Chapter 6 Space, Time, and the Agent of Interactions: Overview 35

Chapter 6 Space, Time, and the Agent of Interactions

Overview This chapter is somewhat different from the other chapters in this text, in that much of the material serves as reference for the following two chapters. We introduce two new models: The Galilean Space-Time Model, which is the basis for developing a useful way of representing variables that are based on spatial dimensions and time. The second is a model of how “things” interact in our physical universe. Forces are the agents of interactions in this model. It is easy to forget that the common and familiar way we talk about distances, speeds, forces and many other variables using these models are not “the way things really are.” They are only this way in the limited range of applicability of these models, which, fortunately, is sufficiently large to include almost all “everyday” phenomena we experience. But whenever we begin to look too closely at the atomic scale, or at systems in which objects travel near the speed of light, or where there are much larger concentrations of matter than we typically experience in our solar system, our familiar notions of space and time and how forces work have to be replaced.

36 Chapter 6 Space, Time, and the Agent of Interactions: Galilean Model

The Galilean Space-Time Model (Summary on foldout #4 at back of text) We live in a world of three spatial dimensions and one time dimension. In our ordinary experience we find that these four dimensions are all independent of each other. We can imagine a reference frame as a set of perpendicular coordinates and a clock. We can measure the velocity of something moving with respect to this first reference frame. We can also imagine a second reference frame moving at a constant velocity with respect to the first. The velocity of that same object as measured in the second reference frame turns out to be given by the vector addition of the velocity of the object as measured in the first reference frame and the velocity of the first reference frame with respect to the second reference frame. This is simply a fancy way of saying that if you walk forward on a moving bus, your velocity with respect to the ground is your velocity as measured on the bus plus the velocity of the bus with respect to the ground. This behavior is referred to as Galilean relativity. Space and time “behave” like “they are supposed to behave.” Time is totally independent of the spatial dimensions and each spatial dimension is independent of the other spatial dimensions. (Recall that we used this notion of Galilean space when we asserted that there were three independent ways atoms could move—and have energy—when we predicted heat capacities using the Particle Model of Thermal Energy.) Most phenomena we encounter in our everyday experience is consistent with the Galilean model of space-time described in the previous paragraph. However, the motion of the electrons moving from the back of your color television picture tube and striking the phosphorescent material on the inside front surface producing the nice color images we like to watch, can not be described by the common-sense notions of the Galilean space-time model! The motion must be described using special relativity. In the special-relativity model of space-time the three spatial dimensions are not independent of time. Time and space get mixed together. The electron—if it had a clock with it—would think it got from the back of the picture tube to the front in a much shorter time than you would measure. And you and the electron would both be correct! The clocks are not messed up. They are giving the correct times. It is just that the times really are different! These kinds of seemingly weird behaviors are really the way space-time “works” when relative velocities get large, meaning appreciable compared to the speed of light, which is 3 x 108m/s. What we think of as normal space-time behavior—Galilean space-time—works only for speeds slow compared to the speed of light. Galilean space-time also breaks down when we get near really large masses, like black holes. Except for a little digression in Part 3 when we try to make sense of the origin of magnetism, we will assume we can use the Galilean space-time model. (Magnetism is a direct manifestation of the effects of special relativity, even when there are no material objects moving at fast speeds. If the speed of light—and other electromagnetic radiation—were infinite, there would be no magnetism.) So, even though we won’t pursue these “weird” effects further, it is useful to keep in mind the limitations to our understanding of space-time as we normally experience it and described in the Galilean Space-Time Model. Describing Motion, Force, and Other Interesting Things in Galilean Space- Time What we are really interested in now is how we describe how objects move in Galilean space-time. (We will omit the phrase Galilean space-time from now on, simply assuming that Chapter 6 Space, Time, and the Agent of Interactions: Galilean Model 37 we are always using this space-time model.) How do we describe and make sense of “things” like velocity, force, momentum and torque that have directional properties as well as a magnitude? Basic Vector Properties A very useful idea (we frequently use the word “construct” for an idea like this) is the concept of a vector. We can give a definition of a vector as something that exhibits both directional properties as well as a magnitude. But beware! The quote below points out a trap that is easy to fall into: "It often does more harm than good to force definitions on things we don't understand. Besides, only in logic and mathematics do definitions ever capture concepts perfectly. The things we deal with in practical life are usually too complicated to be represented by neat, compact expressions. … In any case, one must not mistake defining things for knowing what they are." Marvin Minsky, from The Society Of Mind, 1985, as quoted in The University of Alberta's Cognitive Science Dictionary, on line at http://web.psych.ualberta.ca/~mike/Pearl_Street/Dictionary/dictionary.html The point of the above quote that is relevant to us is that understanding does not come from a definition. As we begin to use the vector concept and talk about vectors in different contexts, we will develop a deeper and richer understanding. Remember how hard it is to define “energy.” Our understanding of energy continually increased as we worked with the idea in more and more contexts. Similarly with “vector,” even though we can give a short one-sentence definition of “vector,” understanding comes as we use the concept and work with it over the remainder of this and the next two chapters. One way to represent vectors is with arrows pointing in the direction of the vector with the length of the arrow representing the magnitude. We refer to the arrowhead as the head of the vector, and the other end as the tail. In print, in order to show the two or three dimensional vector nature of a force (or any other quantity with vector properties,) a bold symbol (such as F, r, v, a) is used or a small arrow is put over the symbol. The magnitude of a vector that could point in any direction in two or three dimensional space is normally printed in plain (not bold) italic type. Thus, the symbol “F” incorporates the vector properties (magnitude and direction), but the symbol “F” means the magnitude of the force only. Note: When a vector can point in only one direction, left or right, for example, a lower case symbol is typically used, and in an algebraic equation involving only one spatial variable (x or y for example), the symbol can take on both positive and negative values, consistent with the convention used for the spatial variable. Vector properties usually do not include the location of the arrow, but only its length and the direction it points. Thus, the arrows, representing 45° vectors, can be slid anywhere around on a drawing as long as the length and J direction are preserved. (Although where a vector is located in space is not significant for the vector properties we are about to discuss, for some of the constructs represented by a vector, such as force, where the forces are located or where they act is very important. In these cases, we need to specify where the construct being represented by an arrow is physically located. We will typically do this using a convenient diagram.) L K In the examples at the right, vector J is about three units in magnitude θ and points in a direction 45° below the +x axis, or we could say points unit length 38 Chapter 6 Space, Time, and the Agent of Interactions: Galilean Model South-East. Note that one unit length is the distance across two squares on the grid. Vector K is four units in magnitude and points in the –y direction, or South. Vector L is about two units in magnitude and points in a direction given by the angle θ with respect to the +x axis. Adding Vectors The figure below shows three vectors, A, B, and C. We add two vectors by sliding one or both around so that the tail of the second is at the head of the first (it doesn't matter which one is which). The vector sum, A + B (or equivalently, B + A) is the arrow that connects the tail of the first to the head of the second. The resulting vector that represents the vector sum A + B is usually represented by a double-line arrow. More than two vectors are added by continuing to add the tail of the next vector to the head of the last one added. It doesn't matter in what order we add these vectors together, or where the location of the vector-sum arrow is. In the figure B is added to A and then C is added, but they could have been added in any order. Subtraction of vectors is accomplished by using the relation that the negative of a vector is a vector of the same magnitude, but pointing in the opposite direction. That is, A – B is obtained by adding – A A B + A A A –B B B B A – B A + B A A+B+C –B C C B adding B to A adding A to B adding A, B and C together subtracting B from A B to A (or A to –B, as shown in the figure). That is, A – B = A + (–B) = –B + A. This is shown in the rightmost construction above. Chapter 6 Space, Time, and the Agent of Interactions: Galilean Model 39 Vector Components The basic idea of vector addition, i.e., that vector A is equal (or equivalent) to the sum of two other vectors, say B and C, is the basis of the concept of vector components. If B and C are perpendicular to each other, then we say that B is the component of A in the direction of B and that C is the component of A in the direction of C. The figure shows one set of components of vector A: B and C and in different directions, another set of components, F and G.

B A = F + G C A = B + C F G

The important thing is that the components are perpendicular to each other. We might associate the directions of the components with an x-y coordinate system. In the figure, we could orient an x-y coordinate system with the positive x direction pointing horizontally to the right. Then, B is the component of A in the x direction and points in the negative direction. Likewise, C is the component in the y direction and also points in the negative direction. The components F and G lie along the perpendicular axes of a tilted coordinate system. The reason vector components are useful is because we are working in Galilean space-time. Each of the three components is independent of the others. Often we are concerned only with the motion of some object in a particular direction; i.e., the component of the motion in a particular direction. We can focus on the motion in this direction (this component of the motion) without worrying about what is happening in the other two perpendicular directions, since the motions are independent of each other. Due to the independence of any three perpendicular directions in space, as well as the independence of what happens in each direction, we should think of vector quantities as really being the combination of three independent “things.” The component representation we are about to develop emphasizes the “threeness” of vector quantities. The “vector” notation—an arrow or a bold symbol—emphasizes their “oneness.” Both representations are very useful. We will frequently use a rectangular coordinate system and the associated vector components when we need to find numerical values involving vectors. Note: Frequently, we y will be dealing with vector quantities in only one or two dimensions, rather than in all three spatial dimensions. That is, we are restricting the vectors in the particular physical situation to lie in only one or two dimensions, but they still represent physical quantities that exist in three dimensions. A A y The components of a vector are the projections of the vector onto some x arbitrarily chosen set of perpendicular coordinate axes. In other words, in A x two dimensions, for example, we visualize the vector sitting in a 2-D coordinate system as shown, with its "tail" at the origin. Perpendiculars are then drawn from the "head" to each of the coordinate axes. These mark off A A y the lengths Ax and Ay which are the coordinates of the vector A. We can θ also relate the magnitude of the vector to its components through A x appropriate trig functions. Ay = Asinθ; Ax = Acosθ. Αs shown in the upper figure to the right, vector A is the vector sum of the two perpendicular components, Ax and 40 Chapter 6 Space, Time, and the Agent of Interactions: Galilean Model

Ay. The components and the magnitude of the vector also satisfy the Pythagorean Theorem, of 2 2 2 course: Ax + Ay = A . The Description of Motion in the Galilean Model The variables describing motion—position, velocity, and acceleration, have direction and magnitude, and so are usefully represented as vectors. The variables we use to specify motion have precise, technical meanings. Often these meanings are different from the everyday meanings we associate with the same words. It is, of course, crucial to understand both the technical and everyday meanings of these words, and how they differ. Position (technical meaning) The position vector represents the position of something with respect to the origin of a particular reference frame. The position vector specifies both the distance (the length of the vector) of the object from the origin and the direction in space with respect to the origin. The position vector is usually represented with a lowercase letter r. Displacement (technical meaning) The displacement of some object is the change in position of the object.

∆r = rf – ri The differential displacement is dr. Velocity (technical meaning) Velocity usually means the instantaneous velocity, rather than an average velocity over some time interval. When we use the word velocity without a modifier, it will mean instantaneous velocity. By definition, velocity is the time rate of change of position: dr v = , dt lim Δr or, v = . Δt → 0 Δt € When the time interval, Δt, is finite, we have the average velocity: Δr € v = . average Δt Acceleration (technical meaning) Acceleration usually means the instantaneous acceleration, rather than an average € acceleration over some time interval. When we use the word acceleration without a modifier, it will mean instantaneous acceleration. By definition, acceleration is the time rate of change of velocity: dv a = , dt lim Δv or, a = . Δt → 0 Δt €

€ Chapter 6 Space, Time, and the Agent of Interactions: Galilean Model 41

When the time interval, Δt, is finite, we have the average acceleration: Δv a = . average Δt Note that acceleration is any change in the velocity. The change can be a change in the magnitude of the velocity vector, either increasing or decreasing in length, that is, either slowing € down or speeding up. The change can also be a change in direction of the velocity, with or without a change in the speed. Everyday or Common Definitions Distance in an everyday sense usually means how far apart two points are, and does not imply a direction, nor does it imply position. It is always a positive number. In terms of the technical motion variables, distance is the magnitude of displacement. However, the term “total distance” does not necessarily mean the magnitude of the displacement. Consider an object that moves back and forth, such as a mass on a spring. The “total distance” usually means how far the mass traveled in going back and forth, while the displacement is simply the distance between where it started and where it ended. Speed in everyday use usually means how fast something is moving. It also does not imply direction and is always positive. In terms of the technical motion variables, speed is the magnitude (the “length”) of the velocity vector. In everyday use, the word acceleration usually means speeding up, but does not mean slowing down. Remember, our technical definition includes both speeding up and slowing down as well as changing direction.

The Force Model (Summary on foldout #5 at back of text) 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 42 Chapter 6 Space, Time, and the Agent of Interactions: Force Model 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 isn’t. 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:

FA on B = −FB on A .

In words, this relationship can be expressed as, “If object A exerts a force A 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 B 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 table an interaction there will be two objects, each with a force acting on it due to the interaction. FB on A 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 object A equal and opposite and act on different objects–the two objects involved in the interaction. The figure to the right shows an object A sitting atop object B. We use of object B 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 FA on B would not be the only forces acting on these objects.) Chapter 6 Space, Time, and the Agent of Interactions: Force Model 43 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 ΣF (Greek sigma right next to a Roman “F”) to represent this construct. ΣF F 3 on 1 is not a fictitious force, but neither is it a “real” force. ΣF is the effect of all the forces acting on an “object.” Operationally, ΣF can be found by constructing the vector sum of the individual forces acting on F 2 on 1 an object. Net force is not connected to a particular interaction with another object. Because of this lack of connection to a particular Two forces that interaction, net force is a rather abstract concept, but one that turns out to act on object 1 be very useful. Force Diagrams and Net Force F 3 on 1 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 object 1 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.” F2 on 1

The figure on the right shows two forces that act on an object, Force diagram for Object 1 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 F 3 on 1

F2 on 1 arrows representing the forces acting on the object pointing out, with their tails at the dot. The net force ΣF 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, ΣF. The head-to-tail vector addition required to obtain ΣF is shown below the force diagram. F F 2 on 1+ F 3 on 1 Σ = The following is a list of common conventions that we will follow The net force on Object 1 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: 44 Chapter 6 Space, Time, and the Agent of Interactions: Force Model

• 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 Ball or 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. € € €

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).

Balanced Forces: When the Net Force is Zero A fundamental aspect of the Galilean model and the way “forces work” is that when we identify all contact and long-range forces acting on a particular object, and if the net force turns out to be zero (the forces balance), then all of those forces acting together have no effect on the object. It is as if there are no forces acting on the object. That is, there is no effect on the object due to its interactions with all the other objects. If there is no effect of the interactions, then there will not be any change occurring to or with the object. Whatever the object “was doing”, it will continue to do. Nothing will change. We need to be more specific here about what we mean by “was doing” and what we mean by “change.” When we are standing on the floor in an airport terminal and then one hour later when standing in the aisle of a jet plane going 500 miles per hour with no turbulence, we experience the same sensations on our body. This illustrates a fundamental aspect of the Galilean model of space and time. If the net force is zero on us when standing on the floor of the airport terminal, and the net force on us is also zero when on the jet traveling at a constant speed with respect to the ground, the effect is the same: no change. We continue to do exactly what we were doing. Chapter 6 Space, Time, and the Agent of Interactions: Force Model 45 From our perspective, we were standing still on the ground and standing still on the airplane. But from a person’s perspective at the airport, we were moving at a constant 500 miles per hour when “standing still” on the plane. Evidently, being motionless and moving with a constant velocity are the same thing with respect to how forces work. So, “no change” really means not changing velocity, which can certainly remain continuing with a zero velocity, if that is what we were doing. But it also means, continuing to move at a constant velocity if that is what we were doing. Because forces work this way, we can turn moving at a constant velocity to being motionless by switching to a reference frame that is moving with the that same velocity with respect to the original reference frame. When a zero net force works this way—no change in velocity if the net force is zero— technically we say we are in an inertial reference frame. For our purposes, it means the reference frame is not accelerating with respect to an inertial reference frame. For most purposes, the surface of the Earth can be considered an inertial reference frame, even though the Earth is rotating at one revolution per day. The resulting acceleration is sufficiently small so that we can usually ignore it. We can summarize the previous discussion regarding balanced forces (net force equal to zero) in the following way: If ΣF = 0, then ∆v = 0 This relationship is traditionally referred to as Newton’s First Law. If for a particular object, ΣF = 0, then there is no change in the motion (magnitude or direction) of that object. No “change in the motion” means that the velocity does not change, ∆v is zero. A traditional statement of the First Law is, “An object at rest will remain at rest and an object in motion will continue to move in a straight line at constant speed unless acted upon by an unbalanced force.” Distinctions Between the First and Third Laws It is sometimes easy to confuse the first and third laws, especially when there are two forces acting on an object. The pairs of forces that balance each other in the First Law act on the same object, so they cannot possibly be Third Law force pairs, even though they are equal and point in opposite directions. A more complete force diagram for object A of the figure showing two blocks on a table is shown at right. Two objects interact with object A: A object B by a contact force and the Earth by the long-range gravitational force. Object A is at rest, so the two forces that act on it are equal in B magnitude and opposite in direction, by virtue of the First Law, not the Third Law. (Also note that we are following the convention of placing on table the force diagram for object A only the forces that can be explicitly labeled as acting on A.) F B on A

object A

F Earth on A

46 Chapter 6 Space, Time, and the Agent of Interactions: Force Model

Newton's First Law is concerned about the situation when all the forces acting on an object are balanced. There are two complementary ways of using the First Law. If we know that there is no change in motion, then we know that the forces/impulses acting on the object must be balanced. If we know all but one of the forces/impulses, we can solve for the magnitude and direction of the unknown force/impulse. The second way to use the First Law is to add up the known forces to see if they balance. If they do, than there can be no resulting change in motion of the object.

Two Fundamental Forces The electric and gravitational forces are said to be long-range forces. (They are long range compared to the atom-atom or molecule-molecule r forces we studied in Part 1 in Chapter 3 on the particle model of matter. F F Unlike those atom-atom forces, the electric and gravitational forces, 2 on 1 1 on 2 q q although decreasing with distance, still exist–and have profound effects– 1 2 even at large separation distances.) The magnitude of the electric and Electric force for charges 2 gravitational force between two particles both decrease as 1 / r . We often of same sign refer to these two forces as inverse-square law forces. In both cases, electric r and gravitational, the force acts along the direction of the vector from one particle to the other. In addition, each force depends on a fundamental F2 on 1 F1 on 2 property of matter: electric charge and gravitational mass, respectively. q q 1 2 These two forces are the manifestations of two of the four fundamental Electric force for charges interactions: gravitational, electromagnetic, weak nuclear, and strong of opposite sign nuclear. r The electric force and gravitational force are similar in many ways, but there is one striking difference: the gravitational force is always attractive, F2 on 1 F1 on 2 while the electric force may be either attractive or repulsive. m m 1 2 There are two kinds of electric charge, which for historical reasons, are Gravitational force referred to as positive and negative. The electric force is repulsive between for two masses like charges (either both positive or both negative) and attractive between unlike charges (one positive and one negative). The magnitudes of the forces between two particles (or chunks of matter) are proportional to the amount of charge (or mass) possessed by each particle and inversely proportional to the square of their separation. Electric Force q q F = k 1 2 . E r2

Gravitational Force m m F = G 1 2 . € G r 2

The constants k and G have fixed values that are dependent on the unit system employed and are characteristic of these two fundamental interactions. Chapter 6 Space, Time, and the Agent of Interactions: Force Model 47 For electric forces, if the charges have the same sign, then the force at either charge points away from the other charge. If the charges have opposite signs, then the force at each charge points toward the other charge. In both cases, the magnitude of the forces F1 on 2 and F2 on 1 are equal, due to Newton's Third Law.

The gravitational force is always attractive. The magnitude of the forces F1 on 2 and F2 on 1 are equal. Our theory (or model) of the electric and gravitational interaction contains two universal constants, G and k. In addition, the mass and charge of the electron and the proton are also universal constants. Masses and charges of objects are not functions of position. It is the forces between them that depend on their positions. The strength of these two universal forces are determined experimentally. They are some of the “givens” that make our universe what it is. (Whether they could be something else in another universe is an interesting question that some physicists ponder.) For the electric force, the value of the constant k is actually set to a specific value and the size of the coulomb is what is actually determined experimentally:

N ⋅m2 N ⋅ m2 k = c2 ×10−7 = 9 ×109 . Coulomb 2 Coulomb 2

Here c is the speed of light, m c = 2.998 ×108 , s and the coulomb is the SI unit of electric charge. The experimentally determined electric charge on a proton is e = +1.602 ×10 −19 coulombs, and the electric charge on an electron is –e. Note carefully that the coulomb is a very large amount of charge compared to the charge on single atomic sized particles. Note: the symbol “q” is generally used to represent an arbitrary charge and the symbol “e” is used to indicate the charge on one electron or proton. In the case of gravitation, the universal gravitational constant has the experimentally determined value, N ⋅ m2 G = 6.672 ×10−11 . kg2 Of all the universal constants, G is known with the least precision; it is hard to “weigh the Earth.”

The Forces We Typically Experience According to the expressions we wrote above for the electric and gravitational forces, we might think that we would experience forces in all directions from all of the electrons and protons and mass distributed all around us. In fact, we experience only the gravitational force between us and the Earth, which is directed toward the center of the Earth. We experience very few electrical forces unless we have matter in contact with other matter. What happened to all the long range gravitational forces and electrical forces that must be acting between all of the electrons, protons, and neutrons in our bodies and all of the matter around us? The answer is fairly simple in both the gravitational case and electrical case, but the reasons for not experiencing the forces are very different in the two cases. 48 Chapter 6 Space, Time, and the Agent of Interactions: Force Model In the case of gravity, all of our atoms do attract all other atoms in our vicinity. However, the gravitational force is a very weak force. Only when there are a very large number of atoms nearby, such as all of the atoms that make up the Earth, is the net force due to the interactions with all of these atoms sufficiently large for us to be aware of it. Newton showed (by inventing calculus) that the attraction of a mass m to a sphere of mass M is the same as if M were concentrated at the center of the sphere, and the distance from the mass m to the center of the sphere is used in the calculation. The net gravitational force between you and the person sitting next to you is too small to feel, because there are not a sufficient number of atoms in the two of you. However, it is interesting to note, that even though it is extremely small, the universal gravitational constant G is experimentally determined by measuring the attraction of two solid metal spheres a foot or so in diameter hung a foot or so apart. So, for all practical purposes, we experience only one gravitational force: the force between an object and the center of the Earth. This same force, described by the same formula, however, also describes the attraction of the planets to our sun, the stars to each other in our galaxy and the attraction of the galaxies to each other in our galaxy cluster. And of course, it describes the attraction of an apple to the Earth, causing it to fall on Newton's head, or so the story goes. Why don't two objects close together attract or repel each other due to the electrical force? Most objects don't do so, because they are composed of very large numbers of electrically neutral molecules. For every positive proton there is a negative electron. Because the charge on a proton is exactly the same magnitude as the charge on the electron, the electrical force that one molecule exerts on another molecule that is more than a short distance away is practically zero. However, because the positive and negative charges—the protons and the electrons—are not exactly in the same locations, molecules do exert attractive forces on nearby molecules. These are the atom-atom forces we encountered in Physics 7A. What are Contact Forces? Recall what happens when molecules get pushed very close together. The electrons then exert very large repulsive forces on each other. This is why substances resist compression. When we push on an object, it is the electrical forces between the electrons in the molecules of our skin and the electrons on the atoms at the surface of what we are pushing that are really doing the pushing. And it is the electrical forces holding molecules together that allow us to establish a tension in a stretched wire or cord. So, we actually do experience electrical forces all the time. But because they are for the most part due to electrically neutral molecules interacting with each other, the net forces are “short range” and are not given by our simple formula. We often describe these electric forces as contact forces. To summarize, the forces that electrically neutral objects exert on each other when they are brought into close proximity really are electric forces, but they are very short range, and are not described by the one-over-r-squared long-range force formula. Weight of objects on the surface of the Earth We directly experience one of the fundamental forces all the time—the gravitational force. We know this force as the weight of an object. In Physics 7A we used the fact that the weight of an object on the surface of the Earth is proportional to the mass of the object with the constant of proportionality, represented by the symbol g, having the value 9.80 N/kg. Now we know where Chapter 6 Space, Time, and the Agent of Interactions: Force Model 49 the constant g comes from. Let’s compare our two expressions for the gravitational force the Earth exerts on an object: m m First, we write the general gravitational force F = G 1 2 in a way that explicitly gives the G r 2 force that the Earth exerts on an object of mass m located at the surface of the Earth:

M Em FEarth on object = G 2 rE Our expression from Physics 7A is

FEarth on object = m g Comparing these expressions, we see that

ME g = G 2 rE

and we also see why the expression FEarth on object = m g with g constant, is valid only near the surface of the Earth. (Note: in Part 3, we will treat g as the magnitude of the gravitational field, when we take up the Field Model of electric, magnetic, and gravitational interactions.) Large Scale Gravity Forces Our solar system is “held together” by the gravitational force. Our solar system is situated in a particular place in our galaxy due to the action of the gravitational force. When dealing with the gravitational force acting between astronomical bodies, we can’t, of course use the convenient expression FEarth on object = mobject g that works for objects on or very near the surface of the Earth. Rather, we have to use the actual masses and separations of the celestial objects in Newton’s universal law of gravitation. The table below lists useful astronomical parameters for our solar system. Mass (kg) Radius (km) Orbital radius (km)

Sun 1.99 ×10 30 696,000 (n/a) Earth 5.98 ×1024 6,370 1.50 ×10 8 Moon 7.35 ×1022 1,738 3.85 ×10 5 Mars 6.42 × 1023 3,407 2.28 ×108 Venus 4.87 ×10 24 6,050 1.08 ×10 8

Everyday forces that result from the electric force Most of the forces we experience, other than weight, are ultimately due to the electrical force. But as we previously mentioned, these forces are due to the collective action of a vast number of oppositely charged particles, whose individual forces almost cancel out. When a spring is stretched, electrical forces acting between atoms of the metal are responsible for the restoring force that causes the spring to resist being stretched or compressed. The forces that bind atoms into molecules are electrical in nature, arising from the interaction of the various electrons and protons of the atoms that make up the molecules. 50 Chapter 6 Space, Time, and the Agent of Interactions: Force Model Historically, many names have been given to different kinds of forces, all of which are fundamentally electrical in nature. In Physics 7A we studied the force due to the pair-wise potential acting between particles. We will mention several others here, along with some useful conventions for dealing with them. It is worthwhile to remember, however, that fundamentally they are all due to the same basic interaction—the electric force between charged particles. Forces Exerted by Springs—Hooke's Law Robert Hooke (a contemporary of Isaac Newton) discovered in 1676 that the force exerted by many stretched springs is proportional to the elongation or compression of the spring and in the opposite direction to the elongation or compression. The constant of proportionality depends on the way the particular spring is made (its material, size, number of coils, etc.). We used this model for springs extensively in Physics 7A. In equation form we express the force exerted by a spring as F kx . by spring = − More on the Contact Force In a certain sense all objects behave like springs. When we place a book on a table, the earth pulls down on the book and the table pushes up. Why does the table push up? Well, the table behaves like a spring. The book pushing down on the table compresses the table a very slight amount, so the table pushes back. As the table becomes more and more compressed, it pushes back harder and harder. This process continues until the force exerted by the table equals the weight of the book. Then the two forces are in balance and the book becomes stationary. It is frequently useful to “break up” the contact force into two components, one perpendicular and one parallel to the surface. Perpendicular Component of the Contact Force: This spring-like force that the table exerts is called the perpendicular contact force, or in some textbooks it is called the normal force. (Here “normal” means “perpendicular.”) This force is always perpendicular to the surface of the object exerting it. In our example of the book on the table, it is the table, pushing up. But if the table surface were not level, the earth would still pull straight down on the mass. The perpendicular component of the contact force would not be straight up, but rather it would be perpendicular to the table surface. The forces acting on the book would then not be in balance, and if there were no friction, the book would slide on the non-level table. We will often add a perpendicular sign to the subscript to designate the perpendicular component of the contact force; e.g., the perpendicular contact force the table exerts on book would be designated as F⊥ table on book. Parallel Component of the Contact Force Frictional forces between solid objects are exerted parallel to their surfaces. These forces arise because the molecules on one surface are attracted to the molecules of the other surface where they come into contact. We call these parallel contact forces. It is sometimes useful (and conventional) to separately talk about the case when the surfaces are not moving with respect to each other and when they are. The former is called static friction; the latter, kinetic or sliding friction. In general, frictional forces act in a direction opposite to the motion, but this is not always the case; the direction can be determined only through analysis of the specific case under study. We Chapter 6 Space, Time, and the Agent of Interactions: Force Model 51 use a parallel symbol to indicate the parallel component of the contact force. Using our previous example the parallel contact force exerted by the table on the book would be written as F|| table on book. Total Contact Force Both perpendicular and parallel contact forces can be present at the same time. They are, after all, simply the total contact force resolved into two perpendicular components. In the case of the mass sitting on a rough-surfaced table whose surface is not horizontal, both perpendicular and parallel contact forces are present; the total contact force balances the force exerted by the Earth, if the mass is indeed stationary. Drag Forces As we will see when we study fluids, when a fluid flows through a pipe, the inside surface of the pipe slows down the fluid adjacent to it. Conversely, when an object moves through a gas or liquid, there are drag or viscous forces retarding the motion of the object. These forces point in a direction opposite to the velocity of the object. Thus, when a car moves down the highway at high speed, the air exerts a drag force on the car, sometimes called air friction. This force exerted on the car by the air is directed “backwards,” opposite to the direction the car is moving.

Wrap Up

The modern understanding of forces and the relation of forces to motion was worked out by Newton and summarized in what has become known as “Newton's Laws of Motion” In this chapter, we have spent considerable time making sense of Newton’s first and third laws. We have also talked a lot about the construct of net force, ΣF. The ideas about force introduced in this chapter will be used extensively in the next two chapters. Don’t forget to come back to this chapter often as you begin dealing with forces when using momentum conservation and Newton’s second law. 52 Chapter 6 Space, Time, and the Agent of Interactions: Force Model