Added en 2.7.6

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en/2.7.6.md +129 −0

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		+	</head>
		+	<body style="">
		+	<header style="text-align:center;">
		+	<h2>Solutions of Savchenko Physics Textbook</h2>
		+	<p class="author">
		+	Aliaksandr Melnichenka <br/>
		+	October 2023
		+	</p>
		+	</header>
		+
		+	<h3 id="back-link"><a href="../#2.7">$\leftarrow$Back</a></h3>
		+
		+	<h3> Statement </h3>
		+	<p>
		+	$2.7.6.$ A thin-walled cylinder of radius $R$ was spun up to an angular velocity $\omega$ and placed in a corner, as shown in the figure. The coefficient of friction between the corner walls and the cylinder is $\mu$. How many revolutions will the cylinder make before it comes to a complete stop?
		+
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="statement.png"
		+	loading="lazy" width="180" />
		+	<figcaption>
		+	For problem $2.7.6$
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+	</p>
		+
		+	<h3>Solution</h3>
		+	<p>
		+
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="draw1.png"
		+	loading="lazy" width="230" />
		+	<figcaption>
		+	Forces acting on the cylinder
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+
		+	Conditions of equilibrium:
		+	$$\left\{\begin{matrix}
		+	O'X\text{: }F_{fr2} = N_1 \\
		+	O'Y\text{: }F_{fr1}+N_2 = mg \\
		+	\end{matrix}\right.$$
		+	From where
		+	$$\left\{\begin{matrix}
		+	\mu N_2=N_1 \\
		+	\mu N_1+N_2=mg \\
		+	\end{matrix}\right.$$
		+	Let's find friction force
		+	$$F_{fr}=F_{fr1}+F_{fr2}$$
		+	$$F_{fr}=\mu (N_1+N_2)=\mu N_2(1+\mu)$$
		+
		+	Considering, $mg = N_2(1+\mu ^2)$:
		+	$$N=\frac{mg}{1+\mu ^2}$$
		+
		+	According to rotational motion dynamics, the sum of moments of external forces can be written as
		+	$$M=I\varepsilon = mR^2=F_{fr} R$$
		+	From where, absolute value of angular acceleration is found as
		+	$$\varepsilon = \frac{M}{I} = \frac{F_{fr}}{mR}$$
		+	$$\varepsilon = \frac{\mu(1+\mu)N_2 }{mR}$$
		+	$$\varepsilon = \frac{\mu(1+\mu)}{1+\mu ^2} \frac{g}{R}$$
		+	The angle by which the cylinder will rotate to the moment of total stop is
		+	$$\varphi=\frac{\omega ^2t}{2\varepsilon}$$
		+	Until the total stop, the cylinder will make $N$ total turnovers
		+	$$\varphi=2\pi N$$
		+	From where
		+	$$N=\frac{\omega ^2t}{4\pi \varepsilon}$$
		+	$$\boxed{N=\frac{1+\mu^{2}}{4\pi \mu(1+\mu)}\frac{\omega^{2}R}{g}}$$
		+	</p>
		+
		+	<h4>Answer</h4>
		+	<p>
		+	$$N=\omega^{2}R(1+\mu^{2})/[4\pi g\mu(1+\mu)]$$
		+	</p>
		+	<p style="text-align: right; font-style: italic; font-size: 14;">
		+	Yuldashev Ulugbek<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>

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			<meta name="keywords" content="solutions, savchenko, physics problems, olympiad physics, physics book">
			<meta name="description" content="A website with solutions to physics problems from Savchenko Textbook">
			<meta property="og:title" content="Savchenko Solutions">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="A website with solutions to physics problems from Savchenko Textbook">
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			<title>Savchenko Solutions</title>
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			<link rel="icon" href="https://savchenko-physics.github.io/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 Physics Textbook</h2>
			<p class="author">
			Aliaksandr Melnichenka <br/>
			October 2023
			</p>
			</header>

			<h3 id="back-link"><a href="../#2.7">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$2.7.6.$ A thin-walled cylinder of radius $R$ was spun up to an angular velocity $\omega$ and placed in a corner, as shown in the figure. The coefficient of friction between the corner walls and the cylinder is $\mu$. How many revolutions will the cylinder make before it comes to a complete stop?

			</p>
			<center>
			<figure>
			<img src="statement.png"
			loading="lazy" width="180" />
			<figcaption>
			For problem $2.7.6$
			</figcaption>
			</figure>
			</center>
			<p>
			</p>

			<h3>Solution</h3>
			<p>

			</p>
			<center>
			<figure>
			<img src="draw1.png"
			loading="lazy" width="230" />
			<figcaption>
			Forces acting on the cylinder
			</figcaption>
			</figure>
			</center>
			<p>

			Conditions of equilibrium:
			$$\left\{\begin{matrix}
			O'X\text{: }F_{fr2} = N_1 \\
			O'Y\text{: }F_{fr1}+N_2 = mg \\
			\end{matrix}\right.$$
			From where
			$$\left\{\begin{matrix}
			\mu N_2=N_1 \\
			\mu N_1+N_2=mg \\
			\end{matrix}\right.$$
			Let's find friction force
			$$F_{fr}=F_{fr1}+F_{fr2}$$
			$$F_{fr}=\mu (N_1+N_2)=\mu N_2(1+\mu)$$

			Considering, $mg = N_2(1+\mu ^2)$:
			$$N=\frac{mg}{1+\mu ^2}$$

			According to rotational motion dynamics, the sum of moments of external forces can be written as
			$$M=I\varepsilon = mR^2=F_{fr} R$$
			From where, absolute value of angular acceleration is found as
			$$\varepsilon = \frac{M}{I} = \frac{F_{fr}}{mR}$$
			$$\varepsilon = \frac{\mu(1+\mu)N_2 }{mR}$$
			$$\varepsilon = \frac{\mu(1+\mu)}{1+\mu ^2} \frac{g}{R}$$
			The angle by which the cylinder will rotate to the moment of total stop is
			$$\varphi=\frac{\omega ^2t}{2\varepsilon}$$
			Until the total stop, the cylinder will make $N$ total turnovers
			$$\varphi=2\pi N$$
			From where
			$$N=\frac{\omega ^2t}{4\pi \varepsilon}$$
			$$\boxed{N=\frac{1+\mu^{2}}{4\pi \mu(1+\mu)}\frac{\omega^{2}R}{g}}$$
			</p>

			<h4>Answer</h4>
			<p>
			$$N=\omega^{2}R(1+\mu^{2})/[4\pi g\mu(1+\mu)]$$
			</p>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			Yuldashev Ulugbek<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>

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