Savchenko Solutions

Solutions of Savchenko Physics Textbook

Aliaksandr Melnichenka
October 2023

    <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?

    <h3>Solution</h3>
    <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
$$N=\frac{\omega ^2t}{4\pi \varepsilon}$$
$$\boxed{N=\frac{1+\mu^{2}}{4\pi \mu(1+\mu)}\frac{\omega^{2}R}{g}}$$

    <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>