Chapter 7 Momentum Conservation: Overview 53

Chapter 7 Momentum Conservation

Overview We began this course by focusing on the idea of physical systems, energy systems, and transfers of energy between different physical systems. In earlier chapters, we concentrated on applying an approach to understanding our physical universe that emphasized the results of interactions. The question we tried to answer was what happened to a physical system from a time before to a time after the system interacted with other systems. We tried to avoid needing to understand the details of the interaction. We discovered that changes of energy of a physical system is a very useful measure of the interaction, not the only measure, but certainly a very useful measure. We couldn’t completely avoid the details of interactions, however. We saw that force, the agent of interaction, was involved in the amount of energy transferred during an interaction. Specifically, the differential amount of energy transferred as work, dW, is equal to the product of the parallel component of an externally applied force and the distance moved: st dW = F||dx. We wrote a conservation of energy expression (a more general form of the 1 law of thermodynamics that allows for all kinds of energy changes) to express how the energy of a system changes in response to energy inputs in the form of heat or work: dE = dQ + dW. We saw how we could apply this energy formalism to more traditional thermodynamic systems (gases, heat engines) as well as to mechanical systems. We also developed a simple particulate model of matter in earlier chapters that involved modeling the bonds between atoms and molecules as analogous to masses hanging on springs, the masses being in continuous random oscillation. This simple model allowed us to explain and predict many of the thermal properties of matter in its various states. Again, we avoided the details of oscillations and focused only on changes in energies. In this chapter, we continue our focus on the results of interactions. We are still trying to address what happens to a physical system from a time before to a time after the system interacted with other systems. We will analyze two new physical quantities (momentum and angular momentum) that round out our understanding of the results of an interaction. We still cannot completely avoid the details of interactions. We will see that force, the agent of interaction, is also involved when either momentum or angular momentum is transferred during an interaction.

x f Instead of calculating an energy transfer called work, W F dx (or, in differential form, = ∫ || xi dW = F||dx), we calculate a momentum transfer called impulse, J = ∫ F dt from t1 to t2 (or, in differential form, dJ = F dt). We will write a conservation of momentum expression, ∆p = J, to express how the momentum of a system changes in response to momentum inputs in the form of impulse. We will see both similarities and differences with energy. One difference is that both momentum and impulse, unlike energy and work, are vector quantities. A physical quantity is a vector if it has both a magnitude and a direction associated with it. We indicate vectors by either making the symbol bold, or using arrows over the symbols. One interesting aspect of forces for us right now is the relationship of forces to the motion of material objects. It is traditional to say that these relationships are governed by Newton's three laws. However, there are many features of forces, some rather subtle, that we need to wrestle a bit with before we can appreciate and use 54 Chapter 7 Momentum Conservation: Overview Newton's Laws to answer interesting questions regarding so much of our everyday experience in the physical world. So initially we avoid the details of the motion during the interaction and focus only on changes in momentum. Then, in Chapter 8: The Relation of Force to Motion, we will explicitly use the time dependence of the impulse to find the detailed time dependence of the motion, rather than just comparing the end result of changes between two points in time. The first model/approach in this chapter, Momentum Conservation, gets us into the meaning of momentum and how changes in momentum are related to forces. We will solidify a lot of learning regarding forces that was introduced in Chapter 6. Then in the second model/approach of this chapter, Angular Momentum Conservation, we explore the fascinating world of rotating objects, from molecules to galaxies. We extend the ideas/constructs of force, impulse, and momentum to their analogous rotational or angular counterparts: torque, angular impulse, and angular momentum. You will have ample opportunity to sharpen your vector manipulation skills that were introduced in Chapter 6.

Chapter 7 Momentum Conservation: Linear Momentum Model 55

(Linear) Momentum Conservation Model (Summary on foldout)

Overview of the Model As you begin this chapter, listening in lecture and working in class, it must seem, at least at first, that you are being introduced to a lot of new concepts. The representation of the motion of an object and the forces acting on an object are necessary ideas to understand before we can fully understand this new conserved quantity called momentum. One goal of this (and the next) chapter is to understand the effects of forces on motion and we begin to do this in this chapter through a discussion of momentum and transfers of momentum. We introduce two concepts which are completely new: momentum and impulse. However, we are taking great pains to help you see how these concepts play roles very similarly to energy and work. So, yes, you have to memorize that momentum, p, is the product of mass and velocity (p = mv). And you have to be careful to not forget that momentum has vector properties, just as velocity does. But, impulse is not an isolated construct you file away in your brain somewhere. Rather, you should really strive to understand impulse in analogy to work. A transfer of energy as work changes the energy of a physical system. Similarly, a transfer of momentum as an impulse changes the momentum of a system. Energy is conserved. Momentum is also conserved. Of course, there are differences between momentum and energy and between impulse and work. As you work in class, as you study this text, as you work the FNT’s, and as you interact with other students and with instructors as you mentally struggle with this material, try to understand these new concepts in relation to what you already know, rather than as simply some more isolated facts that you memorize. Review and Extension of the “Before and After” Interaction Approach In Chapters 1 and 2 we focused on changes in the energy of a physical system. Energy has meaning for one particle or 1023 particles, for objects as small as the nucleus of an atom and as large as a galaxy. It really is a universal concept that applies to any physical system. It turns out that there are two other concepts that are like energy in that they are universally applicable, are transferred among systems as a result of interactions, and the amount transferred gives very useful information that does not depend on the details of the interaction. These are the concepts of momentum and angular momentum. Integrating “the Agent” of Interactions We have called force the “agent of interactions”. Interactions occur between objects as they exert forces on each other. Objects experience changes in energy when other objects exert forces on them and do work on them. We recall that the amount of energy change caused by a force is the integral of the force over a distance. This integral is called the work done on a system. The only component that contributes to the work, however, is the component of the force that is parallel to the motion. We usually indicated this component with the symbol F||: xf W = ∫ F||(x) dx = Ef - Ei = ∆ E xi 56 Chapter 7 Momentum Conservation: Linear Momentum Model

Or, if F is constant, or we define an average force Favg, we can write

W =Favg||∆x = Ef - Ei = ∆E In other words, the parallel component of force integrated over the path of the motion is the work, and this work equals the amount of energy transferred to the system due to the application of the force by an object outside the system. A similar integral of the force is equal to the change in momentum of the system. But instead of integrating over distance, we now integrate over time. This integral is called the impulse of the force, F. We represent the impulse with the symbol J.

tf

J = ∫ F(t) dt = pf - pi = ∆p ti

Or, if F is constant, or we define an average force, Favg, then

J = Favg ∆t = pf - pi = ∆p Impulse is a vector quantity and causes a change in a vector property of a system: specifically, a change in the linear momentum, ∆p. The change in momentum, is of course, independent of what Galilean reference frame we choose to measure the momenta in. Note on units: Force has SI units of newtons, of course. Impulse must therefore have units of newton seconds, N s. Momentum, the product of mass and velocity, must have SI units of kilogram meter per second, kg m/s. Since these two quantities are equated, these units must be equivalent, as you can show using the relation N = kg m/s2.

Linear Momentum The linear momentum of an object is simply the product of the object’s mass and velocity: p = mv Linear momentum incorporates the notion of inertia, expressed as mass, as well as the speed and 1 2 direction of motion. In some ways it is similar to kinetic energy, 2 mv , but an obvious difference is that momentum has a direction; it is described as a vector. (Often, the word “momentum” is used without the modifier “linear,” when talking about linear momentum. Later, however, the modifier “angular” is always used when talking about angular momentum.) Temporary restriction to non-rotating objects and center of mass Until we consider rotation of objects in the next model/approach, Angular Momentum Conservation, we will consider phenomena in which extended objects act only like point particles. A useful construct that will become much more meaningful when we consider rotation, is center of mass. Right now we can simply consider that any extended object acts like a single particle whose mass is equal to the mass of the object, located at the special point, the center of mass.

The Construct of Net Force and net Impulse Be sure to review the discussion of Net Force in Chapter 6. The central point is that the effect of all forces acting on an object can be represented by a single vector construct called the Chapter 7 Momentum Conservation: Linear Momentum Model 57 unbalanced force or the net force, ΣF. When we use the concept of impulse, it is sometimes useful to consider the impulse, J, due to a particular force; we would write this as:

JA = FA∆t (The subscript “A” tells us that the impulse is due to particular force exerted by the object “A”) However, the power of the construct of impulse comes into play when we consider the impulse of the net force; we write this as

Jnet = J = ΣF∆t = ∆p (J without a subscript usually means the impulse due to the net force.) In words, the net impulse is equal to the change in the linear momentum of the system. We explore this relationship further below.

Momentum of a System of Particles The momentum of a single object is simply the product of its mass and velocity. Suppose we define a physical system that contains several particles which move with different velocities. The total linear momentum of this physical system is the vector sum of the individual linear momenta. psystem = p1+p2+p3+...=∑pi If the particles in our system interact with each other, they exert forces on each other, and there will be an impulse associated with each of these forces. Newton’s 3rd law tells us, however, that the impulse that particle a, for example, exerts on particle b is equal in magnitude and opposite in direction to the impulse exerted by particle b on particle a. And using the relation that the impulse is equal to a change in momentum of a particle, we see that the change in momenta of particles a and b due to their interaction will be equal in magnitude, but opposite in direction. Generalizing the above argument to interactions between any of the particles within the system, we see that if the momentum of one particle changes a certain amount, another particle’s momentum changes the same amount in the opposite direction. Thus, when we sum over all the momenta of the system, the total momentum of the system does not change in response to interactions among the particles within the system. However, if the particles of our system interact with particles (objects) outside the system, then the total momentum of the system might change. The figure below shows some of the forces that might be acting on the particles of the system. Some, labeled int (for internal) don’t change the total momentum of the system. The forces labeled ext (for external) do change the momentum of the system. 58 Chapter 7 Momentum Conservation: Linear Momentum Model

F ext on c c F int b on c d

Fint c on b b Fint a on b a Fint b on a

Of the various impulses shown in the figure, only the impulse caused by Fext on c causes a change in momentum of the system of particles.

Statement of Conservation of Momentum So, for a system of particles (objects) it is useful to write the impulse/momentum relation in a way that emphasizes the external interaction:

Net Impulseext = Jext = ∫ ΣFext(t) dt = pf - pi = ∆psystem A system acted on by external forces undergoes a change in total linear momentum equal to the net impulse (total impulse) of the external forces.

← We can rephrase the relationship stated above as a conservation princip←le for the total momentum of a system of particles. ← If the net external impulse acting on a system is zero, then there is no change in the total linear momentum of that system; otherwise, the change in momentum is equal to the net external impulse. This statement is an expression of Conservation of Linear Momentum. The total linear momentum of a system of objects remains constant as long as there is no net impulse due to forces that arise from interactions with objects outside the system. It does not matter that the objects of the system interact with each other and exert impulses on each other. These internal impulses cause changes in the individual momenta of the objects, but not the sum or total momentum of the system of objects. We can rephrase this discussion in terms of open and closed systems: 1) Closed system - A closed system does not interact with its environment so there is no net external impulse. The total momentum of a closed system is conserved. That is, the total momentum of the system remains constant. 2) Open system - An open system interacts with its environment, so that it can exchange both energy and momentum with the environment. For an open system the change in the total momentum is equal to the net impulse added from the environment–from objects outside the system.

Chapter 7 Momentum Conservation: Linear Momentum Model 59

Applications of Momentum Conservation As an example of the use of the application of momentum conservation, consider the collision of two automobiles. The total momentum of the system of two autos is the sum of the individual momenta before the collision. If the forces exerted by the road in the horizontal direction are small compared to the forces exerted by the autos on each other (usually a very good approximation), then the momentum of the system of two autos is conserved. That is, it is the same immediately after the collision as it was before. Why? Because the forces the two cars exert on each other are internal forces and don’t contribute an external impulse. Note that this would not be true if a car hits the proverbial brick wall, since we would typically take the system to be the car, and the brick wall would exert a net external impulse to the system. The diagram shows a not quite head-on collision between car (a) moving to the right with a faster and/or heavier car moving to the left: Before p p a i bi a b

p tot i After p p tot tot p i f af a

p b tot p f bf Regardless of what happens during the collision, as long as the road exerts a negligible impulse during the collision (compared to the impulses exerted by the colliding autos on each other), the total momentum of the two autos immediately before the collision equals the total momentum immediately after the collision. Note that in the example shown in the figure, the total momentum before and after the collision is shown in the right part of the diagram, and the vectors are equal. They are equal in spite of the fact that after the collision, the autos bounce off at an angle wrt (with respect to) the original direction of motion. The components of momenta in the perpendicular direction cancel each other out, since there was no momentum in the perpendicular direction before the collision. But before we explore momentum transfers more closely, we want to examine collisions in general. We especially want to bring energy conservation as well as momentum conservation into the analysis, so we can use these powerful conservation laws together. We will see that together, these conservation laws enable us to answer many (if not most) questions that arise in collisions, whether they be collisions of cars or galaxies or the elementary particles physicists study in the collisions in particle accelerators. And we can do this without having to know any details of the actual forces that act during the collision or the details of how the motion actually changed during the collision. That is, we do it with a “before and after” approach, not a detailed analysis of the forces and motion approach, which we will take up later in this chapter. With the combination of energy and momentum conservation, we have an extremely powerful and general method of analyzing many physical phenomena. There are, however, some important questions that can’t be answered without using a detailed analysis of forces and motion.

60 Chapter 7 Momentum Conservation: Linear Momentum Model Collisions: Momentum & Energy Conservation We just saw that if the external forces are negligible in a collision, the total momentum is conserved. What about energy? If, during a collision, kinetic energy is not converted to thermal energy or into deforming the objects (bond energy), then kinetic energy is also conserved. That is, the kinetic energy of the system before the collision will equal the kinetic energy after the collision. When two cars crash into each other, kinetic energy is usually not conserved (unless they are “bumper cars” with heavy spring bumpers that convert kinetic energy to spring (elastic) potential energy and then back to kinetic energy). In the example shown in the previous figure, the kinetic energy just before the collision is much greater than the kinetic energy immediately after the collision. We can see this, by recognizing that kinetic energy is proportional to the square of the magnitude of the momentum vector (length of the vector). The magnitudes of the two arrows representing initial momenta are much longer than the magnitudes of the two arrows representing final momenta. Much of the kinetic energy must have gone into deformation of the cars and into thermal energy during the collision. If the collision is between two protons or two billiard balls, kinetic energy might be exactly or almost conserved. Collisions in which kinetic energy is conserved are called elastic collisions.

Elastic Collisions In an elastic collision between two objects (particles) both the momentum and kinetic energy are conserved. That is, the values of the total momentum and kinetic Elastic Collision energy of the system before the collision are equal to the For Equal Mass Objects values they have after the collision. This gives us an Before equation relating the squares of the speeds of the objects p 1 = 0 p from KE conservation, and a vector equation (one i 2i equation for each dimension) relating the velocity 1 2 components from linear momentum conservation. After KEi = KEf and ptoti = ptotf p p = 0 As an example, consider the one-dimensional 1f 2f collision of two identical billiard balls. Suppose the first 1 2 ball is at rest and is hit “head on” by a second ball which p = p has velocity v. What are the velocities of the two balls 1f 2i after the collision? Both momentum and energy conservation hold so both equations above must be satisfied. Writing them out in detail we have:

1 2 1 2 1 2 2 mv2i = 2 mv2f + 2 mv1f

m v2i = m v2f + m v1f

The only solution of these two simultaneous equations is v1f = v2i and v2f = 0 That is, the second ball comes to rest and the first moves off with the same velocity the second ball had initially. This is illustrated in the accompanying figure. If the masses are not equal, both objects will have non-zero velocity after the collision. When the collision involves motion in more than one dimension, we can write a momentum conservation equation for each component of the total momentum. The algebra might get a little messy, but the idea is pretty straightforward. Chapter 7 Momentum Conservation: Linear Momentum Model 61

Inelastic Collisions In an inelastic collision between two objects kinetic energy is not conserved, so we can not equate initial and final kinetic energies. However, an interesting special case occurs when the collision is “completely inelastic” so that the objects stick together. Then they both have the same final velocity after the collision.

Momentum Conservation Model Summary One way of summarizing the main ideas in a model/approach is to list the (1) constructs, i.e., the “things” or ideas that are “used” in the model, (2) the relationships–in mathematical or sentence form–that connect the constructs in meaningful ways, and (3) the ways of representing the relationships. During your study of the model/approach you should have developed a good understanding of the meaning of each of the constructs. Some of these constructs probably start out as nothing but memorized definitions, but eventually take on a deeper meaning. The relationships might also start out as nothing more meaningful than a simple equation relating some of the constructs, e.g., J = ΣF ∆t = ∆p. By the time you finish this part of the course, however, you should understand this particular relationship, for example, as expressing one of the most fundamental, universal, and widely applicable principles in all of physics. Developing a deep and rich understanding of the relationships in a model/approach comes slowly. It is absolutely not something you can memorize. This understanding comes only with repeated mental effort over a period of time. A good test you can use to see if you are “getting it” is whether you can tell a full story about each of the relationships. It is the meaning behind the equations, behind the simple sentence relationships, that is important for you to acquire. With this kind of understanding, you can apply a model/approach to the analysis of phenomena you have not thought about before. You can reason with the model.

Listed here are the major, most important constructs, relationships, and representations of the momentum conservation model. Constructs Velocity, v Momentum, p Net Force, ΣF Impulse, J Newton’s 3rd law Conservation of momentum 62 Chapter 7 Momentum Conservation: Linear Momentum Model Relationships dr Δr The velocity is the time derivative of the displacement: v = or v = dt average Δt The linear momentum of an object measured in some coordinate system is simply the product of the object’s mass and velocity: p = mv € € The linear momentum of a system of particles is the vector sum of the individual momenta: psystem = ∑pi The net force acting on an object (physical system) is the vector sum of all forces acting on that object (physical system) due to the interactions with other objects (physical systems).

ΣFA = FB on A + FC on A + FD on A + … The impulse of the total (or net) external force acting on a system equals the product of the average force and the time interval during which the force acted.

Net Impulseext = J = ΣFavg ext∆t = ∫ ΣFext(t) dt The force (impulse) exerted by object A on object B is equal and opposite to the force (impulse) exerted by object B on object A.

FA on B = – FB on A and JA on B = – JB on A

Conservation of Linear Momentum If the net external impulse acting on a system is zero, then there is no change in the total linear momentum of that system; otherwise, the change in momentum is equal to the net external impulse.

Net Impulseext = J = ∫ ΣFext(t) dt = pf - pi = ∆psystem Collisions The momentum of the system of objects (particles) remains constant if the external impulses are negligible. This is true whether the collision is elastic or inelastic

ptoti = ptotf If a collision is elastic, then none of the mechanical energy is transferred to bond or thermal energies and both the total mechanical energy (all kinetic and elastic energies) and the momentum remain constant.

(mechanical energy)i = (mechanical energy)f and ptoti = ptotf Representations Graphical representation of all vector quantities and (vector relationships) as arrows whose length is proportional to the magnitude of the vector and whose direction is in the direction of the vector quantity. Algebraic vector equations. Vectors denoted as bold symbols or with small arrows over the symbol. Component algebraic equations, one equation for each of the three independent directions. Chapter 7 Momentum Conservation: Linear Momentum Model 63 A useful way to organize and use the representations of the various quantities that occur in phenomena involving momentum, change in momentum, and impulse and forces is a momentum chart. The momentum chart, like an energy-system diagram, helps us keep track of what we know about the interaction, as well as helping us see what we don’t know. The boxes are to be filled in with scaled arrows representing the various momenta and changes in momenta.

Closed System Typically used for collisions/interactions involving two or more objects. Closed System pi ∆p pf

Object 1

Object 2

Total 0 System

For total system: ∆p = 0 For each object: pi + ∆p = pf (written as component equations, if useful)

Write expressions for each momentum vector, such as p = mv

Open System Typically used when the phenomenon involves a net impulse acting on the system. Open System pi ∆p pf Total System

For total system: ∆p = J pi + ∆p = pf (and for component equations, if useful)

Write expressions for each momentum vector, such as p = mv

Below the momentum chart draw a force diagram for the object. The net force gives the direction of the impulse and ∆p. 64 Chapter 7 Momentum Conservation: Angular Momentum Model Angular Momentum Conservation Model (Summary on foldout #7 at back of text)

Overview The ideas we have developed for linear momentum and impulse apply to rotational motion as well. But first, we will need to develop the rotational analogs of the various variable and constructs we have been using. Force, momentum, velocity, impulse all have rotational analogs. The concept that impulse equals change in linear momentum has its analog in rotational motion as does the principle of conservation of momentum. In the last model, we focused both on the properties of forces and the momentum transfers governing the connection of force to motion. We found that forces can be rather tricky to deal with, and we, hopefully, began to appreciate the usefulness of being very precise about technical terminology as it relates to force and motion and to the usefulness of representations such as momentum charts and force diagrams. Now we extend the formalism to enable us to analyze and make sense of the motion of extended objects that can rotate as well as translate. We also introduce the last conserved quantity that we will work with, angular momentum (which could also be called rotational momentum). We will introduce a couple of additional concepts: torque and rotational inertia as well as to ways to describe rotational motion. We will then be in a position to answer detailed questions and make specific predictions about the magnitudes of individual forces and the changes in motion caused by the applied forces in a wide variety of situations. Angular momentum is analogous to momentum (translational or linear momentum) even though they are quite different physical quantities. For instance, we found in the last model that the momentum of an object is conserved if there is no net external force acting on it. In this model we will find that the rotational analogue of force is called torque and that the angular momentum of an object is conserved if there is no net external torque acting on it (even if there is a net force). Similarly, a transfer of angular momentum is called angular impulse. Remember from earlier chapters that work is the integral of the applied force over the distance the system moves. In this model we broaden our idea of work a little by including the energy transferred if a torque is applied over the angle that the system rotates. However, translational or linear momentum (usually just called momentum) and angular momentum are clearly very different physical quantities and you will have to work hard and be careful at keeping them separate in your thinking. The difference is obvious when you see a physical situation but, when discussing abstract ideas without a physical picture in mind, it is easy to confuse the two quantities. For instance, a ball may be spinning (i.e. have angular momentum) and flying through the air in a straight line (i.e. have momentum). Or, it may be spinning at any speed (have any angular momentum) and not be flying through the air. Or, it may be flying through the air but not spinning at all. So, you see that the amount of angular momentum the ball has is completely independent of its momentum. The moral of this little story is the same as with all physics problems: try to keep a concrete physical picture in your head as you learn new abstract ideas. Chapter 7 Momentum Conservation: Angular Momentum Model 65 The Center of Mass Idea You may have realized by now that modeling objects as point particles is a rather drastic oversimplification, but often very useful. When does the extended geometry of a non-point object become important? Focusing on just one point of an object can describe perfectly adequately the translational motion of that object, but it does not tell us anything about the object’s rotation. Whether an object rotates or not, depends on where forces are applied to the object. We will not derive or prove the general result, described in the following paragraphs, that we use to handle this situation: combined translation and rotation of rigid objects. We will simply state it. It turns out that we can consider all of the forces acting on the object as if they acted at one point, the center of mass, as far as translation is concerned. That is, if we are concerned only about an object’s translation, it doesn’t matter where the forces act on the object. We can consider them all to act at a single point! This is truly a great simplification. We have been using this result throughout this course without making a “big deal” about it. The special point where we consider the forces to act is called the center of mass. It is the same as the center of gravity (where you can support the object and it won’t rotate) as long as the gravitational force is uniform. Near the surface of the Earth, for all objects of ordinary size, the gravitational force can certainly be considered uniform, so for all problems we consider, the center of mass and center of gravity are the same point. Now, what about rotations? To take into account the effect of applied forces on the rotation of an object, we have to know where the forces are applied. We use a new construct, the torque, τ, which takes into account the magnitude and direction of the applied force as well as its distance from the point or axis about which the object rotates. If objects are constrained to rotate about a particular axis, such as a wheel mounted to an axle, the torques are typically computed about that axis. If there is no constraint, torques should be computed about the center of mass, the point about which the object will rotate. In order to properly discuss the rotational analog of momentum, we need to develop a consistent way to describe rotational motion. We find an analogous set of rotational motion variables to translational motion variables. We will introduce these motional variables by looking at both the circular motion of a point object and the rotational motion of an extended object (an extended object has size, so it is not a point object). By dividing up the general motion of a rigid object into translation plus rotation, we can separately discuss the momentum (actually the translational momentum) and the angular momentum.

The Detailed Description of Rotational Motion—Rotational Kinematics We begin by developing some useful relationships to describe the motion of a point object. Rather than using rectangular coordinates to describe the position P of a particle moving in a circle, we find it convenient to use polar P r coordinates, r and θ. The coordinate r is the distance of the y θ point from an axis of rotation (the origin ϑ ); θ is the angular ϑ x displacement from an arbitrarily chosen axis that defines zero. As in the figure, θ is frequently measured from the positive x axis, but it could be measured from any reference line. 66 Chapter 7 Momentum Conservation: Angular Momentum Model

P When θ changes by an amount ∆θ, the particle moves an f amount ∆s along the circumference of the circle defined by the ∆s radius r. The arc length, ∆s is simply the product of r and ∆θ: r P ∆θ i ∆s = r∆θ r The instantaneous velocity of the point P is always ϑ tangential to the curve at that point. If we differentiate the displacement with respect to time to get the tangential velocity of this object, we get an expression that depends only on the time derivative of θ: ds dθ = v = r . dt tangential dt The time rate of change of the angular position, θ, is called the angular velocity or rotational velocity and is usually represented by the Greek letter ω ("omega"). dθ ω = . dt

The rotational velocity and tangential velocity are related by: v tangential= rω . The Units of θ and ω. The units of θ and ω are respectively an angle unit and an angle unit divided by time. We can use any units we want and that are useful for a particular application for θ and ω. Typical units are degrees, degrees/second; revolutions, revolutions/second or rpm or revolutions/hour, etc. The "natural" units, are, however, radians and radians per second. We must use radians and radians per second when we use the relations connecting v to ω, etc. Note that a “radian” is a rather “funny” kind of unit. For instance, radians multiplied by meters is just meters, not radian·meters. It is a useful word to put into sentences to tell us we are talking about angular motion (and to make phrases “sound right”), but it does not behave like a “real” unit such as meter or second. Note that so far we have been discussing a point object constrained to move in a circle. We can also describe the kinematics of any extended object (e.g. a baseball bat) that is rotating about a fixed origin (where we grip it) by θ and ω, as long as we define the polar coordinates about the fixed axis of rotation. Actually, we can use this same approach for objects that are rotating as well as moving translationally, if we define the polar coordinates about the “center of mass.” The Directions of θ and ω. Just as the translational variables position, Direction of θ and ω is given by the Right-Hand-Rule r, and velocity, v, have both direction and magnitude, so do the angular variables θ and ω. It is useful to treat these variables as vectors, θ and ω. What direction do these variables point? The only unique direction in space associated with a rotation is along the axis of rotation. So, if the axis of rotation rr gives the direction, we need only specify which way along the axis corresponds to a rotation direction Chapter 7 Momentum Conservation: Angular Momentum Model 67 particular direction of rotation. By convention, the direction is specified by the “right-hand- rule.” If you curl the fingers of your right hand in the direction of positive θ or the direction rotation is occurring, your thumb points in the direction (along the axis of rotation) of θ or ω. We will see several more examples of the right-hand-rule (RHR). When forces act on extended objects, they not only cause the object to change its translational motion, but can also cause it to change its rotational motion. That is, these forces can cause an angular acceleration as well as a translational acceleration. It turns out that it is not just the magnitude and direction of the force that is important in causing angular accelerations, but also where the force is applied on F an extended object. Torque is the construct that incorporates both the vector force as well as where it is applied to an object. r

ϑ The Rotational Analogs of Force, Momentum, Mass, and Impulse The Torque Construct; the rotational analog to force F F Consider a force F exerted tangentially on the rim of a wheel or radial disk. The rim is at a distance r from the axis of rotation. We can F formally define torque, represented by the Greek letter τ, in terms of tangential r the force F and the distance r: ϑ τ = r⊥F = rFtangential where r is sometimes referred to as the moment arm of this applied force—the further away from the axis a particular force is applied, the more torque is exerted, producing more change in rotational motion. Torque can be thought of as the “turning effectiveness of a force” or “rotational force.” What happens if the applied force is not purely tangential, as in the second figure? This force can be broken down into its tangential and radial components, Ftangential and Fradial. Note that the radial component Fradial of this force has no effect on the rotational motion of this disk! So, for any general force exerted a distance r from a rotation axis, it is only the tangential component of this force (Ftangential) that will affect rotational motion. The tangential component of the force can always be found with the appropriate trig function. If θ is the angle between the applied force, F, and r, the tangential component is Fsinθ . Torque, along with other angular variables, has vector properties. If we imagine the torque causing the object to rotate about an axis "direction" of τ perpendicular to the plane defined by the force and the moment arm, r, we can use the same right-hand-rule introduced for finding the direction of θ and ω to find the direction of the torque, τ. If you curl the fingers of your right hand in the direction of rotation that the torque would cause, then your thumb points in the direction of the torque. F r ϑ 68 Chapter 7 Momentum Conservation: Angular Momentum Model The Angular Momentum Construct; the rotational analog to momentum It is useful to consider the angular momentum of both a point object as well as the angular momentum of extended objects. In either case, we L need to be clear about the axis (or point) about which we are calculating the angular momentum. A particle with momentum p, located at position r from some point in space has angular momentum L about that point with a magnitude given p by r

L = r⊥p = rptangential

Note that the angular momentum is related to the linear momentum the same way as torque is related to force. Both L and τ depend on the choice of the point in space to which they are referenced. Like torque, angular momentum is a vector. Its direction is perpendicular to both r and p and is given by the RHR. If a system has many parts, its total angular momentum is the vector sum of the angular momenta of all the parts: Direction of θ, ω and L = L1 + L2 + L3 ... = ΣLi L A rigid object with rotational inertia I about some particular axis has an angular momentum about that same axis given by

L = I ω The direction of L is parallel to the direction of ω. These directions are shown in the figure. rr

rotation Rotational Inertia; the rotational analog to mass direction Recall that for translational motion an object with a large amount of inertia has a greater momentum than an object with a small amount of inertia, both moving at the same speed. Mass, m, is the measure of inertia in translational motion. The rotational motion analogy to inertia is rotational inertia (or rotational mass), or in very technical language, moment of inertia. With a given net torque, Στ,, different objects will experience different rotational accelerations. The rotational inertia of an object does not depend solely on the amount of mass in the object, but on how this mass is distributed about the axis of rotation. For the simplest case of a point mass m moving in a circle of radius r, its rotational inertia is given by: I = mr2 . This definition allows us to calculate the rotational inertia of any object, provided we know the position r of every portion of its mass as measured perpendicularly with respect to the axis of rotation:

2 2 2 2 . I = m1r1 + m2r2 + m3r3 +… = ∑ miri i

€ Chapter 7 Momentum Conservation: Angular Momentum Model 69 This looks a lot like calculus (which it is in the limit of infinitesimally small mass increments.) The table below gives the rotational inertia of several simple geometric shapes, as calculated in the limit of infinitesimal increments of mass using this equation.

Object Rotational inertia point mass m moving in radius I = mr2 r r

2 thin ring of mass m, radius r I = mr rotating about center

r

thin rod of mass m, length L 1 2 I = mL rotating about one end 3 perpendicular to the rod L

thin rod of mass m, length L 1 2 I = mL rotating about the center 12 perpendicular to the rod L

disk of mass m, radius r, 1 2 I = mr rotating about an axis 2 perpendicular to disk through r the center

sphere of mass m, radius r, 2 2 I = mr rotating about an axis through 5 the center r

thin hollow spherical shell of 2 2 I = mr mass m, radius r, about an axis 3 through the center r

As seen from the formulas in the table, objects with the same mass can have very different rotational inertias, depending on how the mass is distributed with respect to the axis of rotation. 70 Chapter 7 Momentum Conservation: Angular Momentum Model Also, it is possible for an object to change its rotational inertia (e.g., a gymnast tucking in or extending arms and legs), which can lead to dramatic results as net torques are applied. The rotational inertia of a composite object is the sum of the rotational inertias of each component, all calculated about the same axis. . I total = I1 + I2 + I3 + So for a ring and a disk stacked upon each other and rotating about the symmetry axis of both, the total rotational inertia is: Itotal = Iring + Idisk . € The SI units of rotational inertia are kg·m2.

Angular Impulse; the rotational analog to impulse The angular analog to impulse, is angular impulse:

AngJext =∫ τext(t) dt or, if the torque is constant with time, or we define an average torque, τavg

AngJext = τavg ∆t A Statement of Angular Momentum Conservation:

AngJext =∫ τext(t) dt = ∆Lsystem or

AngJext = τext ∆t = ∆Lsystem

If the net external angular impulse acting on a system is zero, then there is no change in the total angular momentum of that system; otherwise, the change in angular momentum is equal to the net external angular impulse.

Other Angular Counterparts Work We are familiar with the concept of work as a way that the energy of a system is changed. In terms of force and distance, work is:

W = ∫ F||dx where the parallel symbol reminds us that it is only the components of force and displacement in the same direction that contribute to the integral. A similar expression holds for the work done by a torque which acts through an angle:

W = ∫ τ||dθ The energy of a particular system can be changed by the process of a force exerted by an outside object doing work on an object in the system and/or by a torque exerted by an outside Chapter 7 Momentum Conservation: Angular Momentum Model 71 object doing work on an object within the system. In either case, the work can be positive (increases the energy of the system) or negative (decreases the energy of the system). If the force or torque is constant (or we assume an average force or torque), the integral is immediately performed and we have

W = F||∆x and W = τ||∆θ

Energy Systems The total energy of a system is the sum of all of the various energy systems, which can include both translational and rotational energy systems. During collisions among parts of a physical system, energy can be transferred among these separate systems. We have previously mentioned rotational kinetic energy. Another energy system with a rotational counterpart is elastic or spring potential energy. The elastic potential energy of a system described by a spring constant k is: 2 PEelastic = 1/2kx Similarly, the elastic potential energy of a rotating system which has a linear restoring force is given by the expression: 2 PEelastic = 1/2kθ

The Rate of Energy Transfer: Power We previously discussed power as the time derivative of energy transfer, or the rate at which the energy of a system changes. This applies, of course, to any type of energy system. Recall that the SI unit of power is the watt (W) which is equal to a joule per second. In mechanical systems, in which energy is transferred as work, it is often useful to consider the rate of energy transfer, power, associated with a particular force. Since the energy transferred is the work done by the force, the power associated with that force is the time derivative of the work. If the force is constant in time, then:

P = dW/dt = d(F||avgx)/dt = F||avg dx/dt = F||avg v Thus, the power is simply the applied force times the velocity of the object the force is acting on. The rotational counterpart is:

P = τ||ω Putting it all together The chart on the next page shows all of the linear motion and dynamic variables along with their rotational counterparts. Keep this chart out and handy for ready reference to help you from getting “lost” in all the symbols. You should make sure that you recognize the meaning behind the symbols when you see on of these relationships. (Note: acceleration, angular acceleration, and Newton’s second law are treated in detail in the next chapter, but are shown here for completeness and convenience) 72 Chapter 7 Momentum Conservation: Angular Momentum Model Summary Listing Fundamental Concepts Used in Mechanics Emphasizing Translational and Rotational Counterparts

Category Concept Translation Rotation Relation kinematic position x θ θ = arclength /r variables velocity v = dx/dt ω = dθ/dt ω = vt/r acceleration a = dv/dt α = dω/dt α = at/r

fundamental force/torque F τ τ = r⊥F = rFtang dynamic inertia m I I = Σ mr2 variables momentum p = mv L = Iω L = r⊥p = rptang

Elastic Energy 1/2kx2 1/2kθ2 Kinetic 1/2mv2 1/2Iω2 Energy

Energy Work W = ∫ F||dx W = ∫ τ||dθ

System All Kinetic and Potential Energies Energy plus Thermal

Energy ∆E = W + Q Conservation system all

Power P = dE/dt = F||v P = dE/dt =τ||ω

Momentum p = mv L = Iω L = r⊥p = rptang

Momentum Impulse J = ∫ Fdt AngJ = ∫τdt Momentum ∆p = Jext ∆L=angJext Conservation system system

Newton’s if ΣF = 0, then if Στ = 0, then 1st law ∆v = 0, ∆p = 0 ∆L = 0 Newton’s Newton’s ΣF = ma Στ = Iα or, ΣF = dp/dt Laws 2nd law or, Στ = dL/dt Newton’s F = -F 3rd law 1on2 2on1 J1on2 = -J2on1

Chapter 7 Momentum Conservation: Angular Momentum Model 73 Interesting Effects Involving Angular Momentum There are two fascinating aspects of angular motion that don’t exist for linear motion in quite the same way. The first is that the rotational inertia is readily changed, as for example, when a skater extends or pulls in a leg. The second is related to the fact that both p and L are vector quantities and can change in direction without changing in magnitude. When an ice skater begins to spin with a leg extended, there is only a small torque exerted on the skater by the ice. Thus, angular momentum diminishes rather slowly (she can spin for a long time). Now, if she pulls in her leg, her rotational inertia is reduced considerably, and her rotational velocity (spin velocity) increases considerably. This is most easily seen by writing the angular momentum as L = Iω and noting that if L remains almost constant, then the product Iω must remain constant. Another fascinating, and rather startling situation, is the change in direction of the angular momentum of a spinning object when it is acted upon by a torque that is not along the direction of the angular momentum vector itself. This is the weird behavior exhibited by a spinning top or gyroscope. The figure shows a bicycle wheel supported by a rope at the left end of a short axle. The second figure is the extended force diagram. The torque caused by the force of the Earth acting down at the center of gravity of the wheel produces a torque that is perpendicular to this force and the axle; it points into the figure. If the wheel is not spinning, it just falls, rotating about the pivot point, because this is the only point of support. However, when the Side wheel is spinning with angular momentum L0, the situation is much view different.

The third figure is a top view, showing the original angular momentum points into page vector, Li, the new angular momentum vector, Lf and the torque, τ. The torque acts for a time ∆t . We use the angular impulse equation to give the ϑ F change in angular momentum, E on wheel

τ∆t = ∆L = Lf - Li

or Lf = Li + τ∆t That is, the direction of the initial angular momentum, L , is changed by i Top View the presence of the angular impulse, and is moved to the direction shown by Lf. But if L is in a new direction, then the orientation of the wheel must have changed, because L is due to the spinning wheel and points along ω. This turning motion of the orientation of the wheel is called τ ∆t precession. Instead of falling, the wheel precesses. Of course, once the angular momentum (and the wheel) point in a new direction, the torque comes into play again, causing the wheel to precess still farther. In this fashion, the wheel is caused to precess in a horizontal circle about the Lf = Li + τ ∆t pivot point. Precession is analogous to the situation of a ball being twirled around in a circle on the end of a string. Why doesn’t the tension in the string pull the ball in toward the center of the circle? The answer is that it does, but the large tangential velocity also moves the ball in a direction tangent to the circle. The net result is that the ball travels in a circular path. If there were no large tangential velocity, the ball would indeed be pulled directly toward the center of the circle due to the tension in the string. A similar thing happens with the bike wheel. The torque causes 74 Chapter 7 Momentum Conservation: Angular Momentum Model a change in the direction of the large angular momentum of the spinning wheel. If the wheel did not have this large angular momentum, the torque would cause the wheel to tip over, or “fall down.” Angular Momentum Conservation Model Summary Just as we did for linear momentum conservation, we will summarize the main ideas of the angular momentum conservation model/approach by listing the (1) constructs, i.e., the “things” or ideas that are get “used” in the model, (2) the relationships–in mathematical or sentence form– that connect the constructs in meaningful ways, and (3) the ways of representing the relationships. Developing a deep and rich understanding of the relationships in a model/approach comes slowly. It is absolutely not something you can memorize. This understanding comes only with repeated hard mental effort over a period of time. A good test you can use to see if you are “getting it” is whether you can tell a full story about each of the relationships. It is the meaning behind the equations, behind the simple sentence relationships, that is important for you to acquire. With this kind of understanding, you can apply a model/approach to the analysis of phenomena you have not thought about before. You can reason with the model. Listed here are the major, most important constructs, relationships, and representations of the angular momentum conservation model. (Be sure to refer back to the chart on page 34 showing the analogous linear and angular constructs and relationships.) Constructs Angular Velocity, ω Rotational Inertia, I Angular Momentum, L Net Torque, Στ Angular Impulse, angJ Newton’s 3rd law Conservation of angular momentum Relationships The angular velocity is the time derivative of the displacement: dθ Δθ ω = or ω = dt average Δt The angular momentum of an object measured about some fixed axes is simply the product of the object’s rotational inertia and angular velocity: € € L = Iω The angular impulse of the total (or net) external torque acting on an object equals the product of the average torque and the time interval during which the torque acted.

Net Angular Impulseext = angJ = Στ avg ext∆t = ∫ Στext(t) dt The directions of torque, impulse, angular velocity, and angular momentum as determined by the right-hand rule Chapter 7 Momentum Conservation: Angular Momentum Model 75 The torque (angular impulse) exerted by object A on object B is equal and opposite to the torque (angular impulse) exerted by object B on object A.

τ A on B = – τ B on A and angJA on B = – angJB on A Conservation of Angular Momentum If the net external angular impulse acting on a system is zero, then there is no change in the total angular momentum of that system; otherwise, the change in angular momentum is equal to the net external angular impulse.

Net Angular Impulseext = angJ = ∫ Στext(t) dt = Lf - Li = ∆Lsystem Representations Graphical representation of all vector quantities and (vector relationships) as arrows whose length is proportional to the magnitude of the vector and whose direction is in the direction of the vector quantity. Algebraic vector equations. Vectors denoted as bold symbols or with small arrows over the symbol. Component algebraic equations, one equation for each of the three independent directions. A useful way to organize and use the representations of the various quantities that occur in phenomena involving angular momentum, change in angular momentum, and angular impulse and torques is an angular momentum chart, which is totally analogous to the linear momentum chart. The angular momentum chart helps us keep track of what we know about the interaction, as well as helping us see what we don’t know. The boxes are to be filled in with scaled arrows representing the various angular momenta and changes in angular momenta. Closed System Open System Typically used for interactions Typically used when the phenomenon involving two or more objects. involves a net angular impulse acting on the Closed system. System Li ∆L Lf Open System L ∆L L i f Object 1 Total

System Object 2 For total system: ∆L = angJ Total L + ∆L = L 0 i f System (and for component equations, if useful)

For total system: ∆L = 0 Write expressions for each momentum For each object: L + ∆L = L vector, such as L = Iω i f If appropriate, show the extended force (written as component equations, if useful) diagram that determines angJ = Στ∆t

Write expressions for each Make sketches if required to show directions momentum vector, such as L = Iω of various vectors and the application of the Make sketches if required to show right-hand rule. directions of various vectors 76 Chapter 7 Momentum Conservation: Wrap-up

Wrap-up

We have now developed approaches/models to enable us to use the three fundamental conservation laws of all of science: energy, momentum and angular momentum. The “before and after the interaction” approach, which now includes momentum and angular momentum, as well as energy, is extremely general and universally applicable. It is the foundation of equilibrium thermodynamics. It allows us to get answers to most questions we ask regarding the behavior of interacting systems, as long as we don’t need the time dependence of the dynamical variables. What are the limitations to the approaches we have developed these past two chapters? We have mentioned some of these before, but it is good to emphasize them again. We know from our prior studies in chemistry and from some of what we have done in this course, that strange things begin to happen when the systems we are studying get very small, the size of molecules and atoms. Energies become quantized. Atoms and molecules can absorb and emit only certain amounts of energy, not a continuous range. We saw how specific heat modes became frozen out at low temperatures in solids. Things also get weird when speeds become large. In this case, large means moving at speeds that begin to approach the speed of light—3 x 108 m/s. Both at very small scales and when things go fast, our approach breaks down and must be replaced by more complicated theories. But the primary variables in both quantum mechanics and in special relativity turn out to be energy, momentum, and angular momentum. There is something very special about these constructs. They apparently represent some of the most basic aspects of the universe. The fundamental ideas of conservation of energy, momentum, and angular momentum carry through all of the models we use to describe our universe. The concepts of energy, momentum, and angular momentum (and the conservation of energy and momentum) remain as we delve into the details of the microscopic and the realm of very high speeds, but we do have to make changes in our understanding of these constructs. Energy, momentum and angular momentum take on discreet values; i.e., they become quantized. When we go to high speeds momentum and energy become intertwined. Even the separate idea of mass conservation gets pulled into a unified mass-energy conservation principle. We will explore the quantum world a little further in Part 3, but you will need to explore the fascinating world of special and general relativity on your own or in more advanced courses.