Updated spacing between @ latex expressions

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

правка #10265 предыдущая #9962 GitHub c702ebe ← раньше позже →

en/4.2.11.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="With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is
			ho, the length of the stick is l.">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is
			ho, the length of the stick is l.">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is
			ho, the length of the stick is l.">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is
			ho, the length of the stick is l.</title>
			<link rel="stylesheet" href="../../css/css-latex/style.css">
			<link rel="icon" href="../../img/logo.png" type="image/png">
			<script src="../../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="../../#4.2">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>

			<p>
			$4.2.11$
			With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius $r$, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is $\rho$, the length of the stick is $l$.
			</p>
			<center>
			<figure>
			<img src="statement.png"
			loading="lazy" alt="4.2.11" width="200" />
			<figcaption>
			For problem 4.2.11
			</figcaption>
			</figure>
			</center>

			<h3>Solution</h3>
			<p>
			Let's consider the following figure
			</p>
			<center>
			<figure>
			<img src="draw.png"
			loading="lazy" alt="4.2.11" width="200" />
			<figcaption>
			Force analysis
			</figcaption>
			</figure>
			</center>
			<p>
			The system is on equilibrium in above position, so total torque is zero ($\sum \vec{\tau} = \vec{0}$). Considering as a pivot point the center of mass of the stick (located at stick's half),
			$$F_A\left(\frac{l}{2}+r\right)\sin{\left(\frac{\pi}{2}-\alpha\right)}+mg\left(\frac{l}{2}+r\right)\sin{\left(\frac{\pi}{2}+\alpha\right)}+N\frac{l}{2}\sin{\left(\frac{\pi}{2}+\alpha\right)}=0$$
			$$F_A\left(\frac{l}{2}+r\right)\cos{\alpha}-mg\left(\frac{l}{2}+r\right)\cos{\alpha}-N\frac{l}{2}\cos{\alpha}=0$$
		@@ -88,7 +88,7 @@
			$$F_A\left(\frac{l}{2}+r\right)-mg\left(\frac{l}{2}+r\right)-N\frac{l}{2}=0$$
			$$\rho~g~\frac{V}{2}\left(\frac{l}{2}+r\right)-mg\left(\frac{l}{2}+r\right)-N\frac{l}{2}=0$$
			Separating $N$, and considering that ball's mass can be expressed as $m=\rho_0 V$ and ball's volume is $V=\frac{4}{3}\pi r^3$
		−	$$N = \frac{2\pi r^3g}{3}\left(1+2\frac{r}{l}\right)(\rho-2\rho_0)$$
		+	$$N = \frac{2\pi r^3g}{3}\left(1+2\frac{r}{l}\right)(\rho -2\rho_0)$$
			But, since the ball is a hollow sphere, it can be considered massless, so $\rho_0 = 0$, then
			$$\boxed{N = \frac{2\pi r^3\rho g}{3}\left(1+2\frac{r}{l}\right)}$$
			Another way:
			Let's suppose the pivot point at the contact of stick with the bottom. Applying Newton Second Law for rotation ($\sum \vec{\tau} = \vec{0}$) and considering the massless ball
			$$F_A\left(l+r\right)\sin{\left(\frac{\pi}{2}-\alpha\right)}+m_s~g\frac{l}{2}\sin{\left(\frac{\pi}{2}+\alpha\right)}=0$$
			$$F_A\left(l+r\right)\cos{\alpha}-m_s~g\frac{l}{2}\cos{\alpha}=0$$
			Hence, the mass of the stick is
			$$m_s = \rho~V\left(1+\frac{r}{l}\right) \quad(1)$$
			Applying Newton Second Law
			$$F_A+N = m_s~g$$
			$$\rho g \frac{V}{2} + N = m_s g \quad(2)$$
			From $(1)$, $(2)$ and taking $V=\frac{4\pi r^3}{3}$, the above answer is obtained again...
			</p>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			BSc. Luis Daniel Fernández Quintana<br>
			Physics Department (FCNE)<br>
			Universidad de Oriente, Cuba<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>
		ещё строк без изменений 30

	<!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="With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is		<meta name="description" content="With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is
	ho, the length of the stick is l.">		ho, the length of the stick is l.">
	<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="With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is		<meta property="og:title" content="With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is
	ho, the length of the stick is l.">		ho, the length of the stick is l.">
	<meta property="og:image" content="img/logo.png">		<meta property="og:image" content="img/logo.png">
	<meta property="og:description" content="With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is		<meta property="og:description" content="With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is
	ho, the length of the stick is l.">		ho, the length of the stick is l.">
	<meta name="yandex-verification" content="6cfda41f74038368">		<meta name="yandex-verification" content="6cfda41f74038368">
	<title>With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is		<title>With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius r, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is
	ho, the length of the stick is l.</title>		ho, the length of the stick is l.</title>
	<link rel="stylesheet" href="../../css/css-latex/style.css">		<link rel="stylesheet" href="../../css/css-latex/style.css">
	<link rel="icon" href="../../img/logo.png" type="image/png">		<link rel="icon" href="../../img/logo.png" type="image/png">
	<script src="../../js/jquery-1.10.1.min.js"></script>		<script src="../../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="../../#4.2">$\leftarrow$Back</a></h3>		<h3 id="back-link"><a href="../../#4.2">$\leftarrow$Back</a></h3>

	<h3> Statement </h3>		<h3> Statement </h3>

	<p>		<p>
	$4.2.11$		$4.2.11$
	With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius $r$, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is $\rho$, the length of the stick is $l$.		With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius $r$, rigidly connected to the stick, is half immersed in the liquid? The density of the liquid is $\rho$, the length of the stick is $l$.
	</p>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="statement.png"		<img src="statement.png"
	loading="lazy" alt="4.2.11" width="200" />		loading="lazy" alt="4.2.11" width="200" />
	<figcaption>		<figcaption>
	For problem 4.2.11		For problem 4.2.11
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>

	<h3>Solution</h3>		<h3>Solution</h3>
	<p>		<p>
	Let's consider the following figure		Let's consider the following figure
	</p>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="draw.png"		<img src="draw.png"
	loading="lazy" alt="4.2.11" width="200" />		loading="lazy" alt="4.2.11" width="200" />
	<figcaption>		<figcaption>
	Force analysis		Force analysis
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>
	The system is on equilibrium in above position, so total torque is zero ($\sum \vec{\tau} = \vec{0}$). Considering as a pivot point the center of mass of the stick (located at stick's half),		The system is on equilibrium in above position, so total torque is zero ($\sum \vec{\tau} = \vec{0}$). Considering as a pivot point the center of mass of the stick (located at stick's half),
	$$F_A\left(\frac{l}{2}+r\right)\sin{\left(\frac{\pi}{2}-\alpha\right)}+mg\left(\frac{l}{2}+r\right)\sin{\left(\frac{\pi}{2}+\alpha\right)}+N\frac{l}{2}\sin{\left(\frac{\pi}{2}+\alpha\right)}=0$$		$$F_A\left(\frac{l}{2}+r\right)\sin{\left(\frac{\pi}{2}-\alpha\right)}+mg\left(\frac{l}{2}+r\right)\sin{\left(\frac{\pi}{2}+\alpha\right)}+N\frac{l}{2}\sin{\left(\frac{\pi}{2}+\alpha\right)}=0$$
	$$F_A\left(\frac{l}{2}+r\right)\cos{\alpha}-mg\left(\frac{l}{2}+r\right)\cos{\alpha}-N\frac{l}{2}\cos{\alpha}=0$$		$$F_A\left(\frac{l}{2}+r\right)\cos{\alpha}-mg\left(\frac{l}{2}+r\right)\cos{\alpha}-N\frac{l}{2}\cos{\alpha}=0$$
	@@ -88,7 +88,7 @@
	$$F_A\left(\frac{l}{2}+r\right)-mg\left(\frac{l}{2}+r\right)-N\frac{l}{2}=0$$		$$F_A\left(\frac{l}{2}+r\right)-mg\left(\frac{l}{2}+r\right)-N\frac{l}{2}=0$$
	$$\rho~g~\frac{V}{2}\left(\frac{l}{2}+r\right)-mg\left(\frac{l}{2}+r\right)-N\frac{l}{2}=0$$		$$\rho~g~\frac{V}{2}\left(\frac{l}{2}+r\right)-mg\left(\frac{l}{2}+r\right)-N\frac{l}{2}=0$$
	Separating $N$, and considering that ball's mass can be expressed as $m=\rho_0 V$ and ball's volume is $V=\frac{4}{3}\pi r^3$		Separating $N$, and considering that ball's mass can be expressed as $m=\rho_0 V$ and ball's volume is $V=\frac{4}{3}\pi r^3$
	$$N = \frac{2\pi r^3g}{3}\left(1+2\frac{r}{l}\right)(\rho-2\rho_0)$$		$$N = \frac{2\pi r^3g}{3}\left(1+2\frac{r}{l}\right)(\rho -2\rho_0)$$
	But, since the ball is a hollow sphere, it can be considered massless, so $\rho_0 = 0$, then		But, since the ball is a hollow sphere, it can be considered massless, so $\rho_0 = 0$, then
	$$\boxed{N = \frac{2\pi r^3\rho g}{3}\left(1+2\frac{r}{l}\right)}$$		$$\boxed{N = \frac{2\pi r^3\rho g}{3}\left(1+2\frac{r}{l}\right)}$$
	Another way:		Another way:
	Let's suppose the pivot point at the contact of stick with the bottom. Applying Newton Second Law for rotation ($\sum \vec{\tau} = \vec{0}$) and considering the massless ball		Let's suppose the pivot point at the contact of stick with the bottom. Applying Newton Second Law for rotation ($\sum \vec{\tau} = \vec{0}$) and considering the massless ball
	$$F_A\left(l+r\right)\sin{\left(\frac{\pi}{2}-\alpha\right)}+m_s~g\frac{l}{2}\sin{\left(\frac{\pi}{2}+\alpha\right)}=0$$		$$F_A\left(l+r\right)\sin{\left(\frac{\pi}{2}-\alpha\right)}+m_s~g\frac{l}{2}\sin{\left(\frac{\pi}{2}+\alpha\right)}=0$$
	$$F_A\left(l+r\right)\cos{\alpha}-m_s~g\frac{l}{2}\cos{\alpha}=0$$		$$F_A\left(l+r\right)\cos{\alpha}-m_s~g\frac{l}{2}\cos{\alpha}=0$$
	Hence, the mass of the stick is		Hence, the mass of the stick is
	$$m_s = \rho~V\left(1+\frac{r}{l}\right) \quad(1)$$		$$m_s = \rho~V\left(1+\frac{r}{l}\right) \quad(1)$$
	Applying Newton Second Law		Applying Newton Second Law
	$$F_A+N = m_s~g$$		$$F_A+N = m_s~g$$
	$$\rho g \frac{V}{2} + N = m_s g \quad(2)$$		$$\rho g \frac{V}{2} + N = m_s g \quad(2)$$
	From $(1)$, $(2)$ and taking $V=\frac{4\pi r^3}{3}$, the above answer is obtained again...		From $(1)$, $(2)$ and taking $V=\frac{4\pi r^3}{3}$, the above answer is obtained again...
	</p>		</p>
	<p style="text-align: right; font-style: italic; font-size: 14;">		<p style="text-align: right; font-style: italic; font-size: 14;">
	BSc. Luis Daniel Fernández Quintana<br>		BSc. Luis Daniel Fernández Quintana<br>
	Physics Department (FCNE)<br>		Physics Department (FCNE)<br>
	Universidad de Oriente, Cuba<br>		Universidad de Oriente, Cuba<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>
	ещё строк без изменений 30