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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 2-force 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.
Here is the general idea as to what we are interested in for the moment.
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 Fa and Fb,
that means their values (in Newton for example) and their directions
are known. How can we determine how big the third force, Fc,
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.
Multiple Forces on a
Below we will take the position of an experimentalist and discover the
governing laws ourselves.
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 Fc
be, in order for the body A to remain at rest ?
The experimentalist will tell us immediately :
The forces R and Fc 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 ?
Turning now back to the original 3-force problem from section 4.2.
Again, we assume
that the forces Fa and Fb
are given. Question : What is the magnitude and direction of the force
Fc such that the particle A remains at rest?
After some experimenting one would find out that
body A remains at rest whenever the forces
form the sides of a parallelogram the diagonal of which, R
in Figure 2.4b , is equal
and opposite to
Fc of Figure 2.4a.
We call this diagonal force the resultant R
(in this case of Fa and Fb )
It has the same effect on the body A
as the original forces Fa and Fb,
namely to counter-balance Fc in the
same way R
did in the 2-force 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.
Finding the resultant of a given set of forces is one
of the many tools we will use over and over again.
Here are two equations relating the various quantities in Figure 2.5a
to each other :
Resultant of 2 forces
Note where in the figure the angles βa and
βb 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(βa) and sin(βb) and prove that
(βa + βb = α ?
You definitely should look at
An important special case occurs when
Fa and Fb 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 ?
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 Fa,
Fb, and Fc ?
3-force on a particle
In principle this is not too difficult :
We could first replace for example Fa and Fb
by a resultant, say Rab, using
equation 2.5a to find its value and 2.5b
to find its orientation.|
Now we have only two forces left, Rab and Fc.
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.
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
- 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 2nd
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
In statics we want a=0 which requires that F=0.
Multiple Choice Test
This test allows you to ascertain your
knowledge of the definition of terms and your understanding of
Click here to do the test.
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; email@example.com
Last revised: 09/14/13