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CHAPTER 13: Equilibrium
Ian McAteer, Sam Bennett
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*Denotes problem that may be on the Exam |
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EQUILIBRIUM
For any body to be in equilibrium the following must be true:
1) 
Therefore:
α=0, L=Constant
For an object to be in rotational equilibrium the vector sum of the
external torques that act on the body, measured about any possible point, must
equal zero.
2) 
Therefore:
a=0, P=Constant
For an object to be in translational equilibrium, the vector sum of all the
external forces that act on the body must also equal zero.
The main type of problem that is going to be used is a static equilibrium
problem.
*Static Equilibrium is achieved when P=0 and L=0, not constants.
The same equations are used for static equilibrium problems.
And 
Because the equations are vector equations, when solving the problem, break
them up into their scalar components:
FORCE TORQUES
 
 
In class we mostly deals with bodies that lie in the xy-plane, so three of
the equations can eliminated, leaving:
 ,
 , 
Now that having three equations, three unknowns can be solved for. But, if a
problem arises in which there are four or more unknowns, this is called and
indeterminate. Because there are more unknowns than equations, the problem can
not be solved without using principles of elasticity. This subject was not
covered in class, so the problem can be discarded.
Center Of Gravity
If g = Constant at every point in an object then:
This is used in static equilibrium problems to condense the forces (due to
gravity) into one point. This simplifies the problem so that each point in an
object does not have to figured into the equations.
Tips/Advice
- There is a Problem Solving Tactics section on pg. 307 in the book. It
gives a detailed "plan of attack" for solving static equilibrium
problems. A copy of this is provided on this page.
- The most important step is to choose the axis of rotation for the system.
Choose an axis that eliminates the largest amount of unknowns. The best way
to eliminate these rotational unknowns is to choose an axis perpendicular to
the line of action (Rule 6).
- Do several problems. The more practice you have with static equilibrium
problems the easier it will be to solve them in the future. We have provided
several of these problems, for you to try, but try solving them before you
look at the answers.
- There is no substitute for good old fashion STUDYING! The more studying
you do the less time it will take you to approach the problems.
- Get good nights sleep, and do not cram before the exam. Have everything
done before the night before the exam.
Problem Solving Tactics from Page 307
Tactic one: Static Equilibrium Problems
Here is a list of steps for solving static equilibrium problems:
- Draw a nice sketch of the problem.
- Select the system to which you will apply the laws of equilibrium, drawing
a closed curve around it on your sketch to fix it clearly in your mind. In
some situations you can select a single object as the system; it is the
object you wish to be in equilibrium (such as the rock climber in sample
problem 13-6). In other situations, you might include additional objects in
the system if their inclusion simplifies the calculations for the
equilibrium. For example, suppose in Sample Problems 13-3 and 13-4 you
select only the ladder as the system. Then in figure 13-8b you will have to
account for additional unknown forces exerted on the ladder by the hands and
feet of the firefighter. These additional unknowns complicate the
equilibrium calculations. The system in figure 13-8 was chosen to include
the firefighter so that those unknown forces are internal to the system and
thus need not be found in order to solve sample problems 13-3 and 13-4.
- Draw a free body diagram of the system. Show all the forces that act on
the system, labeling them clearly and making sure that their points of
application and lines of action are correctly shown.
- Draw in the x and y axes of a coordinate system. Choose them so that at
least one axis is parallel to one or more unknown force. Resolve into
components the forces that do not lie along one of the axes. In all our
sample problems it made sense to choose the x axis horizontal and the y axis
vertical.
- Write the two balance of forces equations, using symbols throughout.
- Choose one or more rotational axes perpendicular to the plane of the
figure and write the balance of torques equation for each axis. If you
choose an axis that passes through the line of action of an unknown force,
the equation will be simplified because that force will not appear in it.
- Solve your equations algebraically for the unknowns. Some students feel
more confident in substituting numbers with units in the independent
equations at this stage, especially if the algebra is particularly involved.
However, experienced problem solvers prefer the algebraic approach, which
reveals the dependence of solutions on the various variables.
- Finally, substitute numbers with units in you algebraic solutions,
obtaining numerical values for the unknowns.
- Look at your answer – does it make sense? Is it obviously too large or
too small? Is the sign correct? Are the units appropriate?
Links for Chapter 13 web study guide
http://www.scientia.org/cadonline/physics/equilibrium/movies/equilibrium.dcr
http://disney.go.com/disneytelevision/billnye/frames/unyev/episode/e32.html
http://www.pas.rochester.edu/~ygao/phy141/lecture17/sldoo1.htm
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