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# 6.3: The Force model

### Some Properties and Characteristics of Forces

Force is the name we give to a fundamental construct in our theory of interactions. This construct is closely identified with our everyday experience of a push or a pull. Many of the characteristics of pushes and pulls that we are familiar with also apply to forces.

A notion that takes us well beyond pushes and pulls is the idea of force as the agent of the interaction between two objects. The word “object” is used here in a general sense to include any identifiable mass, which might be a particular volume of a fluid or a single electron, as well as ordinary objects such as books and pieces of chalk. The essential idea here is that forces don't exist in the absence of interacting objects. Our understanding of just what a force is will become sharper and more refined as we progress through the next couple of chapters.

#### Forces Act Between Objects

If we can't identify two interacting objects, then there isn't a real force in the scientific sense of the term “force.” For example, when we are the passenger in a moving car and the driver turns sharply, most of us tend to say that we experience a force pushing or pulling us to the outside of the curve. Similarly, when a jet airliner accelerates along the runway during takeoff, most of us would say we experience a force pushing us back into our seat. But in both cases, what we identified as forces are not forces in the scientific sense. There is no object that we are interacting with for which this "force" that we think we experience is the agent of interaction. In fact, the objects we interact with, e.g., the seat, are pushing us in the opposite direction! The term fictitious force is sometimes used to describe what is perceived as a force in situations such as these.

#### Forces Come In Pairs

The starting point for our understanding a very important aspect of forces is that forces come in pairs. We just said in the previous paragraph that force is the agent of interaction between two different objects. But the agent of the interaction, the force, manifests itself two ways, depending on our point of view. Suppose object A is interacting with object B. Focusing on object B, we would say that object A exerts a force on object B. Alternatively, focusing on object A, we would say object B exerts a force on object A. These two forces, associated with the same interaction between the same two objects must be related to each other. But how are they related? It turns out that interactions are very egalitarian. The effect of the interaction on each of the objects, however, does not have to be the same, and usually it is not the same. The effect of the interaction depends on other properties of the specific object, such as its mass and its motion prior to the interaction. This is an important distinction: The interaction is always the same with respect to the two objects, but the effect that the interaction has on each of the two objects does not have to be the same and usually is not. It is very important to keep this distinction in mind when thinking about forces.

What does it mean to say that the interaction is the same with respect to both objects? It means that the agent of interaction, the force, is the same in some sense. Specifically, the magnitude of the force that object A exerts on object B has the same magnitude as the force that object B exerts on object A. Because the interaction doesn’t pick out one direction to favor over any other, the two forces point in opposite directions with respect to each other.

This relationship in terms of forces between two interacting objects, object A and object B, can be represented using vector notation as:

$F_{A on B} = -F_{B on A} . \tag{6.3.1}$

In words, this relationship can be expressed as, “If object A exerts a force on object B, then object B exerts a force equal in magnitude and opposite in direction on object A.” The critically important point is that to have an interaction, we must have two objects that are interacting. Force is the agent of interaction. So to have a force, there must be an interaction, and if there is an interaction there will be two objects, each with a force acting on it due to the interaction.

This fundamental aspect of force has historically been given the name “Newton’s Third Law.” It is common to refer to the two forces involved in an interaction as a “3rd law pair.” The Third Law pair of forces are always equal and opposite and act on different objects–the two objects involved in the interaction.

The figure 6.3.1 shows an object A sitting atop object B. We use of two separate dots to represent the two different objects A and B. Newton's Third Law relates how the same agent of interaction is manifested as separate forces acting on the two interacting objects. (Note, in this example, these would not be the only forces acting on these objects.)

Figure 6.3.1

It is tempting to think that the Third Law applies only to objects “at rest.” However, the Third Law applies whenever there is an interaction between two objects, independently of the motion of the two objects that are interacting.

### The Construct of Net Force

It is important to distinguish between forces exerted by a particular object on other objects and the forces those other objects exert on the original object. Only forces acting ON an object affect that object. Forces that an object itself exerts on other objects do not directly affect itself. They is, of course, a direct relationship between the force that a particular object exerts on a second object and the force the second object exerts back on the first. See the discussion of Newton’s 3rd Law on the previous page.

It turns out that the effect of all forces acting on a particular object can be represented by a single vector construct called the unbalanced force or the net force. We will use the symbol $$\sum F$$ (Greek sigma right next to a Roman “F”) to represent this construct. $$\sum F$$ is not a fictitious force, but neither is it a “real” force. $$\sum F$$ is the effect of all the forces acting on an “object.” Operationally, $$\sum F$$ can be found by constructing the vector sum of the individual forces acting on an object. Net force is not connected to a particular interaction with another object. Because of this lack of connection to a particular interaction, net force is a rather abstract concept, but one that turns out to be very useful.

### Force Diagrams and Net Force

When we want to analyze a physical situation in terms of forces, it is necessary to focus on forces acting on a particular object. (This need arises first when we consider the next model/approach, Momentum Conservation, and then in the next chapter when we consider Newton’s 2nd law.) We will always be interested in the net force on a particular object. To help in identifying forces that act on a particular object it is helpful to pictorially represent the forces as clearly labeled arrows on a diagram. There is a standard convention for representing forces like this, called a “force diagram.”

The figure 6.3.2 shows two forces that act on an object, labeled as “object 1.” F3 on 1 is the force that object 3 exerts on object 1. F2 on 1 is the force that object 2 exerts on object 1. The standard way of representing this situation graphically is to make a force diagram. Regardless of what object 1 is, we represent it as a dot and we draw the arrows representing the forces acting on the object pointing out, with their tails at the dot. The net force $$\sum F$$ is the sum of F3 on 1 and F2 on 1. In this example, the two forces are not balanced and there is a resulting net force, $$\sum F$$. The head-to-tail vector addition required to obtain $$\sum F$$ is shown below the force diagram.

Figure 6.3.2

The following is a list of common conventions that we will follow when making a force diagram. It is important to be familiar with these and strictly follow them; it will make things clearer in the long run:

• A force diagram refers only to one object. (If we have several objects to study, we need a separate force diagram for each one.)
• An object is shown as an enlarged dot in a force diagram. The dot is clearly labeled to indicate what object it refers to.
• All the forces acting on the chosen object are shown on the force diagram. Forces that the object exerts on other objects are not shown on its own force diagram.
• Forces like friction and weight (that act over multiple points) are modeled as a single force acting at a single point.
• To avoid confusion, we usually don't show velocity or other vector quantities on the force diagram.
• A force diagram focuses on a particular “object” and shows only the forces acting on that “object.” However, all forces arise because of interactions between two objects. In order to emphasize this fact, we use the notation FEarth on Ballor FE on B or F1 on 2 to indicate the two objects that are involved and which of the two we are focusing on. Each force shown on a diagram should always have these two subscripts in the correct order with the preposition “on” between them.

Example 6.3.1: Calculating Force

What force is exerted on the tooth in Figure if the tension in the wire is 25.0 N? Note that the force applied to the tooth is smaller than the tension in the wire, but this is necessitated by practical considerations of how force can be applied in the mouth. Explicitly show how you follow steps in the Problem-Solving Strategy for Newton’s laws of motion.

Figure 6.3.1 Braces are used to apply forces to teeth to realign them. Shown in this figure are the tensions applied by the wire to the protruding tooth. The total force applied to the tooth by the wire, $$F_{app}$$, points straight toward the back of the mouth.

Solution

Strategy

1. Use Newton’s laws since we are looking for forces.
2. Draw a free-body diagram:

Figure 6.3.2.

Solution

3. The tension is given as $$T = 25.0 \space N.$$ Fond $$F_{app}.$$ Using Newton’s laws gives: $$\sum{F_y} = 0$$, so that applied force is due to the y-components of the two tensions: $$F_{app} = 2T \space sin \space \theta = 2(25.0 \space N) \space sin (15^o) = 12.9 \space N.$$ The x-components of the tension cancel. $$\sum{F_x} = 0$$.

4. This seems reasonable, since the applied tensions should be greater than the force applied to the tooth.

### Contact and Long Range Forces

An interaction and the accompanying agent of interaction—the force—can occur between objects that are in contact as well as between objects that are widely separated in space. An example of the former is the force you apply on an object when you push it, or the force your bat exerts on a ball as you hit it into left field. An example of the latter is the force of gravity the Earth exerts on an orbiting satellite 200 miles above the surface of the Earth. The gravitational interaction between the Earth and other objects which may or may not be touching the Earth is said to be long range. The interaction and the force, the agent of the interaction, continue to exist, even when there is no direct contact. This is distinctly different from the example of the bat and ball. The force of the bat on the ball is often referred to as a contact force, since there is no interaction or force if the bat and ball are not in actual contact (in a macroscopic sense).