Added 2.6.14

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		+	<meta name="description" content="Two stars of mass m_1 and m_2 form a binary system with a constant distance between the stars R. What is the period of rotation of the stars around the common center of mass?">
		+	<meta name="author" content="Aliaksandr Melnichenka">
		+	<meta name="date" content="2023-10" scheme="YYYY-MM">
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		+	<title>Two stars of mass m_1 and m_2 form a binary system with a constant distance between the stars R. What is the period of rotation of the stars around the common center of mass?</title>
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		+	</head>
		+	<body style="">
		+	<header style="text-align:center;">
		+	<h2>Solutions of Savchenko Problems in Physics</h2>
		+	<p class="author">
		+	Aliaksandr Melnichenka <br/>
		+	October 2023
		+	</p>
		+	</header>
		+
		+	<h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3>
		+
		+	<h3> Statement </h3>
		+	<p>
		+	$2.6.14^*.$ Two stars of mass $m_1$ and $m_2$ form a binary system with a constant distance between the stars $R$. What is the period of rotation of the stars around the common center of mass?
		+	</p>
		+
		+	<h3>Solution</h3>
		+	<p>
		+
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="draw.png"
		+	loading="lazy" width="260" />
		+	<figcaption>
		+	Forces acting on the system
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+
		+	The position of the center of mass (point $O$) could be found as
		+	$$R_1=R\frac{m_2}{m_1+m_2}$$
		+	$$R_2=R-R_1=R\frac{m_1}{m_1+m_2}$$
		+	Since the bodies are stationary, their accelerations are equal, and therefore, according to Newton's first law, the sum of external forces is zero
		+	$$F_{c1}=F_{21} \quad F_{c2}=F_{12}\quad(1)$$
		+	By definition of centripetal acceleration
		+	$$F_{c1}=m_1\omega_1^2R_1\quad(2.1)$$
		+	$$F_{c2}=m_2\omega_2^2R_2\quad(2.2)$$
		+	Since the body $m_1$ and $m_2$ rotate along the same straight line
		+	$$\omega_1=\omega_2=\omega$$
		+	On the other hand, the force of their gravitational attraction is equal to
		+	$$F_{12}=F_{21}=G\frac{m_1m_2}{R^2}\quad(4)$$
		+	Given $(2)$, we equate $(1)$ and $(3)$
		+	$$G\frac{m_1m_2}{R^2}=m_1\omega^2R\frac{m_2}{m_1+m_2}$$
		+	From where
		+	$$\omega=\sqrt{\frac{G(m_1+m_2)}{R^3}}$$
		+	Whence the circulation period is equal to
		+	$$\boxed{T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{R^3}{G(m_1+m_2)}}}$$
		+	</p>
		+
		+	<h4>Answer</h4>
		+	<p>
		+	$$T=2\pi\sqrt{R^3/\gamma(m_1+m_2)}$$
		+	</p>
		+
		+	<p style="text-align: right; font-style: italic; font-size: 14;">
		+	Almaskhan Arsen<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> astrosander01@gmail.com <br></small>
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	@@ -0,0 +1,107 @@
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			<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="Two stars of mass m_1 and m_2 form a binary system with a constant distance between the stars R. What is the period of rotation of the stars around the common center of mass?">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="Two stars of mass m_1 and m_2 form a binary system with a constant distance between the stars R. What is the period of rotation of the stars around the common center of mass?">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="Two stars of mass m_1 and m_2 form a binary system with a constant distance between the stars R. What is the period of rotation of the stars around the common center of mass?">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>Two stars of mass m_1 and m_2 form a binary system with a constant distance between the stars R. What is the period of rotation of the stars around the common center of mass?</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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			ignoreClass: 'html'
			},
			showProcessingMessages: false,
			messageStyle: 'none'
			});
			</script>
			</head>
			<body style="">
			<header style="text-align:center;">
			<h2>Solutions of Savchenko Problems in Physics</h2>
			<p class="author">
			Aliaksandr Melnichenka <br/>
			October 2023
			</p>
			</header>

			<h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$2.6.14^*.$ Two stars of mass $m_1$ and $m_2$ form a binary system with a constant distance between the stars $R$. What is the period of rotation of the stars around the common center of mass?
			</p>

			<h3>Solution</h3>
			<p>

			</p>
			<center>
			<figure>
			<img src="draw.png"
			loading="lazy" width="260" />
			<figcaption>
			Forces acting on the system
			</figcaption>
			</figure>
			</center>
			<p>

			The position of the center of mass (point $O$) could be found as
			$$R_1=R\frac{m_2}{m_1+m_2}$$
			$$R_2=R-R_1=R\frac{m_1}{m_1+m_2}$$
			Since the bodies are stationary, their accelerations are equal, and therefore, according to Newton's first law, the sum of external forces is zero
			$$F_{c1}=F_{21} \quad F_{c2}=F_{12}\quad(1)$$
			By definition of centripetal acceleration
			$$F_{c1}=m_1\omega_1^2R_1\quad(2.1)$$
			$$F_{c2}=m_2\omega_2^2R_2\quad(2.2)$$
			Since the body $m_1$ and $m_2$ rotate along the same straight line
			$$\omega_1=\omega_2=\omega$$
			On the other hand, the force of their gravitational attraction is equal to
			$$F_{12}=F_{21}=G\frac{m_1m_2}{R^2}\quad(4)$$
			Given $(2)$, we equate $(1)$ and $(3)$
			$$G\frac{m_1m_2}{R^2}=m_1\omega^2R\frac{m_2}{m_1+m_2}$$
			From where
			$$\omega=\sqrt{\frac{G(m_1+m_2)}{R^3}}$$
			Whence the circulation period is equal to
			$$\boxed{T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{R^3}{G(m_1+m_2)}}}$$
			</p>

			<h4>Answer</h4>
			<p>
			$$T=2\pi\sqrt{R^3/\gamma(m_1+m_2)}$$
			</p>

			<p style="text-align: right; font-style: italic; font-size: 14;">
			Almaskhan Arsen<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> astrosander01@gmail.com <br></small>
			</p>
			</footer>
			</body>

			</html>