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7.0: Overview

We began this course by focusing on the idea of physical systems, energy systems, and transfers of energy between different physical systems. In Part 1, 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: \(dW = F_{||}dx\).

We wrote a conservation of energy expression (a more general form of the 1st 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:

\[dE = dQ + dW. \tag{7.0.1}\]

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 Part 1 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.

Instead of calculating an energy transfer called work,

\[W = \int _{x_i} ^{x_f} F_{||} dx \tag{7.0.2} \]

or, in differential form

\[dW = F_{||}dx \tag{7.0.3} \]

We calculate a momentum transfer called impulse

\[J = \int_{t_1}^{t_2} F dt \tag{7.0.4} \]

from \(t_1\) to \( t_2\) or, in differential form

\[dJ = F dt. \tag{7.0.5} \]

We will write a conservation of momentum expression, \(\Delta p = J\), 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 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.

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