32 Chapter 2 Force Systems

PROBLEMS 2/4 The line of action of the 3000-lb force runs through
the points A and B as shown in the figure. Determine
Introductory Problems the x and y scalar components of F.

2/1 The force F has a magnitude of 600 N. Express F as a y, ft

vector in terms of the unit vectors i and j. Identify
the x and y scalar components of F. B (8, 6)

y
F = 3000 lb
x, ft
O
A (–7, –2)

x
O 40°

Problem 2/4
F = 600 N
2/5 The 1800-N force F is applied to the end of the
I-beam. Express F as a vector using the unit vectors i
Problem 2/1 and j.

2/2 The magnitude of the force F is 400 lb. Express F as y

F = 1800 N
a vector in terms of the unit vectors i and j. Identify
both the scalar and vector components of F.
4
y, ft A
3
O
F = 400 lb
x

x, ft z
30°
Problem 2/5
A (3, –1)

2/6 The control rod AP exerts a force F on the sector as

shown. Determine both the x-y and the n-t compo-
nents of the force.
Problem 2/2
F
2/3 The slope of the 6.5-kN force F is specified as shown
in the figure. Express F as a vector in terms of the y
β
unit vectors i and j. A

y
n
F = 6.5 kN
P
5 t
12 α
x
x
O r

Problem 2/3 Problem 2/6

 Article 2/3 Problems 33

2/7 The two structural members, one of which is in ten- Representative Problems
sion and the other in compression, exert the indi-
cated forces on joint O. Determine the magnitude of 2/10 Determine the n- and t-components of the force F
the resultant R of the two forces and the angle  which is exerted by the rod AB on the crank OA.
which R makes with the positive x-axis. Evaluate your general expression for F  100 N and
(a)   30⬚,   10⬚ and (b)   15⬚,   25⬚.
3 kN
F

2 kN B β
t
y n
4
3
30° A
x
O

θ
Problem 2/7
O x
2/8 The t-component of the force F is known to be 75 N.
Determine the n-component and the magnitude of F.

n Problem 2/10
t

F 2/11 The two forces shown act at point A of the bent bar.
10° Determine the resultant R of the two forces.
30°
F1 = 3 kips

30°
Problem 2/8 A

2/9 Two forces are applied to the construction bracket as

shown. Determine the angle  which makes the resul- F2 = 7 kips
45°
tant of the two forces vertical. Determine the magni-
tude R of the resultant.

y
F1 = 800 lb y

F2 = 425 lb x

70°
θ
Problem 2/11

x
z

Problem 2/9
34 Chapter 2 Force Systems

2/12 A small probe P is gently forced against the circular T

surface with a vertical force F as shown. Determine
the n- and t-components of this force as functions of
the horizontal position s. 60°
y
F

x
n
P

T
t

s
Problem 2/14

r 2/15 To satisfy design limitations it is necessary to deter-

mine the effect of the 2-kN tension in the cable on
the shear, tension, and bending of the fixed I-beam.
For this purpose replace this force by its equivalent
O of two forces at A, Ft parallel and Fn perpendicular
to the beam. Determine Ft and Fn.
Problem 2/12
A
2/13 The guy cables AB and AC are attached to the top of 20°
the transmission tower. The tension in cable AB is
8 kN. Determine the required tension T in cable AC 2 kN
such that the net effect of the two cable tensions is a 30°
downward force at point A. Determine the magni-
tude R of this downward force.

Problem 2/15
A
2/16 Determine the x- and y-components of the tension T
which is applied to point A of the bar OA. Neglect
30 m
the effects of the small pulley at B. Assume that r
C
and  are known.

20 m
B θ O

40 m 50 m
r
Problem 2/13 n A

2/14 If the equal tensions T in the pulley cable are 400 N, B

express in vector notation the force R exerted on y
the pulley by the two tensions. Determine the mag- t
nitude of R.
x
T
r

Problem 2/16
 Article 2/3 Problems 35

2/17 Refer to the mechanism of the previous problem.

Develop general expressions for the n- and t-compo-
nents of the tension T applied to point A. Then eval-
a
uate your expressions for T  100 N and   35⬚.

2/18 The ratio of the lift force L to the drag force D for O 30°
R = 800 N
the simple airfoil is L/D  10. If the lift force on a
short section of the airfoil is 50 lb, compute the
magnitude of the resultant force R and the angle 
which it makes with the horizontal.
110°

L
b

Problem 2/20

C D 2/21 Determine the components of the 800-lb force F

along the oblique axes a and b. Also, determine the
projections of F onto the a- and b-axes.
Air flow

Problem 2/18 F = 800 lb

2/19 Determine the resultant R of the two forces applied b

to the bracket. Write R in terms of unit vectors
along the x- and y-axes shown. 60°
a
150 N 45°
y' y

x' Problem 2/21

30°
200 N
2/22 Determine the components Fa and Fb of the 4-kN
force along the oblique axes a and b. Determine the
35°
projections Pa and Pb of F onto the a- and b-axes.
x
y
20°
b

Problem 2/19 F = 4 kN

2/20 Determine the scalar components Ra and Rb of the 30°

force R along the nonrectangular axes a and b. Also 40°
determine the orthogonal projection Pa of R onto a
axis a.
15°
x
O

Problem 2/22
36 Chapter 2 Force Systems

2/23 Determine the resultant R of the two forces shown 2/26 The cable AB prevents bar OA from rotating clock-
by (a) applying the parallelogram rule for vector wise about the pivot O. If the cable tension is 750 N,
addition and (b) summing scalar components. determine the n- and t-components of this force act-
ing on point A of the bar.

t
y
n

m
60°

1.5
600 N
B O 60°

400 N
1.2 m
Problem 2/23
Problem 2/26

2/24 It is desired to remove the spike from the timber by

2/27 At what angle  must the 400-lb force be applied in
applying force along its horizontal axis. An obstruc-
order that the resultant R of the two forces have a
tion A prevents direct access, so that two forces, one
magnitude of 1000 lb? For this condition what will
400 lb and the other P, are applied by cables as
be the angle  between R and the horizontal?
shown. Compute the magnitude of P necessary to
ensure a resultant T directed along the spike. Also
find T.

P 400 lb
8″

4″
A
6″

θ
700 lb

O
400 lb

Problem 2/24

2/25 At what angle  must the 800-lb force be applied in

order that the resultant R of the two forces have a
Problem 2/27
magnitude of 2000 lb? For this condition, determine
the angle  between R and the vertical.

1400 lb

800 lb
θ

Problem 2/25
 Article 2/3 Problems 37

2/28 In the design of the robot to insert the small cylin- 䉴2/29 The unstretched length of the spring is r. When pin
drical part into a close-fitting circular hole, the P is in an arbitrary position , determine the x- and
robot arm must exert a 90-N force P on the part y-components of the force which the spring exerts
parallel to the axis of the hole as shown. Determine on the pin. Evaluate your general expressions for
the components of the force which the part exerts r  400 mm, k  1.4 kN/m, and   40⬚. (Note: The
on the robot along axes (a) parallel and perpendicu- force in a spring is given by F  k, where  is the
lar to the arm AB, and (b) parallel and perpendicu- extension from the unstretched length.)
lar to the arm BC.

15° A
B y

x k
D
45° n
P = 90 N
C 2r
A 60°
P
r t
θ

Problem 2/28
Problem 2/29

䉴2/30 Refer to the figure and statement of Prob. 2/29.

When pin P is in the position   20⬚, determine the
n- and t-components of the force F which the spring
of modulus k  1.4 kN/m exerts on the pin. The dis-
tance r  400 mm.
Problem Answers

(When a problem asks for both a general and a specific result, only the specific result might be listed below.)

Chapter 1
1/1 ␪x ⫽ 36.9⬚, ␪y ⫽ 126.9⬚, n ⫽ 0.8i ⫺ 0.6j 1/9 F ⫽ 1.984(1020) N, F ⫽ 4.46(1019) lb
1/2 V ⫽ 16.51 units, ␪x ⫽ 83.0⬚ 1/10 F ⫽ 1.358(10⫺9) N
1/3 V⬘ ⫽ 14.67 units, ␪x ⫽ 162.6⬚ 1/11 ␪ ⫽ 5⬚: ns ⫽ 0.1270%, nt ⫽ ⫺0.254%
1/4 ␪x ⫽ 56.1⬚, ␪y ⫽ 138.0⬚, ␪z ⫽ 68.2⬚ ␪ ⫽ 10⬚: ns ⫽ 0.510%, nt ⫽ ⫺1.017%
1/5 m ⫽ 93.2 slugs, m ⫽ 1361 kg ␪ ⫽ 20⬚: ns ⫽ 2.06%, nt ⫽ ⫺4.09%
1/6 W ⫽ 819 N, W ⫽ 184.1 lb
1/7 W ⫽ 578 N, m ⫽ 4.04 slugs, m ⫽ 58.9 kg
1/8 A ⫹ B ⫽ 8.40, A ⫺ B ⫽ 4.94
AB ⫽ 11.51, A/B ⫽ 3.86

Chapter 2
2/1 Fx ⫽ 460 N, Fy ⫽ ⫺386 N, F ⫽ 460i ⫺ 386j N 2/8 Fn ⫽ ⫺62.9 N, F ⫽ 97.9 N
2/2 F ⫽ ⫺346i ⫹ 200j lb 2/9 ␪ ⫽ 49.9⬚, R ⫽ 1077 lb
Fx ⫽ ⫺346 lb, Fy ⫽ 200 lb 2/10 (a) Fn ⫽ 34.2 N, Ft ⫽ 94.0 N
Fx ⫽ ⫺346i lb, Fy ⫽ 200j lb (b) Fn ⫽ ⫺17.36 N, Ft ⫽ 98.5 N
2/3 F ⫽ ⫺6i ⫺ 2.5j kN 2/11 R ⫽ 2.35i ⫺ 3.45j kips
2/4 Fx ⫽ 2650 lb, Fy ⫽ 1412 lb Fs F冪r2 ⫺ s2
2/12 Ft ⫽ r , Fn ⫽ ⫺ r
2/5 F ⫽ ⫺1080i ⫺ 1440j N
2/6 Fx ⫽ ⫺F sin ␤, Fy ⫽ ⫺F cos ␤ 2/13 T ⫽ 5.83 kN, R ⫽ 9.25 kN
Fn ⫽ F sin (␣⫹␤), Ft ⫽ F cos (␣⫹␤) 2/14 R ⫽ 600i ⫹ 346j N, R ⫽ 693 N
2/7 R ⫽ 3.80 kN, ␪ ⫽ 338⬚ (or ⫺21.6⬚) 2/15 Ft ⫽ 1.286 kN, Fn ⫽ 1.532 kN

507