53.13, 33cos53.13, 1.5 m A 2.4,1.2, 1.5 m , B 0, 5.2, 1.5 m C 0, 3, 0 m , D 4, 0, 0 m , E 0, 3, 1.5 m

 T

 N

53.13°

3m

xA yA

1.5 m

 W

B

      F =W  P  T  N = 0

2.2 m

x

4m

y

 P

A

3m

E

 P C

A

  W = 580  jN  P =  Pk

3m

O 4m

z

x

PROBLEMS

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

      rB / A  2.4i  4 j 1.5k T = TnT = T =T rB / A 4.9

N

53.13° 53.13°

3m

A

  N = Nn N  for nN either        rD / C  rE / C = 4i  3 j  1.5k = 6 j  4.5i      4.5i  6 j nN = = 0.6i  0.8 j 7 . 5      or nN = cos53.13i  sin 53.13 j = 0.6i  0.8 j



 T



1.5 m

3m

E

C

z

 W

 rE / C

2.2 m 3m

 P

 N

y B

   N = N 0.6i  0.8 j 

x

4m

O

 rD / C

 P

A

4m

D

x

PROBLEMS  F = 0

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

 0.6 N  2.4 T i   0.8 N  580  4 T  j    P  1.5 T k = 0       4 . 9 4 . 9 4 . 9        i 0.6 N = 2.4 T , N = 2.4 T 4.9 2.94  j 0.8  N  580  4 T , 1.47T = 580 , T = 395 N 4.9 2.4  k

T 2.94

P = 1.5 T 4.9

,

P =120 N

N = 322 N  y 1.5 m

 T

E

B

 P

C

 W

3m

z

 P

 rE / C

2.2 m

 N

3m

O

 rD / C

A

4m

D

x

PROBLEMS 10. 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

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

PROBLEMS 11. If

WA=WB=1400 N, determine the force P,

TAB and the reactions between the collars and bars.

PROBLEMS

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

PROBLEMS 12. 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 P which must be applied to the collar A to maintain equilibrium. DEFG portion of the rod is parallel to xy-plane.

FBD of collar A

W y

P NAx Four unknowns TAB

NAy x

z

FBD of collar C FBD of collar B

y

y

TBA

W

W z

TCB TBC

x

Four unknowns

NBz NBxy

z

x

NCz Three unknowns

NCx

FBD of collar C y

W TCB z

x

NCz

F

1.5 m 4.5 m

B xB 6m

A (0;9;3) B (2;5;7) C (6;1;0)

1.5 x B = 4.5 6



 F =0



Fy = 0

      4i  4 j  7 k TBC = TBC 9   W = 360 j    E N C = N Cx i  N Cz k

TBC = 810 N



NCx    TBC  W  N C = 0

4 TBC  360 = 0 9

FBD of collar A

W y

P NAx TAB

NAy x

z

     F =0  TAB  W  N A  P = 0      TAB = TBA = 360 i  720 j  720 k   W = 360 j    N A = N Axi  N A y j   P =  Pk



F

z

=0

P = 720 N



720  P = 0

PROBLEMS 13. 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) z and B (0;20;80). Q B

A 20 cm

P 40 cm

40 cm

80 cm

80 cm

30 cm y x

z

z

FBD of collar A

Q

Fspring

B

z

P

A q

NAy NAxz

y x



 F =0

x 

A

NAxz

20 cm

P

q

   Fspring  P  N A = 0

30 cm

40 cm 40 cm

80 cm

80 cm

30 cm y x

         40 i  20 j  40 k Fspring = 120 (60  30 ) AB = (0  40 )i  (20  0) j  (80  40 )k 60         Fspring = 2400 i  1200 j  2400 k AB = 40 i  20 j  40 k  AB = 60 cm   3  4    3  4 P = Pi  Pk N A = N Axz i  N Axz k  N Ay j 5 5 5 5 3 4 Fx = 0  2400  P  N Axz = 0 5 5

 F F

y

=0



1200  N Ay = 0

z

=0



2400 

N Axz = 387 N



4 3 P  N Axz = 0 5 5 P = 3483 .87 N

N Ay = 1200 N

Correct sense