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Chapter 2 : Equilibrium of a particle
 2.1 What is a particle ?
 2.2 The general problem
 2.3 A simple 2force experiment
 2.4 Parallelogram law
 2.5 Resultant of two forces
 2.6 Resultant of several forces
 2.7 Summary and statement of equilibrium
 2.8 Multiple Choice Test and Problems
A particle is body whose rotational aspects are not of interest at the
moment. This is the case of course when the body itself is extremely small
so that we think of it as a single point at which all its mass is concentrated.
Often we take a large body and shift all its mass to its center
of mass and just look at the motion (or the lack thereof) of the center of mass.
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Here is the general idea as to what we are interested in for the moment.
Figure 2.2a Multiple Forces on a Particle

Assume you have
given a small body A
with three forces acting on it as shown in Figure 2.2a. Two
of the forces, let's say F_{a} and F_{b},
are given,
that means their values (in Newton for example) and their directions
are known. How can we determine how big the third force, F_{c},
must be in order for the particle A to remain at rest ?
This question is very representative of the entire subject of statics.
Some forces ( we refer to them often as the loads ) are known while
other forces are unknown but of interest to us.
Below we will take the position of an experimentalist and discover the
governing laws ourselves.
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Figure 2.3a 2force Experiment

Well, having 2 forces given and looking for the third might be more than we
can chew on for the time being. So, let's step back a little.
In Figure 2.3a a body A is acted upon by two forces. Assume
that the force R is given.
What must the magnitude and direction of force F_{c}
be, in order for the body A to remain at rest ?
The experimentalist will tell us immediately :
The forces R and F_{c} must
be exactly equal in value and exactly opposite in direction.

Going back to the business of
action/reaction : can you now think of an experiment
which clearly demonstrates that the force you are exerting on the
chair you are sitting on is equal and opposite to that the chair is exerting
on you ?
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Figure 2.4a 3force Experiment

Turning now back to the original 3force problem from section 4.2.
Again, we assume
that the forces F_{a} and F_{b}
are given. Question : What is the magnitude and direction of the force
F_{c} such that the particle A remains at rest?
After some experimenting one would find out that
body A remains at rest whenever the forces
F_{a}
and
F_{b}
form the sides of a parallelogram the diagonal of which, R
in Figure 2.4b , is equal
Figure 2.4b 3force Experiment

and opposite to
F_{c} of Figure 2.4a.
We call this diagonal force the resultant R
(in this case of F_{a} and F_{b} )
It has the same effect on the body A
as the original forces F_{a} and F_{b},
namely to counterbalance F_{c} in the
same way R
did in the 2force experiment of section 2.3.
I like to point out that this parallelogram relationship can be shown
experimentally to hold true to any desired accuracy, the only caveat being
the amount of money available.
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Finding the resultant of a given set of forces is one
of the many tools we will use over and over again.
Figure 2.5a Resultant of 2 forces

Here are two equations relating the various quantities in Figure 2.5a
to each other :
Note where in the figure the angles β_{a} and
β_{b} appear and how the subscripts change from
Eq. (2.5b) to (2.5c).
Above equations are directly related to the Law of Sines and Cosines .
How about you try to derive them yourself ? We need lots of trigonometry
in the weeks to come and this would be an exercise as good as any.
Speaking of trigonometry. Can you derive an equation for the
sin(β_{a}) and sin(β_{b}) and prove that
(β_{a} + β_{b} = α ?
You definitely should look at
Problem 2.5a.
An important special case occurs when
F_{a} and F_{b} are equal in magnitude and the
angle α between them approaches 180°.
Maybe you can visualize the outcome from Figure 2.5a or remember Newton's
third law ? And what to you get from from Equation 2.5a ?
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Figure 2.6a 3force on a particle

As an example let's look at Figure 2.6a where three forces act on a
small body A.
How can we find their resultant, that is that force which has the
same action on body A as the three given forces F_{a},
F_{b}, and F_{c} ?
In principle this is not too difficult :
We could first replace for example F_{a} and F_{b}
by a resultant, say R_{ab}, using
equation 2.5a to find its value and 2.5b
to find its orientation.
Now we have only two forces left, R_{ab} and F_{c}.
We can find their resultant by using Equations 2.5a and 2.5b again.
Obviously, this concept can be expanded to as many forces
as we like.


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There are several items we have worked out in this chapter :
 The resultant of two forces is that force which has the same
effect on a body as the original two forces. Its value and direction
can be determined using Equations 2.5a and 2.5b, respectively.
 A set of forces acting on a particle can be reduced to a
single resultant by replacing pairs of forces by their resultant.
 The resultant of two equal and oppositely directed forces is
zero.
 A particle remains at rest under influence of two forces if these
two forces are equal in value and opposite in direction.
Here is an alternative statement concerning the equilibrium of
a particle :
A particle remains at rest if the resultant of ALL forces acting
on it is equal to zero.


In essence this is a special case of Newton's 2^{nd}
law :
F = m a
where m is the mass of the particle, F is
the resultant of ALL forces acting on the particle, and a
is the corresponding acceleration (change of velocity) of the
particle.
In statics we want a=0 which requires that F=0.
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Multiple Choice Test
This test allows you to ascertain your
knowledge of the definition of terms and your understanding of
important results.
Click here to do the test.
Problems
Click on any of the problem titles below, the problem statement itself
will contain a link to the solution and/or additional help.
Prob. 2.8a : Resultant of 2 forces
Prob. 2.8b : Resultant of 3 forces
Prob. 2.8c : Equilibrium of particle
Prob. 2.8d : Resultant of 3 forces
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Zig Herzog; hgnherzog@yahoo.com
Last revised: 09/14/13