Translated 3.1.1-3.1.17
en/3.1.8.md
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| + | <title>Two identical balls with charges \pm q, rigidly connected by a weightless rod of length l, are in an electric field, which acts on them with force \pm q E. Determine the mass of the ball if the amplitude of small transverse oscillations of the balls is x_0 and the maximum velocity of the balls is v_0.</title> | ||
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| + | <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | ||
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| + | <p class="author"> | ||
| + | Solutions of Savchenko Problems in Physics <br> | ||
| + | <i><b>knowledge must be free</b></i> | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#3.1">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $3.1.8.$ Two identical balls with charges $\pm q$, rigidly connected by a weightless rod of length $l$, are in an electric field, which acts on them with force $\pm q E$. Determine the mass of the ball if the amplitude of small transverse oscillations of the balls is $x_0$ and the maximum velocity of the balls is $v_0$. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/3/3.1.8/statement.png" | ||
| + | loading="lazy" width="230" /> | ||
| + | <figcaption> | ||
| + | For problem $3.1.8$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | Newton's second law for the normal force of the Coulomb, which creates a restoring moment of force, is found in projection onto the vertical axis | ||
| + | $$ ma_\tau=-qE\sin\varphi $$ | ||
| + | Since the angle $\varphi$ is negligibly small, we will use $\sin\varphi\approx\varphi$ | ||
| + | $$ -qE\varphi=-qE\frac{2x}{l} $$ | ||
| + | From where we obtain the equation of harmonic oscillations | ||
| + | $$ m\ddot{x}(t)+\frac{2qE}{l}x(t)=0 $$ | ||
| + | |||
| + | $$ \ddot{x}(t)+\frac{2qE}{ml}x(t)=0 $$ | ||
| + | The solution to which we know: | ||
| + | $$ \omega=\sqrt{\frac{2qE}{ml}} $$ | ||
| + | From where, similarly to <a href="../3.1.6">3.1.6</a>, we find the maximum speed | ||
| + | $$ \upsilon_0=x_0\omega=x_0\sqrt{\frac{2qE}{ml}} $$ | ||
| + | After mathematical transformations we obtain the mass of the ball | ||
| + | $$ \boxed{m=\frac{2qEx_0^2}{l\upsilon_0^2}} $$ | ||
| + | |||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$m=\frac{2qEx_0^2}{l\upsilon_0^2}$$ | ||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
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| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="Two identical balls with charges \pm q, rigidly connected by a weightless rod of length l, are in an electric field, which acts on them with force \pm q E. Determine the mass of the ball if the amplitude of small transverse oscillations of the balls is x_0 and the maximum velocity of the balls is v_0."> | |||
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| <title>Two identical balls with charges \pm q, rigidly connected by a weightless rod of length l, are in an electric field, which acts on them with force \pm q E. Determine the mass of the ball if the amplitude of small transverse oscillations of the balls is x_0 and the maximum velocity of the balls is v_0.</title> | |||
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| <header style="text-align:center;"> | |||
| <div id = "logo"> | |||
| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | |||
| </div> | |||
| <p class="author"> | |||
| Solutions of Savchenko Problems in Physics <br> | |||
| <i><b>knowledge must be free</b></i> | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#3.1">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $3.1.8.$ Two identical balls with charges $\pm q$, rigidly connected by a weightless rod of length $l$, are in an electric field, which acts on them with force $\pm q E$. Determine the mass of the ball if the amplitude of small transverse oscillations of the balls is $x_0$ and the maximum velocity of the balls is $v_0$. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/3/3.1.8/statement.png" | |||
| loading="lazy" width="230" /> | |||
| <figcaption> | |||
| For problem $3.1.8$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| Newton's second law for the normal force of the Coulomb, which creates a restoring moment of force, is found in projection onto the vertical axis | |||
| $$ ma_\tau=-qE\sin\varphi $$ | |||
| Since the angle $\varphi$ is negligibly small, we will use $\sin\varphi\approx\varphi$ | |||
| $$ -qE\varphi=-qE\frac{2x}{l} $$ | |||
| From where we obtain the equation of harmonic oscillations | |||
| $$ m\ddot{x}(t)+\frac{2qE}{l}x(t)=0 $$ | |||
| $$ \ddot{x}(t)+\frac{2qE}{ml}x(t)=0 $$ | |||
| The solution to which we know: | |||
| $$ \omega=\sqrt{\frac{2qE}{ml}} $$ | |||
| From where, similarly to <a href="../3.1.6">3.1.6</a>, we find the maximum speed | |||
| $$ \upsilon_0=x_0\omega=x_0\sqrt{\frac{2qE}{ml}} $$ | |||
| After mathematical transformations we obtain the mass of the ball | |||
| $$ \boxed{m=\frac{2qEx_0^2}{l\upsilon_0^2}} $$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$m=\frac{2qEx_0^2}{l\upsilon_0^2}$$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Dzikan Mikita<br> | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <p> | |||
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> | |||
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