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.