Updated spacing between @ latex expressions

astrosander edited Oct 6, 2024, 12:48 AM

View solution History

revision #10486 parent #9816 GitHub c702ebe ← older newer →

en/7.1.23.md +1 −1

			<!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 of Savchenko Problems in 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
		@@ -91,7 +91,7 @@
			$$ma = -mg\frac{x}{R} - \frac{x}{2R} \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$
			$$a = -x\left(\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}\right)$$
			Solving the equation of harmonic oscillations $(\ddot{x}+\omega^2x(t)=0)$, we could obtain the value for angular velocity
		−	$$\omega=\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}$$
		+	$$\omega =\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}$$
			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>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			Lutfulloyev Shukurullo<br>
			Physics Department<br>
			National University of Uzbekistan<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>
			</p>
			</footer>
			</body>

			</html>
		unchanged lines 29

	<!DOCTYPE html>		<!DOCTYPE html>
	<html lang="en">		<html lang="en">

	<head>		<head>
	<meta charset="utf-8">		<meta charset="utf-8">
	<meta name="viewport" content="width=device-width, initial-scale=1.0">		<meta name="viewport" content="width=device-width, initial-scale=1.0">
	<meta http-equiv="content-language" content="en">		<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="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="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="author" content="Aliaksandr Melnichenka">
	<meta name="date" content="2023-10" scheme="YYYY-MM">		<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: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: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 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">		<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>		<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="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css">
	<link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png">		<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 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 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">		<script type="text/x-mathjax-config">
	MathJax.Hub.Config({		MathJax.Hub.Config({
	extensions: ['tex2jax.js'],		extensions: ['tex2jax.js'],
	jax: ['input/TeX', 'output/HTML-CSS'],		jax: ['input/TeX', 'output/HTML-CSS'],
	tex2jax: {		tex2jax: {
	inlineMath: [['$', '$'], ['$', '$']],		inlineMath: [['$', '$'], ['$', '$']],
	processEscapes: true,		processEscapes: true,
	processClass: 'tex2jax',		processClass: 'tex2jax',
	ignoreClass: 'html'		ignoreClass: 'html'
	},		},
	showProcessingMessages: false,		showProcessingMessages: false,
	messageStyle: 'none'		messageStyle: 'none'
	});		});
	</script>		</script>
	</head>		</head>
	<body style="">		<body style="">
	<header style="text-align:center;">		<header style="text-align:center;">
	<a href="../../" style="text-decoration: none;">		<a href="../../" style="text-decoration: none;">
	<div id="logo">		<div id="logo">
	<span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>		<span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>
	</div>		</div>
	</a>		</a>
	<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>
	</p>		</p>
	</header>		</header>

	<h3 id="back-link"><a href="../../#7.1">$\leftarrow$Back</a></h3>		<h3 id="back-link"><a href="../../#7.1">$\leftarrow$Back</a></h3>

	<h3> Statement </h3>		<h3> Statement </h3>
	<p>		<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.		$7.1.23^*.$ 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>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="7.1.23.png"		<img src="7.1.23.png"
	loading="lazy" width="200" />		loading="lazy" width="200" />
	<figcaption>		<figcaption>
	For problem $7.1.23^*$		For problem $7.1.23^*$
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>
	</p>		</p>

	<h3>Solution</h3>		<h3>Solution</h3>
	<p>		<p>

	</p>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="7.1.23_1.png"		<img src="7.1.23_1.png"
	loading="lazy" width="200" />		loading="lazy" width="200" />
	<figcaption>		<figcaption>
	Forces acting on the system of two charges		Forces acting on the system of two charges
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>

	Considering the fact that oscillations are small, $\varphi \ll 1$, we could use the approximations for small angle $\varphi$		Considering the fact that oscillations are small, $\varphi \ll 1$, we could use the approximations for small angle $\varphi$
	$$\sin\varphi \approx \varphi; \quad\cos\varphi \approx 1\quad(1)$$		$$\sin\varphi \approx \varphi; \quad\cos\varphi \approx 1\quad(1)$$
	From where, we could approximate that the distance between charges remains approximately the same		From where, we could approximate that the distance between charges remains approximately the same
	$$r\approx2R$$		$$r\approx2R$$
	Thus, the сoulomb's law for two charges		Thus, the сoulomb's law for two charges
	$$F_c = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$		$$F_c = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$
	Let's write Newton's second law on a tangential trajectory, in the direction of motion of a charge of mass $m$		Let's write Newton's second law on a tangential trajectory, in the direction of motion of a charge of mass $m$
	$$ma = mg \sin\varphi + F_c \sin\frac{\varphi}{2}$$		$$ma = mg \sin\varphi + F_c \sin\frac{\varphi}{2}$$
	Considering the approximation $(1)$ and the expression		Considering the approximation $(1)$ and the expression
	@@ -91,7 +91,7 @@
	$$ma = -mg\frac{x}{R} - \frac{x}{2R} \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$		$$ma = -mg\frac{x}{R} - \frac{x}{2R} \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$
	$$a = -x\left(\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}\right)$$		$$a = -x\left(\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}\right)$$
	Solving the equation of harmonic oscillations $(\ddot{x}+\omega^2x(t)=0)$, we could obtain the value for angular velocity		Solving the equation of harmonic oscillations $(\ddot{x}+\omega^2x(t)=0)$, we could obtain the value for angular velocity
	$$\omega=\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}$$		$$\omega =\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}$$
	From here the period of oscillation		From here the period of oscillation
	$$\boxed{T=\frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}}}$$		$$\boxed{T=\frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}}}$$
	Since the expression under the square root cannot be less than zero, the ratio at which there will be no oscillations		Since the expression under the square root cannot be less than zero, the ratio at which there will be no oscillations
	$$\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3} > 0 \Leftrightarrow \boxed{\frac{qQ}{32\pi\varepsilon_0R^2}>-mg}$$		$$\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3} > 0 \Leftrightarrow \boxed{\frac{qQ}{32\pi\varepsilon_0R^2}>-mg}$$
	</p>		</p>

	<h4>Answer</h4>		<h4>Answer</h4>
	<div class="scroll-wrapper">		<div class="scroll-wrapper">
	<p>		<p>
	$$T=2\pi\left(\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}\right)^{-1/2}\text{ with }\frac{qQ}{32\pi\varepsilon_0R^2}>-mg$$		$$T=2\pi\left(\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}\right)^{-1/2}\text{ with }\frac{qQ}{32\pi\varepsilon_0R^2}>-mg$$
	</p>		</p>
	</div>		</div>
	<p style="text-align: right; font-style: italic; font-size: 14;">		<p style="text-align: right; font-style: italic; font-size: 14;">
	Lutfulloyev Shukurullo<br>		Lutfulloyev Shukurullo<br>
	Physics Department<br>		Physics Department<br>
	National University of Uzbekistan<br>		National University of Uzbekistan<br>
	</p>		</p>



	<footer class="row container">		<footer class="row container">
	<br>		<br>
	<p>		<p>
	<small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>		<small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
	</p>		</p>
	<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>		<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>		</p>
	</footer>		</footer>
	</body>		</body>

	</html>		</html>
	unchanged lines 29