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<div id="content" class="content">
<h1 class="title"> <br> Uniform circular motion: the motion of an infinite number of circles
<br />
<span class="subtitle">Yuhao <br></span>
</h1>
<div id="outline-container-orgfa5422c" class="outline-2">
<h2 id="orgfa5422c">Intro</h2>
<div class="outline-text-2" id="text-orgfa5422c">
<ul class="org-ul">
<li>Along with the uniform motion, the uniform <i>circular</i> motion is one of the most amazing motions of all the motions.</li>
<li>The uniform motion is a motion of one straight line.</li>
<li>In contrast, the uniform circular motion is a motion of an infinite number of circles.</li>
</ul>
</div>
</div>
<div id="outline-container-org1b0e02e" class="outline-2">
<h2 id="org1b0e02e">The infinite number of circles</h2>
<div class="outline-text-2" id="text-org1b0e02e">
</div>
<div id="outline-container-org8b754c1" class="outline-3">
<h3 id="org8b754c1">The first circle</h3>
<div class="outline-text-3" id="text-org8b754c1">
<ul class="org-ul">
<li>Suppose an object is moving along a circle at a constant speed (by speed we mean the magnitude of the velocity, which includes not only the magnitude but also the direction)</li>
<li>In this case, we say the object is in uniform circular motion</li>
<li>The trajectory of the object forms the first circle</li>
<li><p>
To calculate the magnitude of the velocity, we divide the circumference of the first circle by the time it takes the object to complete one cycle of the circular motion
</p>
\begin{equation}
\label{org2454b7b}
v = \frac{2\pi r}{T}
\end{equation}</li>
</ul>
</div>
</div>
<div id="outline-container-org6935126" class="outline-3">
<h3 id="org6935126">The second circle</h3>
<div class="outline-text-3" id="text-org6935126">
<ul class="org-ul">
<li>The direction of the object's velocity is along the tangential at each point on the first circle</li>
<li>If we move the tail of the velocity vector to the origin, the head of the velocity vector traces a circle</li>
<li>This is the second circle</li>
<li>The acceleration is the instaneous change of the velocity</li>
<li>The magnitude of the acceleration is constant, since the motion is uniform</li>
<li>To calculate the magnitude of the acceleration, we divide the circumference of the second circle by the time it takes the object to complete one cycle of the circular motion</li>
</ul>
\begin{equation}
\label{org79ab77e}
a_c = \frac{2\pi v}{T}
\end{equation}
</div>
<div id="outline-container-orgcca1b6e" class="outline-4">
<h4 id="orgcca1b6e">Other forms of the formula</h4>
<div class="outline-text-4" id="text-orgcca1b6e">
<ul class="org-ul">
<li><p>
If we plug in \(v\) from Eq.\(~\)\eqref{org2454b7b} to Eq.\(~\)\eqref{org79ab77e}
</p>
\begin{equation}
a_c = \frac{2\pi v}{T} = \frac{2\pi \frac{2\pi r} {T}}{T} = \frac{4\pi^2 r}{T^2}
\end{equation}</li>
<li>From Eq.\(~\)\eqref{org2454b7b}, \(T = 2\pi r/ v\)</li>
<li><p>
Plugging this in Eq.\(~\)\eqref{org79ab77e}, we have
</p>
\begin{equation}
a_c = \frac{2\pi v}{T} = \frac{2\pi v}{\left(\frac{2\pi r}{v}\right)}{T} = \frac{v^2}{r}
\end{equation}</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-org8f1be99" class="outline-3">
<h3 id="org8f1be99">The thrid circle</h3>
<div class="outline-text-3" id="text-org8f1be99">
<ul class="org-ul">
<li>The direction of the acceleration of the object is along the tangential fo the second circle at each point on the second circle</li>
<li>If we move the tail of the acceleration vector to the origin, the head of the velocity vector traces a circle</li>
<li>This is the third circle</li>
<li>The acceleration of the acceleration, which we may call \(\overrightarrow{b_c}\), is the instantaneous change of the acceleration</li>
<li>The magnitude \(b_c\) is constant since the motion is uniform</li>
<li><p>
To calcualte \(b_c\), we devide the circumference of the second circle by the time it takes the object to complete one cycle of the circular motion
</p>
\begin{equation}
\label{org8f07024}
b_c = \frac{2\pi a_c}{T}
\end{equation}</li>
</ul>
</div>
</div>
<div id="outline-container-org93f0b92" class="outline-3">
<h3 id="org93f0b92">An infinite number of circles</h3>
<div class="outline-text-3" id="text-org93f0b92">
<ul class="org-ul">
<li>When we keep on with the process we get an infinite number of circles</li>
<li>The acceleration of the acceleration of the acceleration … can be calcualted</li>
<li><p>
The expresion for the nth one \(a_{cn}\) would be
</p>
\begin{equation}
\label{orgb9ce6f0}
a_{cn} = \frac{(2\pi)^n r}{T^n}
\end{equation}</li>
</ul>
<p>
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</p>
</div>
</div>
</div>
<div id="outline-container-org019c4cf" class="outline-2">
<h2 id="org019c4cf">Acknowledgement</h2>
<div class="outline-text-2" id="text-org019c4cf">
<p>
<i><b>Thanks for reading</b></i>
</p>
<div id="orgd1db018" class="figure">
<p><img src="./figures/Thank_you.jpeg" alt="Thank_you.jpeg" width="70%" />
</p>
</div>
</div>
</div>
</div>
</body>
</html>