Translated 3.1.1-3.1.17

astrosander edited Sep 24, 2024, 2:16 AM

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		+	<meta name="description" content="A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius r, and to the right, on a surface of radius R. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.">
		+	<meta name="author" content="Aliaksandr Melnichenka">
		+	<meta name="date" content="2023-10" scheme="YYYY-MM">
		+	<meta property="og:title" content="A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius r, and to the right, on a surface of radius R. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.">
		+	<meta property="og:image" content="img/logo.png">
		+	<meta property="og:description" content="A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius r, and to the right, on a surface of radius R. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.">
		+	<meta name="yandex-verification" content="6cfda41f74038368">
		+	<title>A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius r, and to the right, on a surface of radius R. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.</title>
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		+	</head>
		+	<body style="">
		+	<header style="text-align:center;">
		+	<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>
		+	$3.1.7.$ A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius $r$, and to the right, on a surface of radius $R$. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="https://savchenkosolutions.com/3/3.1.7/statement.png"
		+	loading="lazy" width="150" />
		+	<figcaption>
		+	For problem $3.1.7$
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+	</p>
		+
		+	<h3>Solution</h3>
		+	<p>
		+	According to the law of conservation of body energy, the maximum potential energy will be conserved in both cases
		+	$$ E_{p1}=E_{p2} $$
		+	And therefore the height $h$ will rise relative to the lower point of the trajectory to which the body will rise
		+	will be equal
		+
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="https://savchenkosolutions.com/3/3.1.7/3.1.7_1.png"
		+	loading="lazy" width="230" />
		+	<figcaption>
		+	The position of the ball at the extreme points of the trajectory
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+	We will use the expression $\Delta U$ for small oscillations obtained in <a href="../3.1.5">3.1.5</a>
		+	$$ \Delta U = \frac{mgx_1^2}{2r}=\frac{mgx_2^2}{2R} $$
		+	Whence it follows that
		+	$$ \boxed{\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}} $$
		+
		+	</p>
		+
		+	<h4>Answer</h4>
		+	<p>
		+	$$\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}$$
		+	</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>
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		+	<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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	@@ -0,0 +1,110 @@
			<!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="A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius r, and to the right, on a surface of radius R. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius r, and to the right, on a surface of radius R. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius r, and to the right, on a surface of radius R. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius r, and to the right, on a surface of radius R. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.</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">
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			<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;">
			<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>
			$3.1.7.$ A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius $r$, and to the right, on a surface of radius $R$. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.
			</p>
			<center>
			<figure>
			<img src="https://savchenkosolutions.com/3/3.1.7/statement.png"
			loading="lazy" width="150" />
			<figcaption>
			For problem $3.1.7$
			</figcaption>
			</figure>
			</center>
			<p>
			</p>

			<h3>Solution</h3>
			<p>
			According to the law of conservation of body energy, the maximum potential energy will be conserved in both cases
			$$ E_{p1}=E_{p2} $$
			And therefore the height $h$ will rise relative to the lower point of the trajectory to which the body will rise
			will be equal

			</p>
			<center>
			<figure>
			<img src="https://savchenkosolutions.com/3/3.1.7/3.1.7_1.png"
			loading="lazy" width="230" />
			<figcaption>
			The position of the ball at the extreme points of the trajectory
			</figcaption>
			</figure>
			</center>
			<p>
			We will use the expression $\Delta U$ for small oscillations obtained in <a href="../3.1.5">3.1.5</a>
			$$ \Delta U = \frac{mgx_1^2}{2r}=\frac{mgx_2^2}{2R} $$
			Whence it follows that
			$$ \boxed{\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}} $$

			</p>

			<h4>Answer</h4>
			<p>
			$$\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}$$
			</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>
			</p>
			</footer>
			</body>

			</html>