3.2: Particles

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We'll begin our study of Equilibrium with the simplest possible object in the simplest possible situation — a particle in a one-dimensional coordinate system. Also, in this chapter and the next all forces will be represented as concentrated forces. In later sections the we will address more complicated situations, higher dimensions, and distributed forces, but beginning with very simple situations will help you to develop engineering sense and problem solving skills which will be useful later.

The defining characteristic of a particle is that all forces that act on it are coincident¹Two lines are coincident when one lies on top of the other. or concurrent²Two or more lines are concurrent if they intersect at a single point., not that it is small. Forces are coincident if they have the same line of action, and concurrent if they intersect at a point. The moon, earth and sun can all be treated as particles, but we probably won’t encounter them in statics since they’re not in equilibrium. Forces are coincident/concurrent if their lines of action all intersect at a single, common point. Two or more forces are also considered concurrent if they share the same line of action. One practical consequence of this is that particles are never subjected to forces which cause rotation. So a see-saw, for example, is not a particle because the weights of the children tend ³We say tend to cause rotation because in a statics context all objects are static — so no actual rotation occurs. to cause rotation.

Another consequence of concurrent forces is that Equation (3.1.1) is the only equilibrium equation that applies. This vector equation can be used to solve for a maximum of one unknown per dimension. If you find yourself trying to solve a two-dimensional particle equilibrium problem and you are seeking more than two unknowns, it’s likely that you have missed something and need to re-read the question.

Another simplification we will be making is to treat all forces as concentrated. Concentrated forces act at a single point, have a well defined line of action, and can be represented with an arrow — in other words, they are vectors. Real forces don’t actually act at a single mathematical point but concentrating them is intuitive and will be justified in a later chapter 7.8. You’re already familiar with the concept if you have ever placed all the weight of an object at its center of gravity.