Updated spacing between @ latex expressions

astrosander правка от 6 окт. 2024 г., 00:48

правка #10287 предыдущая #9260 GitHub c702ebe ← раньше

en/3.2.8.md +2 −2

			<!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="In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.</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="../../#3.2">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$3.2.8.$ In the vicinity of the ore deposit the pendulum oscillation period changed by $0.1 \%$. The density of the ore in the deposit is $8 \cdot 10^3$ $\frac{kg}{m^3}$ Estimate the size of the deposit if the average density of the Earth is $5.6 \cdot 10^3$ $\frac{kg}{m^3}$ and its radius is $6400$ km.
			</p>

			<h3>Solution</h3>
			<p>
			Let us write down, using the law of gravity, the equations for determining the acceleration of free fall, taking into account that the acceleration above the deposit will be greater than far from it.
			$$ g_{0}=\frac{4}{3}\pi RG\rho_{0}\approx 4\pi RG\rho_{0} $$

			$$ g=g_{0}+g_{k}\approx g_{0} + 4\pi RG(\rho - \rho_{0}) $$
		@@ -62,9 +62,9 @@
			Let us write down the ratio of the periods of oscillation of the pendulum given by the conditions of the problem
			$$ \alpha =\frac{T-T_{0}}{T_{0}} = 0,1;\quad\xi = \frac{T}{T_{0}}=1+10^{-3} $$
			The ratio of the periods is expressed through the values of the acceleration of gravity
		−	$$ \xi ^{2} = \frac{g_{0}+g}{g_{0}}=1+\frac{r(\rho -\rho _{0})}{R\rho _{0}} $$
		+	$$ \xi^{2} = \frac{g_{0}+g}{g_{0}}=1+\frac{r(\rho -\rho_{0})}{R\rho_{0}} $$

		−	$$ \boxed{r=\frac{(\xi ^{2}-1)R\rho _{0}}{\rho -\rho _{0}}\approx 30\text{ km}} $$
		+	$$ \boxed{r=\frac{(\xi^{2}-1)R\rho_{0}}{\rho -\rho_{0}}\approx 30\text{ km}} $$
			</p>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			Dzikan Mikita<br>
			Aliaksandr Kanashenka
			</p>

			<h4>Answer</h4>
			<p>
			$$r\approx30\text{ km}$$
			</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>
		ещё строк без изменений 21

	<!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="In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.">		<meta name="description" content="In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.">
	<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="In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.">		<meta property="og:title" content="In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.">
	<meta property="og:image" content="img/logo.png">		<meta property="og:image" content="img/logo.png">
	<meta property="og:description" content="In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.">		<meta property="og:description" content="In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.">
	<meta name="yandex-verification" content="6cfda41f74038368">		<meta name="yandex-verification" content="6cfda41f74038368">
	<title>In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.</title>		<title>In the vicinity of the ore deposit the pendulum oscillation period changed by 0.1 \%. The density of the ore in the deposit is 8 \cdot 10^3 \frac{kg}{m^3} Estimate the size of the deposit if the average density of the Earth is 5.6 \cdot 10^3 \frac{kg}{m^3} and its radius is 6400 km.</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="../../#3.2">$\leftarrow$Back</a></h3>		<h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3>

	<h3> Statement </h3>		<h3> Statement </h3>
	<p>		<p>
	$3.2.8.$ In the vicinity of the ore deposit the pendulum oscillation period changed by $0.1 \%$. The density of the ore in the deposit is $8 \cdot 10^3$ $\frac{kg}{m^3}$ Estimate the size of the deposit if the average density of the Earth is $5.6 \cdot 10^3$ $\frac{kg}{m^3}$ and its radius is $6400$ km.		$3.2.8.$ In the vicinity of the ore deposit the pendulum oscillation period changed by $0.1 \%$. The density of the ore in the deposit is $8 \cdot 10^3$ $\frac{kg}{m^3}$ Estimate the size of the deposit if the average density of the Earth is $5.6 \cdot 10^3$ $\frac{kg}{m^3}$ and its radius is $6400$ km.
	</p>		</p>

	<h3>Solution</h3>		<h3>Solution</h3>
	<p>		<p>
	Let us write down, using the law of gravity, the equations for determining the acceleration of free fall, taking into account that the acceleration above the deposit will be greater than far from it.		Let us write down, using the law of gravity, the equations for determining the acceleration of free fall, taking into account that the acceleration above the deposit will be greater than far from it.
	$$ g_{0}=\frac{4}{3}\pi RG\rho_{0}\approx 4\pi RG\rho_{0} $$		$$ g_{0}=\frac{4}{3}\pi RG\rho_{0}\approx 4\pi RG\rho_{0} $$

	$$ g=g_{0}+g_{k}\approx g_{0} + 4\pi RG(\rho - \rho_{0}) $$		$$ g=g_{0}+g_{k}\approx g_{0} + 4\pi RG(\rho - \rho_{0}) $$
	@@ -62,9 +62,9 @@
	Let us write down the ratio of the periods of oscillation of the pendulum given by the conditions of the problem		Let us write down the ratio of the periods of oscillation of the pendulum given by the conditions of the problem
	$$ \alpha =\frac{T-T_{0}}{T_{0}} = 0,1;\quad\xi = \frac{T}{T_{0}}=1+10^{-3} $$		$$ \alpha =\frac{T-T_{0}}{T_{0}} = 0,1;\quad\xi = \frac{T}{T_{0}}=1+10^{-3} $$
	The ratio of the periods is expressed through the values of the acceleration of gravity		The ratio of the periods is expressed through the values of the acceleration of gravity
	$$ \xi ^{2} = \frac{g_{0}+g}{g_{0}}=1+\frac{r(\rho -\rho _{0})}{R\rho _{0}} $$		$$ \xi^{2} = \frac{g_{0}+g}{g_{0}}=1+\frac{r(\rho -\rho_{0})}{R\rho_{0}} $$

	$$ \boxed{r=\frac{(\xi ^{2}-1)R\rho _{0}}{\rho -\rho _{0}}\approx 30\text{ km}} $$		$$ \boxed{r=\frac{(\xi^{2}-1)R\rho_{0}}{\rho -\rho_{0}}\approx 30\text{ km}} $$
	</p>		</p>
	<p style="text-align: right; font-style: italic; font-size: 14;">		<p style="text-align: right; font-style: italic; font-size: 14;">
	Dzikan Mikita<br>		Dzikan Mikita<br>
	Aliaksandr Kanashenka		Aliaksandr Kanashenka
	</p>		</p>

	<h4>Answer</h4>		<h4>Answer</h4>
	<p>		<p>
	$$r\approx30\text{ km}$$		$$r\approx30\text{ km}$$
	</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>
	ещё строк без изменений 21