PROBLEMS ON EQUILIBRIUM OF RIGID BODIES IN TWO DIMENSIONS
PROBLEMS 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. (3/8)
PROBLEMS Free Body Diagram of crate + tailgate:
M
+
0 264.87(0.24) 529.74(0.35) 2T sin 44.07(0.31) 0 63.57 185.41 0.43T 0 248.98 0.43T T 579 N
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.
O
F 0 x
2T cos 44.07 Ox 0 Ox 832 N Fy 0 529.74 264.87 2T sin 44.07 Oy 0 Oy 10.79 N O Ox Oy 832.07 N 2
2
54(9.81)=529.74 N
2T Ox Oy 27(9.81)=264.87 N
tan 300 0.968 , 44.07 310
PROBLEMS
2) A pipe P is being bent by the pipe bender as shown. If the hydraulic cylinder applies a force of magnitude F = 24 kN to the pipe at C, determine the magnitude of the roller reactions at A and B. (3/23)
PROBLEMS
F = 24 kN, determine the magnitude of the roller reactions at A and B.
Free Body Diagram of pipe:
F 0 x
Ax Bx
,
Ax Bx tan 15 Ay B y 0.268
F
0 24 2 Ay 0 Ay 12 kN Ax 3.216 kN
A
,
A B
0.268 Ay Ax
Ay
By
15°
15°
B
y
Ax
N A N B 12.42 kN
Bx
2T
F = 24 kN
PROBLEMS 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 vertically elongated hole in the handle at B, so that negligible vertical force is transmitted there.
(3/28)
PROBLEMS
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. Free Body Diagram of OAB:
M
+
0 , 35(18) F cos10(38) F sin 10(44) 0 630 37.42 F 7.64 F 45.06 F O
B = 35 N
F 13.98 N
F 0
Ox
x
F cos10 35 Ox 0 Ox 48.77 N
Oy
F
0 F sin 10 O y 0 O y 2.39 N O 48.83 N y
15°
PROBLEMS 4) The device shown is designed to apply pressure when bonding laminate to each side of a countertop near an edge. If a 120 N force is applied to the
handle, determine the force which each roller exerts on its corresponding surface. (3/35)
PROBLEMS Free Body Diagram of ACB:
M
F = 120 N, determine the force which each roller exerts on its corresponding surface.
+
0 120(148.49 165)sin 45 C (105) 0 B
C 253.34 N
F
0 120 253.34 B 0 y
B 133.34 N
B
C 105 mm
30+45+30=105 mm
PROBLEMS
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 rockerarm shaft O. (3/49)
PROBLEMS
Fspring = 900 N. Determine B and O.
Free Body Diagram of AOB:
Fspring 900 N ,
M
+ O
Fspringx 900sin 10 156.28 N ,
0
Fspringy 900cos10 886.33 N
B(30) 886.33(48) 156.28(7) 0
B 1381.66 N
F
0 1381.66 886.33 O y 0 O y 2268 N Fx 0 156.28 Ox 0 Ox 156.28 N y
Fspring x
O 2273.38 N Fspring y
Fspring
Oy Ox
B
PROBLEMS
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. (3/54)
PROBLEMS
2m
0.5 m
mbeam = 100-kg, mboy = 50-kg, T = 150 N, O = ?
Wbeam 100 (9.81) 981 N Wbeam
Wboy 50 (9.81) 490.5 N
Ox
Free Body Diagram of beam + boy:
Fx 0
,
Ox 0
150 N
150 N
Wboy
NA
M O 0 , N A (0.75) 981(2) 490.5(2.5) 150(3.25) 150(4) 0 N A 5701 N Fy 0 , O y 5701 981 490.5 150 150 0 O y 3929.5 N 3.93 kN O O y 3.93 kN
Oy
PROBLEMS
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
Free Body Diagram of rod:
25(9.81) = 245.25 N
45° 45°
Ax 30°
A
Bx
B By
By Bx B y tan 45 Bx A tan 30 x , Ay Ax Ay tan 30
30°
Ay
M A 0 , 245.25(0.5 cos q ) Bx 1sin q B y 1cos q 0 122.625 cos q Bx sin q Bx cos q 0
122.625 89.77 0.37 89.77 q 20.10 tan q
cos q (122.625 Bx ) Bx sin q 122.625 Bx sin q tan q cos q Bx
Fy 0
A y 245.25 B y 0
Fx 0
A x Bx 0
B y tan 30 155.48 89.77 N
tan 30 A y B y 0
Bx B y 89.77 N
A y B y 245.25
tan 30 1Ay 245.25
Ax 155.48 tan 30 89.77 N
, Ay
245.25 155.48 N tan 30 1
A 179.53 N
B 126.95 N
PROBLEMS 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 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. A cable B 36 cm 3 4
C 32 cm
24 cm
PROBLEMS
mcylinder = 150 kg, mbar = 20 kg, determine reaction at C and tension T in the cable.
A cable B 36 cm
3 4
C 32 cm
24 cm
PROBLEMS
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 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.
300 mm 30o 100 mm
300 mm
k=50 N/m
PROBLEMS
Free Body Diagram of lever:
Cosine theorem 2
2
2
C
DC AD AC 2 AD AC cos150 300 2 400 2 2300400 cos150 300 mm 30o 100 mm
DC x final 676.64 mm Sine theorem
300 mm
Fsp
Ax
AC DC , sin 0.2956 , sin sin 150 17.19
k=50 N/m
Fsp kx k x final xinitial
xinitial 200 mm
Fsp 500.677 0.2 23.85 N
M A 0 , Fspx AD N B AB 0
30°
B
150°
NB
A Ay
300 mm
Fspx Fsp
D
Fspy
23.85 sin 17.19(300) N B (100) N B 21.15 N
Fx 0 , Ax N B cos 30 Fspx 0 Ax 21.15 cos 30 23.85 sin 17.19 ,
Ax 25.37 N
Fy 0 , Ay N B sin 30 Fspy 0 Ay 21.15 sin 30 23.85 cos17.19 , Ay 12.21 N A Ax2 Ay2
A 28.16 N
