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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> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.5.8.$ Plot the trajectories of the points of the wheel rolling without slipping on the rail. Consider cases where the points are at a distance from the wheel axis: $r > R$, $r = R$, $r < R$. Find the acceleration of these points if the wheel axis is moving at a constant speed $v$. Find the radius of curvature of the trajectory of a point located in the highest and lowest positions at a distance $r \ne R$ from the wheel axis. | | $1.5.8.$ Plot the trajectories of the points of the wheel rolling without slipping on the rail. Consider cases where the points are at a distance from the wheel axis: $r > R$, $r = R$, $r < R$. Find the acceleration of these points if the wheel axis is moving at a constant speed $v$. Find the radius of curvature of the trajectory of a point located in the highest and lowest positions at a distance $r \ne R$ from the wheel axis. |
| </p> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.8/Cycloid_animated_.gif" | | <img src="https://savchenkosolutions.com/1/1.5.8/Cycloid_animated_.gif" |
| loading="lazy" width="500" /> | | loading="lazy" width="500" /> |
| <figcaption> | | <figcaption> |
| Cycloid - the sum of translational and rotational motion | | Cycloid - the sum of translational and rotational motion |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
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| $$ u = \omega r + v = v(1 + \frac{r}{R}) $$ | | $$ u = \omega r + v = v(1 + \frac{r}{R}) $$ |
| </p> | | </p> |
| <p>Knowing the acceleration $a$ and the velocity $u$ at a given point, we find its radius of curvature</p> | | <p>Knowing the acceleration $a$ and the velocity $u$ at a given point, we find its radius of curvature</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ r = \frac{u^2}{a} $$ | | $$ r = \frac{u^2}{a} $$ |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \fbox{$r = \frac{(R+r)^2}{r}$} $$ | | $$ \fbox{$r = \frac{(R+r)^2}{r}$} $$ |
| </p> | | </p> |
| <p>For better understanding, I offer an illustration of the movement:</p> | | <p>For better understanding, I offer an illustration of the movement:</p> |
| <div class="video-container"> | | <div class="video-container"> |
| <iframe class="video" src='https://www.youtube.com/embed/XgqOLWkicFc' frameborder="0" allowfullscreen></iframe> | | <iframe class="video" src='https://www.youtube.com/embed/XgqOLWkicFc' frameborder="0" allowfullscreen></iframe> |
| </div> | | </div> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$a=(v^2/R^2)r; r_{in}=(R+r)^2/r, r_{out}=(R-r)^2/r$$ | | $$a=(v^2/R^2)r; r_{in}=(R+r)^2/r, r_{out}=(R-r)^2/r$$ |
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