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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2.2 The general problem
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Figure 2.2a Multiple Forces on a Particle

2.3 A simple 2-force experiment
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Figure 2.3a 2-force Experiment

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 :

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

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 is equal

Figure 2.4b 3-force 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 counter-balance F_c in the same way R did in the 2-force experiment of section 2.3.
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2.5 Resultant of two forces
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Figure 2.5a Resultant of 2 forces

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 tan(β) ?
You have to be careful when applying equation 2.5b because it gives you always two possible values for the angle β because in the ambiguity in determination of an angle with only its cosine is given. (example cos(β) = 0.5 has as solutions β=60° and β=300 °)

The rule is that the resultant R has to lie always inbetween the given forces F₁ and F₂. Hence, you choose the angle β such that :

You definitely should look at Problem 2.5a.
An important special case occurs when F₁ and F₂ are equal in magnitude and the angle α between them approaches 180°.
Maybe you can visualize the outcome from Figure 2.5a. Alternatively, from Equation 2.5a we obtain that R=0.

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2.6 Resultant of several forces forces
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Figure 2.6a 3-force on a particle

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2.7 Summary and statement of equilibrium
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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 :

In essence this is a special case of Newton's 2^nd law :

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