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Rigid Body Statics

Central Force Systems: Forces with a Common Point of Application

In engineering mechanics, the analysis of forces acting on an object plays a central role. An important special case is the central force system, where all these forces have a common point of application. In this chapter, we will explore the properties and analysis of central force systems and explore:

  • Definition and Properties: What is a central force system? What special properties distinguish it?
  • Basic Tasks of Rigid Body Statics for Central Force Systems:
    • Reduction: Reducing a system of multiple forces to a single point (center of force).
    • Equilibrium: Determining whether a body remains at rest or moves under the action of forces.
    • Decomposition: Decomposing a force into several individual forces.

Are you ready to unlock these secrets?

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Practice Exercises
Table of Contents

4. Second Basic Task: Equilibrium

Graphical Solution
Mission: Find equilibrium!

Imagine you're dealing with a rigid body under siege by four mysterious forces, F1, F2, F3 and F4. These pesky fellows want to upset the body's balance, but you're the clever hero who saves the day!

In our layout diagram in Fig. 3.4.1, you can see the whole situation.

This illustration 3.4.1 shows the layout plan for the task. The orientation and directions of the forces, as well as a scale for length measurement, are indicated.
Fig. 3.4.1: Layout Diagram

The lines of action of the forces all meet at one point, so we're dealing with a central force system. Your goal: Find the secret condition that keeps these nasty forces in equilibrium!

Step 1: Divide and conquer!

To understand how we proceed, let's first divide the evil four into two teams: F1 and F2 against F3 and F4. In Fig. 3.4.2 in the force diagram, you can see the two teams:

This illustration 3.4.2 shows the force diagram for the task.
Fig. 3.4.2: Force Diagram

Both teams have their own starting point S12 or S34 and their own endpoint E.

Now, for each team, we use the force parallelogram to calculate the resultant force, which can be imagined as a superpower that combines all the other forces of the team. We call these superpowers R12 and R34.

This illustration 3.4.3 shows the force diagram for the task with its resultants.
Fig. 3.4.3: Force Diagram with Resultants
Step 2: The Grand Unification!

According to the equilibrium axiom (sounds cool, right?), the two superpowers R12 and R34 must be exactly opposite and equal in magnitude for the body to remain in equilibrium.

This illustration 3.4.4 shows the force diagram for the task with its resultants, which are equal in magnitude and act in opposite directions.
Fig. 3.4.4: Equilibrium of the Resultants

Mathematically expressed:

$$ \begin{align} \tag{1} \vec{R}_{12} &= -\vec{R}_{34} \end{align} $$

Now comes the trick: We add the two superpowers vectorially. The result is the total resultant force R acting on the body. If R12 and R34 are equal in magnitude and opposite in direction, then the total force R = 0 and the body is in equilibrium.

Since

$$ \begin{align} \tag{2} \vec{R}_{12} &= \vec{F}_{1} + \vec{F}_{2} \end{align} $$

and

$$ \begin{align} \tag{3} \vec{R}_{34} &= \vec{F}_{3} + \vec{F}_{4} \end{align} $$

we get the condition for equilibrium:

$$ \begin{align} \tag{4} \vec{R} &= \vec{R}_{12} + \vec{R}_{34} \\[7pt] \tag{5} &= \vec{F}_{1} + \vec{F}_{2}+\vec{F}_{3} + \vec{F}_{4} \\[7pt] \tag{6} &=\sum\limits^4_{i=1} \vec{F}_{i} \\[7pt] \tag{7} &= 0 \end{align} $$

Aha! So the condition for equilibrium is that the vectorial sum of all forces is zero. This means that all forces cancel each other out and the body remains happily at rest.

Step 3: Draw the victory!

In the graphical representation in the force diagram in Fig. 3.4.5, the forces F1, F2, F3 and F4 form a closed force polygon. This means: S = E, the endpoint E meets exactly the starting point S. It's like a magic circle that keeps the forces in equilibrium.

This illustration 3.4.5 shows the closed force polygon of the forces F1, F2, F3, and F4.
Fig. 3.4.5: Closed Force Polygon
Summary
  • A central force system is in equilibrium if the vectorial sum of all forces is zero:
    $$ \begin{align} \tag{8} R &=\sum\ \vec{F}_{i} = 0 \end{align} $$
  • Graphically: The force polygon is closed and S = E.

That's it! With these rules, you can handle any central force system.

Bonus: This rule applies not only to 4 forces, but generally to any number of forces!