Translated 3.1.1-3.1.17
en/3.1.10.md
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| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
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| + | <title>A bead with charge q can move without friction along a stretched string of length 2 L with charges Q attached to its ends. Find the increment of potential energy when the bead moves x along the string from its center. Find that for small displacements the dependence of the increment of potential energy on x is quadratic. Find how much the bead of mass m will be displaced if it is given a small velocity v at the equilibrium position.</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> | ||
| + | $3.1.10^*.$ A bead with charge $q$ can move without friction along a stretched string of length $2 L$ with charges $Q$ attached to its ends. Find the increment of potential energy when the bead moves $x$ along the string from its center. Find that for small displacements the dependence of the increment of potential energy on $x$ is quadratic. Find how much the bead of mass $m$ will be displaced if it is given a small velocity $v$ at the equilibrium position. | ||
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| + | loading="lazy" width="230" /> | ||
| + | <figcaption> | ||
| + | For problem $3.1.10^*$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | Initial energy of the electric field | ||
| + | $$ U_1=\frac{2kQq}{L} $$ | ||
| + | Energy of the system after the bead is displaced | ||
| + | $$ U_2=\frac{kQq}{L-x}+\frac{kQq}{L+x} $$ | ||
| + | Change in energy of the system | ||
| + | $$ \Delta U=U_2-U_1=kQq\left(\frac{1}{L-x}+\frac{1}{L+x}-\frac{2}{L}\right) $$ | ||
| + | Let us consider the transformation for small values of $(1+\xi)^\alpha=1+\alpha\xi$, where $\xi\ll1$ | ||
| + | $$ \boxed{\Delta U \approx\frac{Qqx^2}{2\pi\varepsilon_0L^3}} $$ | ||
| + | From the principle of conservation of energy it follows that the kinetic energy will be equal to the change in electrical energy $\Delta U$ | ||
| + | $$ E_k=\Delta U\Leftrightarrow \frac{m\upsilon^2}{2}=\frac{Qq\Delta x^2}{2\pi\varepsilon_0L^3} $$ | ||
| + | Where do we find the change in length $\Delta x$ | ||
| + | $$ \boxed{\Delta x=\upsilon\sqrt{\frac{m\pi\varepsilon_0L^3}{Qq}}} $$ | ||
| + | |||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$\Delta U=\frac{Qqx^2}{2\pi\varepsilon_0L^3};\quad\Delta x=\upsilon\sqrt{\frac{m\pi\varepsilon_0L^3}{Qq}}$$ | ||
| + | </p> | ||
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| <meta name="description" content="A bead with charge q can move without friction along a stretched string of length 2 L with charges Q attached to its ends. Find the increment of potential energy when the bead moves x along the string from its center. Find that for small displacements the dependence of the increment of potential energy on x is quadratic. Find how much the bead of mass m will be displaced if it is given a small velocity v at the equilibrium position."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A bead with charge q can move without friction along a stretched string of length 2 L with charges Q attached to its ends. Find the increment of potential energy when the bead moves x along the string from its center. Find that for small displacements the dependence of the increment of potential energy on x is quadratic. Find how much the bead of mass m will be displaced if it is given a small velocity v at the equilibrium position."> | |||
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| <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> | |||
| $3.1.10^*.$ A bead with charge $q$ can move without friction along a stretched string of length $2 L$ with charges $Q$ attached to its ends. Find the increment of potential energy when the bead moves $x$ along the string from its center. Find that for small displacements the dependence of the increment of potential energy on $x$ is quadratic. Find how much the bead of mass $m$ will be displaced if it is given a small velocity $v$ at the equilibrium position. | |||
| </p> | |||
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| <img src="https://savchenkosolutions.com/3/3.1.10/3.1.10.png" | |||
| loading="lazy" width="230" /> | |||
| <figcaption> | |||
| For problem $3.1.10^*$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| Initial energy of the electric field | |||
| $$ U_1=\frac{2kQq}{L} $$ | |||
| Energy of the system after the bead is displaced | |||
| $$ U_2=\frac{kQq}{L-x}+\frac{kQq}{L+x} $$ | |||
| Change in energy of the system | |||
| $$ \Delta U=U_2-U_1=kQq\left(\frac{1}{L-x}+\frac{1}{L+x}-\frac{2}{L}\right) $$ | |||
| Let us consider the transformation for small values of $(1+\xi)^\alpha=1+\alpha\xi$, where $\xi\ll1$ | |||
| $$ \boxed{\Delta U \approx\frac{Qqx^2}{2\pi\varepsilon_0L^3}} $$ | |||
| From the principle of conservation of energy it follows that the kinetic energy will be equal to the change in electrical energy $\Delta U$ | |||
| $$ E_k=\Delta U\Leftrightarrow \frac{m\upsilon^2}{2}=\frac{Qq\Delta x^2}{2\pi\varepsilon_0L^3} $$ | |||
| Where do we find the change in length $\Delta x$ | |||
| $$ \boxed{\Delta x=\upsilon\sqrt{\frac{m\pi\varepsilon_0L^3}{Qq}}} $$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$\Delta U=\frac{Qqx^2}{2\pi\varepsilon_0L^3};\quad\Delta x=\upsilon\sqrt{\frac{m\pi\varepsilon_0L^3}{Qq}}$$ | |||
| </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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