Translated 3.2.1-3.2.17

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+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
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+ <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span>
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+ Solutions&nbsp;of&nbsp;Savchenko Problems&nbsp;in&nbsp;Physics <br>
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+ <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $3.2.11.$ Find the frequency of small oscillations of the system described in problem <a href="../3.1.10">3.1.10</a>
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+ <h3>Solution</h3>
+ <p>
+ Newton's Second Law
+$$ m\ddot{x}(t)-F=0 $$
+Where we find the total external force from Coulomb's law
+$$ F=kqQ\left(\frac{1}{(L-x)^2}-\frac{1}{(L+x)^2}\right) $$
+Using the approximation for a small value
+$$ x =\frac{h}{R} \ll 1; \quad(1+x)^\alpha\approx 1+\alpha x $$
+
+$$ F=-\frac{4kqQLx}{(L^2-x^2)^2}\approx-\frac{4kqQx}{L^3} $$
+Harmonic oscillation equation
+$$ \ddot{x}(t)+\frac{4kqQx}{mL^3}x(t) $$
+We obtain the required frequency of small oscillations
+$$ \boxed{\omega=\sqrt{\frac{4kqQ}{mL^3}}=\sqrt{\frac{qQ}{m\pi\varepsilon_0L^3}}} $$
+</p>
+ <p style="text-align: right; font-style: italic; font-size: 14;">
+ Dzikan Mikita<br>
+Aliaksandr Kanashenka
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $$\omega=\sqrt{\frac{qQ}{m\pi\varepsilon_0L^3}}$$
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