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| <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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> | | $3.2.11.$ 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> | | <h3>Solution</h3> |
| <p> | | <p> |
| Newton's Second Law | | Newton's Second Law |
| $$ m\ddot{x}(t)-F=0 $$ | | $$ m\ddot{x}(t)-F=0 $$ |
| Where we find the total external force from Coulomb's law | | Where we find the total external force from Coulomb's law |
| $$ F=kqQ\left(\frac{1}{(L-x)^2}-\frac{1}{(L+x)^2}\right) $$ | | $$ F=kqQ\left(\frac{1}{(L-x)^2}-\frac{1}{(L+x)^2}\right) $$ |
| Using the approximation for a small value | | Using the approximation for a small value |
| $$ x =\frac{h}{R} \ll 1; \quad(1+x)^\alpha\approx 1+\alpha x $$ | | $$ x =\frac{h}{R} \ll 1; \quad(1+x)^\alpha\approx 1+\alpha x $$ |
| | | |
| $$ F=-\frac{4kqQLx}{(L^2-x^2)^2}\approx-\frac{4kqQx}{L^3} $$ | | $$ F=-\frac{4kqQLx}{(L^2-x^2)^2}\approx-\frac{4kqQx}{L^3} $$ |
| Harmonic oscillation equation | | Harmonic oscillation equation |
| $$ \ddot{x}(t)+\frac{4kqQx}{mL^3}x(t) $$ | | $$ \ddot{x}(t)+\frac{4kqQx}{mL^3}x(t) $$ |
| We obtain the required frequency of small oscillations | | We obtain the required frequency of small oscillations |
| $$ \boxed{\omega=\sqrt{\frac{4kqQ}{mL^3}}=\sqrt{\frac{qQ}{m\pi\varepsilon_0L^3}}} $$ | | $$ \boxed{\omega=\sqrt{\frac{4kqQ}{mL^3}}=\sqrt{\frac{qQ}{m\pi\varepsilon_0L^3}}} $$ |
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| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Dzikan Mikita<br> | | Dzikan Mikita<br> |
| Aliaksandr Kanashenka | | Aliaksandr Kanashenka |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\omega=\sqrt{\frac{qQ}{m\pi\varepsilon_0L^3}}$$ | | $$\omega=\sqrt{\frac{qQ}{m\pi\varepsilon_0L^3}}$$ |
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