Lecture slide on Equilibrium of a Particle.pdf

Learning objectives
• distinguish reactive and active forces
• state the necessary and sufficient
conditions for equilibrium of a particle
• develop the equations of equilibrium for
a particle

3.1 Equilibrium & Free-body Diagrams
• A body exhibit signs of deformation when it is
subjected to external forces and couple
moments.
1.1 Basic Concepts

3.1 Cont…
• If the resultant force FR and resultant of couple
moments M are both equal to zero, the body or
particle will resist deformation.
• Equilibrium is a condition in which a body or
particle, subjected to a force system remains at
rest if originally at rest, or continue moving at a
constant velocity if originally in motion.

velocity v = 0
  0
r F
  
 
o
M
1. The object must be at rest or continue
moving at constant speed
0;

F
2. The resultant force from all external forces
must give us zero
3. The resultant (net) moment must be zero
1.2 Necessary and sufficient Conditions for Equilibrium
acceleration a = 0
3.1 Cont…
velocity v > 0 = Constant acceleration a = 0
0;

 x
F 0;

 y
F 0

 z
F

• Concept of equilibrium can be applied in
mechanics to determine internal forces and
reactive forces for any given system.
1.3 Application of Equilibrium
In an example of a 2-D or
coplanar force system shown
Fig.1 Coplanar Force System
We can determine the internal
force in cables AB and AD by
applying the concept of
equilibrium once we isolate the
system at the common point A.
3.1 Cont…

• A free-body diagram (FBD) is a sketch or skeleton
of the object without supports that shows various
forces and couple moments acting on the object
Point of isolation
FBD
x
y
30̊
FB
FD
FC
A
1.4 Free Body Diagrams FBD
3.1 Cont…

• Forces acting on a body can be divided into
two categories.
1. Reactive forces (Reactions)
These are forces that are exerted on a body by a support to
which it is attached.
2. Active forces (Applied forces)
These are forces acting on a body that are not provided by
the supports.
1.5 Reactions at Supports & Connections
3.1 Cont…

1.5-1 Common supports and their reactions
• A body subjected to coplanar force system yields
various types of reactions at point of supports
and within connections.
3.1 Cont…
1. Spring
0

 y
F
0
 
s mg
F

s ks
F

k stiffness

s change in length
0
 
s l l
 If a spring is subjected to a force F, there will be a
reaction in the spring equal in magnitude but opposite
in direction
 This reaction is called the spring force Fs
0
 
s mg

FBD

3.1 Cont…
2. Cables, cords and short member
• Cables and cords can support only a tension or “pulling” force,
while short members support both tension and compressive force
• The force along the cable, cord and short member always act in
the direction of the cable, cord or short members
FBD FBD

3.1 Cont…
3. Smooth or Rough Surface
• If an object rests on a smooth surface, then the
surface will exert a force on the object that is normal
to the surface at the point of contact.
FBD

Other types of supports and connections
3.1 Cont…

• If a particle is subjected to a coplanar force system,
say x–y plane, each force can be resolved into its
components
1.5 Equilibrium in Two Dimensions “2-D”
0

F
0
x y
i j
 
 
F F
0
x 
F scalar form
and 0
y 
F
3.1 Cont…
To satisfy the above condition, we require;

Determine the internal force in cables AB and AD for
the given coplanar force system
3.1 Cont…
Example 1

0
x 
F 0

M
0
y 
F
3.1 Cont…
Solution
1. Create FBD
2. Establish the x, y axes
3. Apply equations of equilibrium
cos30 sin30 0
 
B D
F F
0
x 
F 0
y 
F
sin30 0
 
B C
F F  sin30 392.4 0
 
B
F
392.4
784.8
sin30
 
B
F N
cos30 784.8 cos30
sin30 sin30

 
B
D
F
F
1359.3

D
F N

• If a particle is subjected to a space force system, each
force can be resolved into x, y and z components
1.5 Equilibrium in Three Dimensions “3-D”
0

F
0
  
  
x y z
i j k
F F F
0
x 
F and
0
y 
F
3.1 Cont…
To satisfy the above condition, we require;
0

 z
F

• A 200 kg cylinder is hung by means of two cables AB
and AC , which are attached to the top of a vertical
wall. A horizontal force P perpendicular to the wall
holds the cylinder in the position shown. Determine
the magnitude of P and the tension in each cable.
3.1 Cont…
Example 2
y
x

3.1 Cont…
Solution
1. Create FBD
2. Establish the x, y, z axes
3. Resolve each force into components

Force P
 
ˆ ˆ
200 9.81 1962
      
w mgj N j
weight W
Force TAB
ˆ
ˆ ˆ
1.2 10 8
   
AB
r i j k
 
1.2,2,0
A  
0,12,8
B
ˆ
ˆ ˆ
1.2 10 8 ˆ
ˆ ˆ
0.09 0.78 0.62
12.862
  
     
AB
AB
AB
r i j k
u i j k
r
ˆ

P Pi

ˆ
ˆ ˆ
0.09 0.78 0.62
    
AB AB AB AB AB AB
T T u T i T j T k

3.1 Cont…
Solution
3. Resolve each force into components
Force TAC
ˆ
ˆ ˆ
1.2 10 10
   
AC
r i j k
 
1.2,2,0
A  
0,12, 10

B
ˆ
ˆ ˆ
1.2 10 10 ˆ
ˆ ˆ
0.08 0.70 0.70
14.193
  
     
AC
AC
AC
r i j k
u i j k
r
ˆ
ˆ ˆ
0.08 0.70 0.70
    
AC AC AC AC AC AC
T T u T i T j T k

3.1 Cont…
Solution
4. Apply equations of equilibrium
ˆ ˆ ˆ
0.09 0.08 0
   
AB AC
T i T i Pi
0
x 
F 0
y 
F

0

 z
F
0
x 
F
0.09 0.08 0
   
AB AC
T T P

0

 y
F  ˆ ˆ ˆ
0.78 0.70 1962 0
  
AB AC
T j T j j
0.78 0.70 1962 0
  
AB AC
T T



0

 z
F
ˆ ˆ
0.62 0.70 0
 
AB AC
T k T k
0.62 0.70 0
 
AB AC
T T
Hence 235

P N 1402

AB
T N 1238

AC
T N

Q1
Two cables are tied together at C and are loaded as shown.
Determine the tension in cable AC and cable BC.
Problems

Q2
Determine the required length of cord AC in the figure below so
that the 8 kg lamp can be suspended in the position shown. The
undeformed length of the spring AB is LAB = 0.4 m, and the spring
has a spring constant “stiffness” of kAB = 300 N/m.
Problems

Q3
Determine the stretch in each spring for equilibrium of the 2kg
block as shown below. The springs are shown in the equilibrium
position.
Problems

Q4
Cable ABC has a length of 5 m. Determine the position x and the
tension developed in ABC required for equilibrium of the 100-kg
sack. Neglect the size of the pulley at B.
Problems

Q5
Three cables are used to tether a balloon as shown. Determine the
vertical force P exerted by the balloon at A knowing that the
tension in cable AB is 259 N..
Problems

Q6
Determine the magnitude of forces F1, F2, F3, so that the
particle is held in equilibrium
Problems

Q7
The three cables are used to support the 40kg flowerpot.
Determine the force developed in each cable for the system to
remain in equilibrium.
Problems

1. Meriam J. L & Kraige L. G (2002) Engineering
Mechanics: Statics, John Wiley & Sons, Inc: New York
Acknowledgements
2. Malhotra M.M &Subramanian R (1994) Textbook of
Applied Mechanics, New Age International: New Delhi
3. Beer F.P, Johnson E.R, Elsenberg E.R & Mazuke D.F
(2010) Vector Mechanics for Engineers, 9th edition,
McGraw hill education: New York

Lecture slide on Equilibrium of a Particle.pdf

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