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| <title>The rod, with one end pivotally fixed on a horizontal plane, lies on the cylinder. Angular velocity of the rod \omega. There is no slippage between the cylinder and the plane. Find the dependence of the angular velocity of the cylinder on the angle \alpha between the rod and the plane.</title> | | <title>The rod, with one end pivotally fixed on a horizontal plane, lies on the cylinder. Angular velocity of the rod \omega. There is no slippage between the cylinder and the plane. Find the dependence of the angular velocity of the cylinder on the angle \alpha between the rod and the plane.</title> |
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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
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| <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.5.18.$ The rod, with one end pivotally fixed on a horizontal plane, lies on the cylinder. Angular velocity of the rod $\omega$. There is no slippage between the cylinder and the plane. Find the dependence of the angular velocity of the cylinder on the angle $\alpha$ between the rod and the plane. | | $1.5.18.$ The rod, with one end pivotally fixed on a horizontal plane, lies on the cylinder. Angular velocity of the rod $\omega$. There is no slippage between the cylinder and the plane. Find the dependence of the angular velocity of the cylinder on the angle $\alpha$ between the rod and the plane. |
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| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| For problem $1.5.18$ | | For problem $1.5.18$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
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| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.18/draw.png" | | <img src="https://savchenkosolutions.com/1/1.5.18/draw.png" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| Cylinder inscribed in angle $\alpha$ | | Cylinder inscribed in angle $\alpha$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p>NO: Before viewing the solution to this problem, I advise you to familiarize yourself with the solution <a href="../1.5.17">1.5.17</a></p> | | <p>NO: Before viewing the solution to this problem, I advise you to familiarize yourself with the solution <a href="../1.5.17">1.5.17</a></p> |
| <p>By the time $t$ the rod will form an angle $\alpha$</p> | | <p>By the time $t$ the rod will form an angle $\alpha$</p> |
| <p>Consider the change in coordinate $x = AC$ over a short period of time $dt$</p> | | <p>Consider the change in coordinate $x = AC$ over a short period of time $dt$</p> |
| <p> | | <p> |
| $$ dx = d\left(\frac{R}{\tan (\frac{\alpha}{2})}\right) $$ | | $$ dx = d\left(\frac{R}{\tan (\frac{\alpha}{2})}\right) $$ |
| </p> | | </p> |
| <p>I share both honors with $dt$</p> | | <p>I share both honors with $dt$</p> |
| <p> | | <p> |
| $$ \frac{dx}{dt} = \frac{d\left(\frac{R}{\tan \alpha/2}\right)}{dt} $$ | | $$ \frac{dx}{dt} = \frac{d\left(\frac{R}{\tan \alpha/2}\right)}{dt} $$ |
| </p> | | </p> |
| <p>Considering that $\omega = \frac{d \alpha}{dt}$ and $v = \frac{dx}{dt}$</p> | | <p>Considering that $\omega = \frac{d \alpha}{dt}$ and $v = \frac{dx}{dt}$</p> |
| <p> | | <p> |
| $$ v = \frac{\omega R}{2 \sin ^2 \left(\frac{\alpha}{2}\right)} $$ | | $$ v = \frac{\omega R}{2 \sin ^2 \left(\frac{\alpha}{2}\right)} $$ |
| </p> | | </p> |
| <p>Let's write down the condition of no slippage</p> | | <p>Let's write down the condition of no slippage</p> |
| <p> | | <p> |
| $$ \omega ' R = v $$ | | $$ \omega ' R = v $$ |
| </p> | | </p> |
| <p>From here</p> | | <p>From here</p> |
| <p> | | <p> |
| $$ \fbox{$\omega ' = \frac{\omega}{2 \sin ^2 \left(\frac{\alpha}{2}\right)}$} $$ | | $$ \fbox{$\omega ' = \frac{\omega}{2 \sin ^2 \left(\frac{\alpha}{2}\right)}$} $$ |
