3.1.10"> 3.1.10"> 3.1.10"> Find the frequency of small oscillations of the system described in problem <a href="/%3Ca%20class="problem-ref" href="/en/3.1.10">3.1.10">3.1.10</a>

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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>

    <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
$$\boxed{\omega=\sqrt{\frac{4kqQ}{mL^3}}=\sqrt{\frac{qQ}{m\pi\varepsilon_0L^3}}}$$

Dzikan Mikita
Aliaksandr Kanashenka

    <h4>Answer</h4>
    <p>
        $$\omega=\sqrt{\frac{qQ}{m\pi\varepsilon_0L^3}}$$
    </p>


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