1.5 Conservation laws

class: center, middle

# Engineering Physics - Fall 2024
## <p style="color: tomato;">(Energy)</p>
---

<div style="text-align: center;">
<h3 style="color: black; font-size: 20px;"> Momentum <strong>conservation</strong> is a consequence of:</h3>
<div style="color: darkgreen;"> \(\vec{F}_\text{net}=m\,\vec{a}\)   <a style="color: black">and</a>  
actio = reactio
</div>
</div>

<h4 style="color: tomato;text-align: center;margin-top: 20px;">Formula:</h4>  
<div style="color: black;text-align: center">\(\vec{P}_\text{initial}=\vec{P}_\text{final}\)</div>

<h4 style="color: tomato;text-align: center;margin-top: 20px;">Prose:</h4>  
<div style="color: black;text-align: center;font-size:12px">
  Regardless what happens in a closed box between <i>initial</i> and <i>final</i>, the sum of all momenta in
  the box equals a constant. Three equations which allow us to relate <strong style="color:dodgerblue;">states
   in motion</strong> without having to know the time dependence of the velocities, accelerations, and/or positions.
</div>

<div style="color: black;text-align: center">\(\vec{P}=\vec{\text{function}}(m_1,m_2,\ldots,\vec{v}_1,\vec{v}_2,\ldots)\) 
  <strong style="color:red">not</strong> of time!</div>

<div style="color: black;text-align: center;font-size:12px;margin-top:20px">
 and <strong style="color:red">not</strong> of location. Hence, we <strong style="color:dodgerblue;">cannot</strong> 
 use the equations to deduce any information about the position of masses!
</div>

<h3>The function which does that:</h3>

<div style="color: black;text-align: center">\(E=\text{function}(m_1,m_2,\ldots,\vec{v}_1,\vec{v}_2,\ldots,\vec{r}_1,\vec{r}_2,\ldots)\)

<div style="color: black;text-align: center;font-size:12px;margin-top:20px">
 is the <strong style="font-size: 150%;color: dodgerblue;">energy</strong>.
</div>

<ul style="font-size: 12px;">
<li> the energy function is a <strong>number</strong>, i.e., the conservation law is a single equation: \(E_\text{initial}=E_\text{final}\)</li>
<li> regardless of what changes nature undergoes inside an insulated box, this number E exists and does never change from the moment we "filled" the box.</li>
</ul>

---
##So what is this mysterious function?

<div class="l1">Let us classify:<div>

<div class="l2">energy of motion: <strong style="font-size150%">kinetic</strong> energy:   <span style="color:dodgerblue;">\(T(\vec{v}\text{elocities})\)</span></div>

<div class="l2">energy at a location <i>relative</i> to something: <strong style="font-size150%">potential</strong> energy:   <span style="color:dodgerblue;">\(U(\vec{r},\vec{v})\)</span></div>

<div class="l1">A pendulum swinging in an evacuated and insulated box close to the earth's surface.<div>

<div class="grid-container" style="grid-template-columns: auto auto;margin-top: 25px;">
  <div class="grid-item"><img src="./images/Pendulum.svg" style="display: block;margin-right: auto;max-height: 250px;"></div>
  <div class="grid-item" style="font-size: 18px;color: green;background-color: rgb(255, 255, 255);">
    <div class="l3">\(\vec{v}=0\)  at the turning/highest points  →  \(T=0\)</div>
    <div class="l3">maximal speed for  \(\theta=0\)  →  \(U=\text{max}\)</div>
<div class="l2">⇓</div>
<div class="l2">force transfers energy</div>
<div class="l2" style="font-size:10px">\(\begin{pmatrix}\Delta\text{Energy}_{T\leftrightarrow U} \text{when}\\\text{moved over distance}\end{pmatrix}=\begin{pmatrix}\text{force at each point}\\\text{over the distance}\end{pmatrix}\times
\begin{pmatrix}\text{distance}\end{pmatrix}\)</div>
  </div>
</div>

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<div style="text-align:center">
<h1>Conjecture:  <span style="color:tomato;">\(T=\frac{1}{2}m\vec{v}^2\)</span></h1>

<l2>and using   \(\vec{F}=m\vec{a}\)   we obtain   \(\frac{dT}{dt}=\vec{F}\cdot\vec{v}\)</l2>
<br><br>
<l3>Now <strong>You</strong> can show:</l3>

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<div class="grid-item" style="background-color: rgb(255, 255, 255);"><img src="./images/freefall.png" style="display: block;margin-right: auto;margin-left: auto; max-height: 250px;">
  <div class="l2">for a mass \(m\) falling freely vertically:</div>
  <div class="l2" style="color:dodgerblue;">\(\frac{1}{2}mv^2+mgy=\text{const.}\)</div>
</div>
<div class="grid-item" style="background-color: rgb(255, 255, 255);"><img src="./images/spring.png" style="display: block;margin-right: auto;max-height: 250px;">
  <div class="l2">for a mass \(m\) compressing a spring \(\kappa\) horizontally:</div>
  <div class="l2"  style="color:dodgerblue;">\(\frac{1}{2}mv^2+\frac{1}{2}\kappa x^2=\text{const.}\)</div>
</div>
</div>
</div>

---
<div style="text-align:center">

<h3>The classical work-energy relation:</h3>
<img src="./images/initial_final.svg" style="max-height: 60px;">

<div class="l1">
<span style="color:tomato;font-size: 150%;">
\[
T_\text{f(inal)}-T_\text{i(nitial)}=\int_i^f\vec{F}_\text{total}\cdot d\vec{s}
\]
</span>

<span style="font-size: 150%;color: black;">
\[
\Downarrow
\]
</span>

<span style="color:black;font-size: 120%;">
\[
T_\text{i}+\color{dodgerblue}{U_\text{i}^\text{grav}}+\color{dodgerblue}{U_\text{i}^\text{Hooke}}=T_\text{f}+\color{dodgerblue}{U_\text{f}^\text{grav}}+\color{dodgerblue}{U_\text{f}^\text{Hooke}}+\int_i^f\vec{F}_\text{external}\cdot d\vec{s}
\]
</span>

<span style="font-size: 95%;color: black;">
where we <strong>defined</strong> the <em style="color:dodgerblue;">potentials</em> as the change in energy induced <strong>solely</strong> by the respective force:
\begin{align}
\Delta E_\text{grav}&=&W\text{(ork)}_\text{grav}\quad\stackrel{\text{1-dim}}{=}&\quad\int_{y_i}^{y_f}(mg)dy=mg(y_f-y_i)=:\color{dodgerblue}{U^\text{grav}(y_f)}-\color{dodgerblue}{U^\text{grav}(y_i)}\\
\Delta E_\text{Hooke}&=&W\text{(ork)}_\text{Hooke}\quad\stackrel{\text{1-dim}}{=}&\quad\int_{x_i}^{x_f}(\kappa x)dx=\frac{\kappa}{2}(x_f^2-x_i^2)=:\color{dodgerblue}{U^\text{Hooke}(x_f)}-\color{dodgerblue}{U^\text{Hooke}(x_i)}\\
\end{align}
</span>

</div>

---
<div style="text-align:center">
<h3>A <span style="color:green;">greener</span>/<span style="color:blue;">more blue</span> earth?</h3>

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<div class="grid-item" style="background-color: rgb(220, 220, 220);">
<span style="font-size: 14px;">Your mission, should you be willing to accept it, will be to estimate the <em style="color:darkgreen;font-size:120%">amount</em> of water needed to be pumped from
the Gulf of Bengal up to the <em style="color:darkgreen;font-size:120%">altitude</em> of the Everest base camp such that the <em style="color:darkgreen;font-size:120%">potential energy gained</em> by the water supplies the <strong>world</strong> for <em style="color:darkgreen;font-size:120%">one</em> day. </span>
</div>
<div class="grid-item" style="background-color: rgb(255, 255, 255);">
  <figure>
<img src="./images/Schluchsee.jpg" style="max-height: 230px;">
<figcaption style="font-size:10px"><a href="https://en.wikipedia.org/wiki/Schluchsee">Schluchsee</a> storage reservoir, near St. Blasien, Germany.</figcaption>
</figure>
</div>
</div>
<br>
<span style="font-size: 75%;color: gray;">
<ul>
<li>Make appropriate assumptions which make the calculation as simple as possible</li>
<li> gather the data you need</li>
</ul>
</span>
</div>
---
<h3>Energy conserved?</h3>

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<div class="grid-item" style="background-color: rgb(255, 255, 255);">
<span style="color:dodgerblue;">state 1: </span>
</div>
<div class="grid-item" style="background-color: rgb(255, 255, 255);">
<img src="./images/h-bomb.png" style="max-height: 150px;">
</div>
<div class="grid-item" style="background-color: rgb(255, 255, 255);">
<span style="color:tomato;">state 2:</span>
</div>
<div class="grid-item" style="background-color: rgb(255, 255, 255);">
 <img src="./images/cloud.jpg" style="max-height: 250px;">
</div></div>