Technical University of Sofia
Branch Plovdiv
Theoretical Mechanics
STATICS
KINEMATICS
* Navigation:
Right (Down) arrow – next slide
Left (Up) arrow – previous slide
Esc – Exit
Notes and Recommendations: ruschev@tu-plovdiv.bg
Lecture 3
Equilibrium of a Concurrent System of Forces
Equilibrium refers to a condition in which an object (particle or body) is at rest if originally at rest, or has a
constant velocity if originally in motion. Most often however, the term “equilibrium” or more specifically “static
equilibrium” is used to describe an object at rest.
Vector condition for equilibrium
The necessary and sufficient vector condition for equilibrium of a concurrent force system is the resultant
force of all forces acting on the particle to be zero.
N
R   Fk  0
k 1
Analytical conditions for equilibrium
Vector condition for equilibrium (after projecting it on three coordinate axes) may be written in scalar form as:
N
F
k 1
x
0
N
F
k 1
y
0
N
F
k 1
z
0
Therefore, for a given concurrent force system to be in equilibrium it is necessary and sufficient the algebraic
sums of components of all forces acting on the particle to be equal to zero.
These scalar equations can be used for obtaining three scalar components of one unknown force
equilibrating the system of concurrent forces.
Lecture 3
Free-Body Diagrams
Space Diagram:
Free-Body Diagram:
A sketch showing the physical
conditions of the problem.
A sketch showing only the forces on the
selected particle.
Lecture 3
Sample Problem 1
SOLUTION:
• Construct a free-body diagram for the particle
at the junction of the rope and cable.
• Apply the conditions for equilibrium by
creating a closed polygon from the forces
applied to the particle.
• Apply trigonometric relations to determine the
unknown force magnitudes.
In a ship-unloading operation, a 3500-N
automobile is supported by a cable. A
rope is tied to the cable and pulled to
center the automobile over its intended
position. What is the tension in the
rope?
Lecture 3
Sample Problem 1 Contd.
SOLUTION:
• Construct a free-body diagram for the particle
at A.
• Apply the conditions for equilibrium.
• Solve for the unknown force magnitudes.
T
TAB
3500 N
 AC 
sin 120 sin 2 sin 58
TAB  3570 N
TAC  144 N
Lecture 3
Sample Problem 2
As part of the design of a new sailboat, it is desired to determine the drag force which
may be expected at a given speed. To do so, a model of the proposed hull is placed in a test
channel and three cables are used to keep its bow on the centerline of the channel.
Dynamometer readings indicate that for a given speed, the tension is 40 lb in cable AB and
60 lb in cable AE . Determine the drag force exerted on the hull and the tension in cable AC.
1 ft = 30.48 cm
TAB  40 lb
TAE  60 lb
Lecture 3
Sample Problem 2
Determination of the Angles
tan  
7 ft
 1.75
4 ft
  60.260
tan  
1.5 ft
 0.375
4 ft
  20.560
Lecture 3
Sample Problem 2
Free-Body Diagram
Choosing the hull as a free body, we
draw the freebody diagram shown. It
includes the forces exerted by the three
cables on the hull, as well as the drag
force FD exerted by the flow.
Equilibrium Condition
We express that the hull is in
equilibrium by writing that the resultant of
all forces is zero:
TAB  TAC  TAE  FD  0
We resolve the forces into x and y components:
F
ix
0
TAB sin 60.260  TAC sin 20.560  FD  0
 Fiy  0
TAB cos60.26  TAC cos 20.56  TAE  0
0
0
TAC  42.9 lb
FD  19.66 lb
Lecture 3
Sample Problem 3
A sailor is being rescued using a boatswain’s chair
that is suspended from a pulley that can roll freely on
the support cable ACB and is pulled at a constant speed
by cable CD. Knowing that a 30° and b 10° and
that the combined weight of the boatswain’s chair and
the sailor is 900 N, determine the tension
(a) in the support cable ACB,
(b) in the traction cable CD.