ch.03 equilibrium of a particle
TRANSCRIPT
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03. Equilibrium of a Particle
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.01 Equilibrium of a Particle
Chapter Objectives
To introduce the concept of the free-body diagram for a particle
To show how to solve particle equilibrium problems using the equations of equilibrium
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.02 Equilibrium of a Particle
1. Condition for Equilibrium of a Particle
- Newtons First Law of Physics: If the resultant force on a particle is zero, the particle will remain at rest or will continue
at constant speed in a straight line
= 0 Equilibrium
- Equilibrium: a key concept in statics is that of equilibrium. If an
object is at rest, we will assume that it is in equilibrium and that
the sum of the forces acting on the object equal zero
= = 0
Equations for 2D Equilibrium: = 0, = 0
Equations for 3D Equilibrium : = 0, = 0, = 0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.03 Equilibrium of a Particle
2. The Free-Body Diagram
- A free-body diagram (FBD): drawing that shows the particle
with all the forces that act on it
- Two types of connections often encountered in particle
equilibrium problems
Springs: can be used to apply forces of tension/compression
Cables and Pulleys
+ Ideal pulleys simply change the direction of a force
+ The tension on each side of an ideal pulley is the same
+ The tension is the same everywhere in a given rope or cable if ideal pulleys are used
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.04 Equilibrium of a Particle
2. The Free-Body Diagram
- A Procedure for Drawing a Free-body Diagram
Imagine the particle to be isolated or cut free from its surroundings
Show all the forces that act on the particle
+ Active forces: they want to move the particle
+ Reactive forces: they tend to resist the motion
Identify each force and show all known magnitudes and directions. Show all unknown magnitudes and/or directions
as variables
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.05 Equilibrium of a Particle
2. The Free-Body Diagram
- Example 3.1 The sphere has a mass of 6 and is supported as shown. Draw a free-body diagram of the sphere,
the cord , and the knot at Solution
- Knot
- Cord
- Sphere
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.06 Equilibrium of a Particle
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3. Coplanar Force Systems
- If a particle is subjected to a system of coplanar forces that lie
in the plane, then each force can be resolved into its and components
= 0
+ = 0
= 0 = 0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.07 Equilibrium of a Particle
3. Coplanar Force Systems
- Procedure for Analysis
+ Free-Body Diagram
Establish the , axes in any suitable orientation
Label all the known/unknown force magnitudes and directions
The sense of an unknown magnitude force can be assumed
+ Equations of Equilibrium
Apply the equations of equilibrium, = 0 and = 0
If more than two unknowns exist and the problem involves a spring, apply = to relate the spring force to the deformation of the spring
Since the magnitude of a force is always (+), then if the solution for a force yields a () result, this indicates its sense is the reverse of that shown on the free-body diagram
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.08 Equilibrium of a Particle
3. Coplanar Force Systems
- Example 3.2 Determine the tension in cables and necessary to support the 60 cylinder
Solution
Free-body diagram
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.09 Equilibrium of a Particle
3. Coplanar Force Systems
Equation of equilibrium
+ = 0: 450
4
5 = 0
+ = 0: 450 +
3
5 60 9.81 = 0
The tension in cables
: = 420
: = 475.66
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.10 Equilibrium of a Particle
3. Coplanar Force Systems
- Example 3.3 The 200 crate is suspended using the ropes and . Each rope can withstand a maximum force of 10 before it breaks. If always remains horizontal, determine the smallest angle to which the crate can be suspended before one of the ropes breaks
Solution
Equation of equilibrium
+ = 0: + = 0
+ = 0: 200 9.81 = 0
The smallest angle
= 11962
10 103= 11.310 11.30
= 10 10311.310 = 9.81 103
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.11 Equilibrium of a Particle
3. Coplanar Force Systems
- Example 3.4 Determine the required length of cord so that the 8 lamp can be suspended in the position shown. The undeformed length of spring is = 0.4
Solution
Equation of equilibrium
300 = 0
300 78.5 = 0
= 157.0
= 135.9
The stretch of spring
= = 0.453
The stretched length = + = 0.4 + 0.453 = 0.853
Distance from to 2 = 300 + 0.853 = 1.32
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.12 Equilibrium of a Particle
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Fundamental Problems
- F3.1: The crate has a weight of 550. Determine the force in each supporting cable
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.13 Equilibrium of a Particle
Fundamental Problems
- F3.2: The beam has a weight of 700. Determine the shortest cable that can be used to lift it if the maximum force the cable can sustain is 1500
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.14 Equilibrium of a Particle
Fundamental Problems
- F3.3: If the 5 block is suspended from the pulley and the sag of the cord is = 0.15, determine the force in cord . Neglect the size of the pulley
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.15 Equilibrium of a Particle
Fundamental Problems
- F3.4: The block has a mass of 5 and rests on the smooth plane. Determine the unstretched length of the spring
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.16 Equilibrium of a Particle
Fundamental Problems
- F3.5: If the mass of cylinder is 40, determine the mass of cylinder in order to hold the assembly in the position shown
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.17 Equilibrium of a Particle
Fundamental Problems
- F3.6: Determine the tension in cables , , and , necessary to support the 10 and 15 traffic lights at and , respectively. Also, find the angle
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.18 Equilibrium of a Particle
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4. Three-Dimensional Force Systems
- The necessary and sufficient condition for particle equilibrium
= 0
+ + = 0
= 0 = 0
= 0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.19 Equilibrium of a Particle
4. Three-Dimensional Force Systems
- Procedure for Analysis
+ Free-Body Diagram
Establish the , , axes in any suitable orientation
Label all the known/unknown force magnitudes and directions
The sense of an unknown magnitude force can be assumed
+ Equations of Equilibrium
Apply the equations of equilibrium, = 0, = 0, = 0
If the 3D geometry appears difficult, express each force on the free-body diagram as a Cartesian vector, substitute these
vectors into = 0, set the , , components equal to zero
Since the magnitude of a force is always (+), then if the solution for a force yields a () result, this indicates its sense is the reverse of that shown on the free-body diagram
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.20 Equilibrium of a Particle
4. Three-Dimensional Force Systems
- Example 3.5 A 90 load is suspended from the hook. If the load is supported by two cables and a
spring having a stiffness = 500/, determine the force in the cables and
the stretch of the spring for equilibrium.
Cable lies in the plane and
cable lies in the plane
Solution
Free-body diagram
Equation of equilibrium
= 0: 300
4
5 = 0
= 0: 300 + = 0
= 0: 3
5 90 = 0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.21 Equilibrium of a Particle
4. Three-Dimensional Force Systems
Equation of equilibrium
= 0: 300
4
5 = 0
= 0: 300 + = 0
= 0: 3
5 90 = 0
Solving the above equations
= 150
= 240
= 207.8
The stretch of the spring
=
==207.8
500= 0.416
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.22 Equilibrium of a Particle
4. Three-Dimensional Force Systems
- Example 3.6 The 10 lamp is suspended from the three equal-length cords. Determine its smallest vertical distance
from the ceiling if the force developed in any cord is not allowed to exceed 50
Solution
Free-body diagram
Equation of equilibrium
= 0: 3 50
109.81= 0
= 198.1
150= 49.160
From the shaded triangle
=600
49.160= 519
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.23 Equilibrium of a Particle
4. Three-Dimensional Force Systems
- Example 3.7 Determine the force in each cable used to
support the 40 crate
Solution
Free-body diagram
Equation of equilibrium
= 3 4 + 8
(3)2+(4)2+82
= 0.318 0.424 + 0.848
= 3 + 4 + 8
(3)2+42 + 82
= 0.318 + 0.424 + 0.848
= , = 40
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.24 Equilibrium of a Particle
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4. Three-Dimensional Force Systems
Equilibrium required
+ + + = 0
0.318 0.318 +
+ 0.424 +0.424
+ 0.848 +0.848 40 = 0
Equating the respective , , components to zero yields
= 0: 0.318 0.318 + = 0
= 0: 0.424 +0.424 = 0
= 0: 0.848 +0.848 40 = 0
Solving the above equations give
= = 23.6
= 15.0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.25 Equilibrium of a Particle
4. Three-Dimensional Force Systems
- Example 3.7 Determine the tension in each cord used to support
the 100 crate
Solution
Free-body diagram
Equation of equilibrium
=
= 1200 + 135
0 + 600
= 0.5 0.707 + 0.5
= + 2 + 2
(1)2+22 + 22
= 0.333 + 0.667 + 0.667
= 981
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.26 Equilibrium of a Particle
4. Three-Dimensional Force Systems
Equilibrium required
+ + + = 0
0.5 0.333
+ 0.707 +0.667
+ 0.5 +0.667 981 = 0
Equating the respective , , components to zero yields
= 0: 0.5 0.333 = 0
= 0: 0.707 + 0.667 = 0
= 0: 0.5 + 0.667 981 = 0
Solving the above equations give
= 694, = 813, = 862
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.27 Equilibrium of a Particle
Fundamental Problems
- F3.7: Determine the magnitude of forces 1, 2 , 3 so that the particle is held in equilibrium
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.28 Equilibrium of a Particle
Fundamental Problems
- F3.8: Determine the tension developed in cables , ,
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.29 Equilibrium of a Particle
Fundamental Problems
- F3.9: Determine the tension developed in cables , ,
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.30 Equilibrium of a Particle
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Fundamental Problems
- F3.10: Determine the tension developed in cables , ,
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.31 Equilibrium of a Particle
Fundamental Problems
- F3.11: The 150 crate is supported by cables , , and . Determine the tension in these wires
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 3.32 Equilibrium of a Particle