# 01. Concepts and Principles

- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)# What is the Conservation Law?

In general, a conservation law is a statement that a certain quantity does not change over time. If you know how much of this quantity you have today, you can be assured that the exact same amount of the quantity will be available tomorrow. A famous (at least to physicists) explanation of the nature of a conservation law was given by Richard Feynman.

*Imagine your child has a set of 20 wooden blocks. Every day before bedtime, you gather up your child's blocks to put them away. As you gather up the blocks, you keep count in your head. Once you reach 20, you know you have found all of the blocks and it is unnecessary for you to search any longer. This is because the number of blocks is conserved. It is the same today as it was yesterday. *

*If one day you only find 18 blocks, you know to keep looking until you find the missing 2 blocks. Also, with experience, you discover the typical hiding places for the blocks. You know to check under the sofa, or behind the curtains. *

*If your child is rambunctious, you may even have to look outside of the room. Perhaps he threw a block or two out of the window. Even though blocks can disappear from inside of the room, and appear out in the yard, if you search everywhere you will always find the 20 blocks.*

Physicists have discovered a number of quantities that behave exactly like the number of wooden blocks. We will examine two of these quantities, energy and momentum, below.

# The Impulse-Momentum Relation

While Newton’s Second Law directly relates the total force that acts on an object at a specific time to the object’s acceleration at that exact same time, conservation laws relate the amount of a certain quantity present at one time to the amount present at a later time.

The first conserved quantity we will investigate is *momentum*. Of course, just because momentum is conserved doesn't mean that the momentum of any particular object is always constant. The momentum of a single object, like the number of blocks in the playroom, can change. Just as blocks can be thrown out of the window of the playroom, the momentum of a single object can be changed by applying *impulse* to it. The relationship between impulse and momentum is, conceptually,

*initial momentum + impulse = final momentum*

Momentum: P = mv [kg m/s]

Impulse: J = Ft [N sec]

Mathematically this is written as:

\[ mv_{i} +\sum(F(\Delta t))=mv_{f} \]

The product of mass and velocity is the momentum of the object, typically denoted P, and the product of force and the time interval over which the force acts is impulse, J. The summation symbol, \(\sum\), indicates that you must sum all of the impulses acting on the object to find the change in momentum.

In short, if no impulse is applied to an object, its momentum will remain constant. However, if an impulse is applied to the object, its momentum will change by an amount exactly equal to the impulse applied. This momentum does not disappear; however, it is simply transferred to the object *supplying* the impulse. In this sense, impulse is the transfer of momentum from one object to another, analogous to tossing blocks out of the playroom and into the yard.

# The Work-Energy Relation

The second conserved quantity we will investigate is *energy*. Energy can be defined as the ability to do work and has units of Joules (J) where a 1 J = 1 N m. Just like momentum, or wooden blocks, the conservation of energy doesn't mean that the energy of any particular object is always constant. The energy of a single object can be changed by doing *work *to it. The relationship between work and energy is, conceptually,

*initial energy + work = final energy*

The similarity between momentum and energy is not complete, however. While there is only one form of momentum (i.e., one hiding place for momentum “blocks”) there are several forms of energy. These different forms of energy will be introduced as you progress through more and more complicated models of the physical world. For now, the only “hiding place” I want to discuss is *kinetic energy*. In terms of kinetic energy, the above conceptual relationship between work and energy becomes, expressed mathematically,

\[ \frac{1}{2}mv^{2}_{i}+\sum(\left |F \right |\left |\Delta r \right | cos \theta )=\frac{1}{2}mv^{2}_{f} \]

Unlike anything we’ve studied up to this point, the work-energy relation is a *scalar* equation. This will become especially important when we study objects moving in more than one dimension. For now, all this means is that in the expression for work, \(\left |F \right |\left |\Delta r \right | cos \theta \), we should use the *magnitude* of the force and the *magnitude* of the change in position. This product is then multiplied by \(cos \theta\) where \(\theta\) is defined to be the angle between the applied force and the displacement of the object. If the force and displacement are in the same direction \(\theta = 0^{\circ}\) and the work is positive (the object gains energy). If the force and displacement are in the opposite direction \(\theta =180^{\circ}\) ,and the work is negative (the object loses energy). Note that the actual directions of the force and the displacement are unimportant, only their directions *relative to each other* affect the work.

In general, if no work is done to an object, its total (not simply kinetic) energy will remain constant. However, if work is done to the object, its total energy will change by an amount exactly equal to the work done. In this sense, work is the transfer of energy from one object to another, again analogous to tossing blocks out of the playroom and into the yard.