Transcript Section 3.1 Statics

Ch. 3 Equilibrium of Particles
Looking ahead:
§3.1 Equilibrium of Particles in 2–D.
§3.2 Behavior of Cables, Bars and Springs.
§3.3 Equilibrium of Particles in 3–D.
§3.4 Engineering design.
§3.1 Equilibrium of particles 2–D
Recall: Newton's laws of motion:
1st law: A particle remains at rest, or continues to move in a
straight line with uniform velocity, if there is no unbalanced
force acting on it.
2nd law: The acceleration of a particle is proportional to the
resultant force acting on the particle, and is in the direction of
this force.
r
r
F  ma
3rd law: The forces of action and reaction between interacting
bodies are equal in magnitude, opposite in direction, and
collinear.


Static equilibrium
r r
With a  0, Newton's 2nd law of motion in 2-D becomes:

 

 0 and
F
r
 Fx iˆ   Fy ˆj  0 , or
(1)
F
(2)
x
y
 0.
Remarks:
• Summations are included above to emphasize that all forces applied to a
particle must be included.
• Eq. (1) is a vector equation.
• Eq. (2) is two scalar equations.
• Both Eqs. (1) and (2) are equivalent.
• Analysis of equilibrium using Eq. (1) is often called a vector approach.
• Analysis of equilibrium using Eq. (2) is often called a scalar approach.
Cables and pulleys
• In prior discussions, we have assumed that cables
can support tensile forces only.
• In addition, in most of our work, we will idealize
cables as being inextensible and weightless, and
pulleys as being frictionless (that is, the bearing of a
pulley is frictionless). The consequence of this is
that the tensile force in a continuous cable is the
same everywhere.
Free Body Diagram
A Free Body Diagram (FBD) is a sketch of a particle that
shows all of the forces that are applied to the particle. A FBD is an
essential aid for the application of Newton’s laws! The forces that
are typically applied to a particle have a number of sources,
including:
• Forces from the environment (e.g., weight, wind force, etc.)
• Forces from structural members that are attached to (or contain)
the particle.
• Forces from supports (these are called support reactions, or
simply reactions).
Procedure for drawing FBDs
1) Decide on the particle whose equilibrium you want to analyze.
2) Imagine that this particle is "cut" completely free (separated)
from the structure and/or its environment.
• In 2-D, think of a closed line that completely encircles the point.
• In 3-D, think of a closed surface that completely surrounds the point.
3) Sketch the particle (i.e., draw a point).
4) Sketch the forces:
• Forces from the environment (e.g., weight).
• Forces where a cut passes through a structural member.
• Forces where a cut passes through a support.
5) Select a coordinate system, show dimensions.
Modeling
Modeling refers to the process of idealizing a real life structure
by a mathematical model. The mathematical model includes
idealizations such as use of point forces, assuming a cable to be
weightless, inextensible and perfectly flexible, assumptions on
geometry, and so on. A FBD is one of the results of modeling.
Many real-life problems may be idealized as a particle (or system
of particles) in equilibrium.
• Some structures or objects, even if
very large, may be idealized as a
particle. This is the case when an object
is subjected to a concurrent force
system – all forces intersect at a
common point.
Example: An aircraft flies in a straight
line at constant speed. The lines of
action of all forces may be idealized to
intersect at a common point, thus
giving a concurrent force system.
W = weight.
L = lift.
T = thrust.
D = drag.
• Sometimes, equilibrium of an entire
structure may boil down to equilibrium
of a single point within the structure.
Example: In the cable-bar structure,
where points A, B, and C are pin
connections, the pin at A has a
concurrent force system. Thus,
equilibrium of the entire structure can
be determined by analyzing the
equilibrium of just point A.
Problem solving
(1) Once the FBD is drawn, equations of equilibrium can be
applied:
r r
vector approach
F  0
F
x
 0 and
F
y
0
scalar approach
2In two dimensions, there are two equilibrium equations
available to determine the unknowns in the the FBD.

(3) In some problems, a FBD may have more than two unknowns:
• In such cases, drawing more FBDs and writing more
equilibrium equations may provide a determinate system of
equations, or
• The problem may be statically indeterminate (more on this
later).
FBDs for cables and pulleys
Q: Can equilibrium of a
pulley and cable be
idealized as equilibrium of
a particle?
Click the image to play
A: Yes! This movie shows how the cable forces applied to the
pulley may be "shifted" to the bearing of the pulley, thus giving a
concurrent force system at point A.
Example 1: The structure consists of bar AB and cable
AC. Determine the force supported by the bar and cable.
A: TAB = - 275 N,
TAC = 292 N.
Example 2: The structure
shown is used to lift an
engine with weight W. The
structure consists of bar
AB and cables AC and
ADE.
Determine the
largest weight that may be
lifted if the bar and cables
have the following failure
strengths:
member
AB
AC
ADE
strength
6000 lb tension,
2000 lb compression.
3000 lb.
600 lb.
A: W = 503 lb
Example 3: For each case, determine the cable tension in
terms of W. Assume all cable segments are vertical.
A:
(a) T = W,
(b) T = W/2,
(c) T = W/4.
Support reactions
More on this
in Ch. 5.
Example 4: The driving mechanism for a steam locomotive
is shown. Piston A is acted upon by steam pressure p, which
drives piston rod AB. Plate B (called the cross head) slides without
friction on guide DE, and is connected to the wheel by bar
BC. Idealize plate B as a particle and draw its FBD. Find the
force in bar BC and the reaction between B and guide DE.
A: TBC = -5350 lb,
R = -1830 lb