Translated 9.1.6-9.1.15
en/9.1.8.md
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| + | <title>A square frame with current is located in a uniform magnetic field of induction B. The frame mass is m, and the current in it is I. Determine the frequency of free oscillations of the frame around the OO^\prime axis.</title> | ||
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| + | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | ||
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| + | Solutions of Savchenko Problems in Physics <br> | ||
| + | <i><b>knowledge must be free</b></i> | ||
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| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#9.1">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $9.1.8^*.$ A square frame with current is located in a uniform magnetic field of induction $B$. The frame mass is $m$, and the current in it is $I$. Determine the frequency of free oscillations of the frame around the $OO^\prime$ axis. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/9/9.1.8/statement.png" | ||
| + | loading="lazy" width="200" /> | ||
| + | <figcaption> | ||
| + | For problem $9.1.8^*$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | Let's consider a small deviation from equilibrium by an angle $\alpha$: </p><p> | ||
| + | The moment acting on the frame from the magnetic field: | ||
| + | $$ M = BIS\cdot \sin\alpha = BIa^2 \sin\alpha $$ | ||
| + | where $a$ is the side of the frame</p><p> | ||
| + | gravity does not create a moment as it passes through the point of rotation</p><p> | ||
| + | From the basic equation of rotational motion: | ||
| + | $$ J\ddot{\alpha} = \sum\pm{M_i} $$ | ||
| + | for our case: | ||
| + | $$ J\ddot{\alpha} = -BIa^2 \sin\alpha\quad (1) $$ | ||
| + | now we will find the moment of inertia $J$ of the frame during the observed oscillations</p><p> | ||
| + | The moment of inertia of the upper and lower sides of the frame is | ||
| + | $$ J_1=J_2 = \frac{m}{4}\cdot \left(\frac{a}{2}\right)^2 = \frac{m a^2}{16}\quad (2) $$ | ||
| + | where $a/2$ is the distance to this side.</p><p> | ||
| + | Moment of inertia of the lateral sides: | ||
| + | $$ J_3 = J_4 = \frac{1}{12} \cdot \frac{m}{4}\cdot a^2 = \frac{m a^2}{48}\quad (3) $$ | ||
| + | We find the moment of inertia $J$ by substituting $(2)$ and $(3)$ | ||
| + | $$ J = J_1 + J_2 + J_3 + J_4 $$ | ||
| + | |||
| + | $$ J = 2\cdot \frac{m a^2}{16} + 2 \cdot \frac{m a^2}{48} = \frac{m a^2}{6} $$ | ||
| + | then equation $(1)$ will look like: | ||
| + | $$ \frac{m a^2}{6}\ddot{\alpha} = -BIa^2 \sin\alpha $$ | ||
| + | |||
| + | $$ \ddot{\alpha} = -\frac{6BI}{m} \alpha (t) $$ | ||
| + | Solving the equation of harmonic oscillations, we find the oscillation frequency | ||
| + | $$ \boxed{\omega = \sqrt{\frac{6BI}{m}}} $$ | ||
| + | |||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$\omega = \sqrt{\frac{6BI}{m}}$$ | ||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Andrei Yersh<br> | ||
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| <meta name="description" content="A square frame with current is located in a uniform magnetic field of induction B. The frame mass is m, and the current in it is I. Determine the frequency of free oscillations of the frame around the OO^\prime axis."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
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| <div id="logo"> | |||
| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | |||
| </div> | |||
| </a> | |||
| <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="../../#9.1">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $9.1.8^*.$ A square frame with current is located in a uniform magnetic field of induction $B$. The frame mass is $m$, and the current in it is $I$. Determine the frequency of free oscillations of the frame around the $OO^\prime$ axis. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/9/9.1.8/statement.png" | |||
| loading="lazy" width="200" /> | |||
| <figcaption> | |||
| For problem $9.1.8^*$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| Let's consider a small deviation from equilibrium by an angle $\alpha$: </p><p> | |||
| The moment acting on the frame from the magnetic field: | |||
| $$ M = BIS\cdot \sin\alpha = BIa^2 \sin\alpha $$ | |||
| where $a$ is the side of the frame</p><p> | |||
| gravity does not create a moment as it passes through the point of rotation</p><p> | |||
| From the basic equation of rotational motion: | |||
| $$ J\ddot{\alpha} = \sum\pm{M_i} $$ | |||
| for our case: | |||
| $$ J\ddot{\alpha} = -BIa^2 \sin\alpha\quad (1) $$ | |||
| now we will find the moment of inertia $J$ of the frame during the observed oscillations</p><p> | |||
| The moment of inertia of the upper and lower sides of the frame is | |||
| $$ J_1=J_2 = \frac{m}{4}\cdot \left(\frac{a}{2}\right)^2 = \frac{m a^2}{16}\quad (2) $$ | |||
| where $a/2$ is the distance to this side.</p><p> | |||
| Moment of inertia of the lateral sides: | |||
| $$ J_3 = J_4 = \frac{1}{12} \cdot \frac{m}{4}\cdot a^2 = \frac{m a^2}{48}\quad (3) $$ | |||
| We find the moment of inertia $J$ by substituting $(2)$ and $(3)$ | |||
| $$ J = J_1 + J_2 + J_3 + J_4 $$ | |||
| $$ J = 2\cdot \frac{m a^2}{16} + 2 \cdot \frac{m a^2}{48} = \frac{m a^2}{6} $$ | |||
| then equation $(1)$ will look like: | |||
| $$ \frac{m a^2}{6}\ddot{\alpha} = -BIa^2 \sin\alpha $$ | |||
| $$ \ddot{\alpha} = -\frac{6BI}{m} \alpha (t) $$ | |||
| Solving the equation of harmonic oscillations, we find the oscillation frequency | |||
| $$ \boxed{\omega = \sqrt{\frac{6BI}{m}}} $$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$\omega = \sqrt{\frac{6BI}{m}}$$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Andrei Yersh<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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