Added 7.1.12, 7.1.23, 7.3.9 & 11.5.11

astrosander edited
View solution History
revision #9814 GitHub 5aa3af1 newer →
en/7.1.23.md +120 −0
@@ -0,0 +1,120 @@
+<!DOCTYPE html>
+<html lang="en">
+
+<head>
+ <meta charset="utf-8">
+ <meta name="viewport" content="width=device-width, initial-scale=1.0">
+ <meta http-equiv="content-language" content="en">
+ <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
+ <meta name="description" content="Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere.">
+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
+ <meta property="og:title" content="Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere.">
+ <meta property="og:image" content="img/logo.png">
+ <meta property="og:description" content="Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere.">
+ <meta name="yandex-verification" content="6cfda41f74038368">
+ <title>Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere.</title>
+ <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css">
+ <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png">
+ <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script>
+ <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script>
+ <script type="text/x-mathjax-config">
+ MathJax.Hub.Config({
+ extensions: ['tex2jax.js'],
+ jax: ['input/TeX', 'output/HTML-CSS'],
+ tex2jax: {
+ inlineMath: [['$', '$'], ['$', '$']],
+ processEscapes: true,
+ processClass: 'tex2jax',
+ ignoreClass: 'html'
+ },
+ showProcessingMessages: false,
+ messageStyle: 'none'
+ });
+ </script>
+</head>
+ <body style="">
+ <header style="text-align:center;">
+ <a href="../../" style="text-decoration: none;">
+ <div id="logo">
+ <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>
+ </div>
+ </a>
+ <p class="author">
+ Solutions&nbsp;of&nbsp;Savchenko Problems&nbsp;in&nbsp;Physics <br>
+ <i><b>knowledge must be free</b></i>
+ </p>
+ </header>
+
+ <h3 id="back-link"><a href="../../#7.1">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $7.1.23^*.$ Find the period of small vibrations of a body of mass $m$, whose charge is $q$, inside a smooth sphere of radius $R$, if the charge $Q$ is fixed at the top point of the sphere.
+</p>
+<center>
+ <figure>
+ <img src="7.1.23.png"
+ loading="lazy" width="200" />
+ <figcaption>
+ For problem $7.1.23^*$
+ </figcaption>
+ </figure>
+</center>
+<p>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+
+</p>
+<center>
+ <figure>
+ <img src="7.1.23_1.png"
+ loading="lazy" width="200" />
+ <figcaption>
+ Forces acting on the system of two charges
+ </figcaption>
+ </figure>
+</center>
+<p>
+
+Considering the fact that oscillations are small, $\varphi \ll 1$, we could use the approximations for small angle $\varphi$
+$$\sin\varphi \approx \varphi; \quad\cos\varphi \approx 1\quad(1)$$
+From where, we could approximate that the distance between charges remains approximately the same
+$$r\approx2R$$
+Thus, the сoulomb's law for two charges
+$$F_c = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$
+Let's write Newton's second law on a tangential trajectory, in the direction of motion of a charge of mass $m$
+$$ma = mg \sin\varphi + F_c \sin\frac{\varphi}{2}$$
+Considering the approximation $(1)$ and the expression
+$$ma = -mg\frac{x}{R} - \frac{x}{2R} \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$
+$$a = -x\left(\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}\right)$$
+Solving the equation of harmonic oscillations $(\ddot{x}+\omega^2x(t)=0)$, we could obtain the value for angular velocity
+$$\omega=\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}$$
+From here the period of oscillation
+$$\boxed{T=\frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}}}$$
+Since the expression under the square root cannot be less than zero, the ratio at which there will be no oscillations
+$$\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3} > 0 \Leftrightarrow \boxed{\frac{qQ}{32\pi\varepsilon_0R^2}>-mg}$$
+ </p>
+
+ <h4>Answer</h4>
+ <div class="scroll-wrapper">
+ <p>
+ $$T=2\pi\left(\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}\right)^{-1/2}\text{ with }\frac{qQ}{32\pi\varepsilon_0R^2}>-mg$$
+ </p>
+ </div>
+
+
+ <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>
+ </p>
+ </footer>
+ </body>
+
+</html>