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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
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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> |
| | | |
| <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 |
| $$ x =\frac{h}{R} \ll 1; \quad(1+x)^\alpha\approx 1+\alpha x $$ | | $$ x =\frac{h}{R} \ll 1; \quad(1+x)^\alpha\approx 1+\alpha x $$ |
| | | |
| $$ F=-\frac{4kqQLx}{(L^2-x^2)^2}\approx-\frac{4kqQx}{L^3} $$ | | $$ F=-\frac{4kqQLx}{(L^2-x^2)^2}\approx-\frac{4kqQx}{L^3} $$ |
