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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

2.1 What is a particle ?
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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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2.2 The general problem
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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 Fa and Fb, are given, 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.
Below we will take the position of an experimentalist and discover the governing laws ourselves.
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2.3 A simple 2-force experiment
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Figure 2.3a
2-force 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 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 ?

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2.4 Parallelogram Law
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Figure 2.4a
3-force Experiment
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 Fa and Fb form the sides of a parallelogram the diagonal of which, R in Figure 2.4b , is equal

Figure 2.4b
3-force Experiment
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.
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2.5 Resultant of two forces
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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 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 Problem 2.5a.
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 ?

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2.6 Resultant of several forces forces
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Figure 2.6a
3-force 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 Fa, Fb, and Fc ?
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.

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2.7 Summary and statement of equilibrium
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There are several items we have worked out in this chapter :
  1. 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.
  2. A set of forces acting on a particle can be reduced to a single resultant by replacing pairs of forces by their resultant.
  3. The resultant of two equal and oppositely directed forces is zero.
  4. 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 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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2.8 Self-Test and Problems
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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