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<meta name="description" content="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?">
<meta property="og:title" content="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?">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="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?">
<title>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?</title>
$2.7.6.$ 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
<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>
<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="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?">
<meta name="description" content="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?">
<meta property="og:title" content="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?">
<meta property="og:title" content="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?">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="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?">
<meta property="og:description" content="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?">
<title>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?</title>
<title>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?</title>
$2.7.6.$ 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?
$2.7.6.$ 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>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" width="180" />
loading="lazy" width="180" />
<figcaption>
<figcaption>
For problem $2.7.6$
For problem $2.7.6$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="draw1.png"
<img src="draw1.png"
loading="lazy" width="230" />
loading="lazy" width="230" />
<figcaption>
<figcaption>
Forces acting on the cylinder
Forces acting on the cylinder
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
Conditions of equilibrium:
Conditions of equilibrium:
$$\left\{\begin{matrix}
$$\left\{\begin{matrix}
O'X\text{: }F_{fr2} = N_1 \\
O'X\text{: }F_{fr2} = N_1 \\
O'Y\text{: }F_{fr1}+N_2 = mg \\
O'Y\text{: }F_{fr1}+N_2 = mg \\
\end{matrix}\right.$$
\end{matrix}\right.$$
From where
From where
$$\left\{\begin{matrix}
$$\left\{\begin{matrix}
\mu N_2=N_1 \\
\mu N_2=N_1 \\
\mu N_1+N_2=mg \\
\mu N_1+N_2=mg \\
\end{matrix}\right.$$
\end{matrix}\right.$$
Let's find friction force
Let's find friction force
$$F_{fr}=F_{fr1}+F_{fr2}$$
$$F_{fr}=F_{fr1}+F_{fr2}$$
$$F_{fr}=\mu (N_1+N_2)=\mu N_2(1+\mu)$$
$$F_{fr}=\mu (N_1+N_2)=\mu N_2(1+\mu)$$
Considering, $mg = N_2(1+\mu ^2)$:
Considering, $mg = N_2(1+\mu ^2)$:
$$N=\frac{mg}{1+\mu ^2}$$
$$N=\frac{mg}{1+\mu ^2}$$
According to rotational motion dynamics, the sum of moments of external forces can be written as
According to rotational motion dynamics, the sum of moments of external forces can be written as
$$M=I\varepsilon = mR^2=F_{fr} R$$
$$M=I\varepsilon = mR^2=F_{fr} R$$
From where, absolute value of angular acceleration is found as
From where, absolute value of angular acceleration is found as
<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>
