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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?">
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<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
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<meta name="description" content="The largest dataset of solutions of 'Savchenko. Problems in Physics'. Savchenko’s Problems in General Physics is widely used to prepare for olympiads and it is a useful tool to
<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?">
master and sharpen your skills and techniques in comptetitive problem solving. Some of these problems were a source
of inspiration for Jaan Kalda’s handouts and to some NBPhO problems. You may find problems from old IPhO
<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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<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?
$2.7.6.$ 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> astrosander01@gmail.com <br></small>
<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>
