| <!DOCTYPE html> | | <!DOCTYPE html> |
| <html lang="en"> | | <html lang="en"> |
| | | |
| <head> | | <head> |
| <meta charset="utf-8"> | | <meta charset="utf-8"> |
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | | <meta name="viewport" content="width=device-width, initial-scale=1.0"> |
| <meta http-equiv="content-language" content="en"> | | <meta http-equiv="content-language" content="en"> |
| <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="The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of v. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length L."> | | <meta name="description" content="The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of v. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length L."> |
| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
| <meta property="og:title" content="The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of v. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length L."> | | <meta property="og:title" content="The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of v. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length L."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of v. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length L."> | | <meta property="og:description" content="The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of v. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length L."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of v. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length L.</title> | | <title>The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of v. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length L.</title> |
| <link rel="stylesheet" href="../../css/css-latex/style.css"> | | <link rel="stylesheet" href="../../css/css-latex/style.css"> |
| <link rel="icon" href="../../img/logo.png" type="image/png"> | | <link rel="icon" href="../../img/logo.png" type="image/png"> |
| <script src="../../js/jquery-1.10.1.min.js"></script> | | <script src="../../js/jquery-1.10.1.min.js"></script> |
| <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> | | <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> |
| <script type="text/x-mathjax-config"> | | <script type="text/x-mathjax-config"> |
| MathJax.Hub.Config({ | | MathJax.Hub.Config({ |
| extensions: ['tex2jax.js'], | | extensions: ['tex2jax.js'], |
| jax: ['input/TeX', 'output/HTML-CSS'], | | jax: ['input/TeX', 'output/HTML-CSS'], |
| tex2jax: { | | tex2jax: { |
| inlineMath: [['$', '$'], ['$', '$']], | | inlineMath: [['$', '$'], ['$', '$']], |
| processEscapes: true, | | processEscapes: true, |
| processClass: 'tex2jax', | | processClass: 'tex2jax', |
| ignoreClass: 'html' | | ignoreClass: 'html' |
| }, | | }, |
| showProcessingMessages: false, | | showProcessingMessages: false, |
| messageStyle: 'none' | | messageStyle: 'none' |
| }); | | }); |
| </script> | | </script> |
| </head> | | </head> |
| <body style=""> | | <body style=""> |
| <header style="text-align:center;"> | | <header style="text-align:center;"> |
| <a href="../../" style="text-decoration: none;"> | | <a href="../../" style="text-decoration: none;"> |
| <div id="logo"> | | <div id="logo"> |
| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
| </div> | | </div> |
| </a> | | </a> |
| <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> |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <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> |
| $1.5.16.$ The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of $v$. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length $L$. | | $1.5.16.$ The rod rests its ends on the sides of a right angle. The upper end of the rod is lifted at a speed of $v$. Find how the speed of its lower end depends on time. Take the moment when the upper end is at the top of the corner as the beginning of the time reference. Rod length $L$. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.16/statement.png" | | <img src="https://savchenkosolutions.com/1/1.5.16/statement.png" |
| loading="lazy" width="180" /> | | loading="lazy" width="180" /> |
| <figcaption> | | <figcaption> |
| For problem $1.5.16$ | | For problem $1.5.16$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.16/draw.png" | | <img src="https://savchenkosolutions.com/1/1.5.16/draw.png" |
| loading="lazy" width="240" /> | | loading="lazy" width="240" /> |
| <figcaption> | | <figcaption> |
| Distribution of rod end velocities | | Distribution of rod end velocities |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| $$ dx = \frac{ydy}{\sqrt{L^2 - y^2}} $$ | | $$ dx = \frac{ydy}{\sqrt{L^2 - y^2}} $$ |
| </p> | | </p> |
| <p>Considering that $v = \frac{dy}{dt}$ and $v_A = \frac{dx}{dt}$</p> | | <p>Considering that $v = \frac{dy}{dt}$ and $v_A = \frac{dx}{dt}$</p> |
| <p> | | <p> |
| $$ v_A = \frac{vy}{\sqrt{L^2 - y^2}} $$ | | $$ v_A = \frac{vy}{\sqrt{L^2 - y^2}} $$ |
| </p> | | </p> |
| <p>Substitute $(1)$</p> | | <p>Substitute $(1)$</p> |
| <p> | | <p> |
| $$ \fbox{$v_A = \frac{v^2t}{\sqrt{L^2 - v^2t^2}}$} $$ | | $$ \fbox{$v_A = \frac{v^2t}{\sqrt{L^2 - v^2t^2}}$} $$ |
| </p> | | </p> |
| | | |
| <h3> Alternative solution: </h3> | | <h3> Alternative solution: </h3> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.16/draw1.png" | | <img src="https://savchenkosolutions.com/1/1.5.16/draw1.png" |
| loading="lazy" width="240" /> | | loading="lazy" width="240" /> |
| <figcaption> | | <figcaption> |
| Velocity distribution of the rod ends | | Velocity distribution of the rod ends |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p>The coordinate of the upper end can be found as</p> | | <p>The coordinate of the upper end can be found as</p> |
| <p> | | <p> |
| $$ y=\upsilon t $$ | | $$ y=\upsilon t $$ |
| </p> | | </p> |
| <p>The projections of the velocities of both ends onto the direction of the rod are:</p> | | <p>The projections of the velocities of both ends onto the direction of the rod are:</p> |
| <p> | | <p> |
| $$ \upsilon \cos\alpha=u\cos(90^{\circ}-\alpha) $$ | | $$ \upsilon \cos\alpha=u\cos(90^{\circ}-\alpha) $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ \upsilon \cos\alpha=u\sin\alpha $$ | | $$ \upsilon \cos\alpha=u\sin\alpha $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ u=\upsilon \operatorname{ctg}\alpha=\upsilon\frac{y}{x} $$ | | $$ u=\upsilon \operatorname{ctg}\alpha=\upsilon\frac{y}{x} $$ |
| </p> | | </p> |
| <p>By the Pythagorean theorem</p> | | <p>By the Pythagorean theorem</p> |
| <p> | | <p> |
| $$ x^2+y^2=L^2 $$ | | $$ x^2+y^2=L^2 $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ x=\sqrt{L^2-y^2}=\sqrt{L^2-\upsilon^2t^2} $$ | | $$ x=\sqrt{L^2-y^2}=\sqrt{L^2-\upsilon^2t^2} $$ |
| </p> | | </p> |
| <p>Since the velocity $u$ is directed against the $x$ axis, then</p> | | <p>Since the velocity $u$ is directed against the $x$ axis, then</p> |
| <p> | | <p> |
| $$ x’=-u $$ | | $$ x’=-u $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ u=-\frac{\frac{1}{2}\cdot 2t\cdot (-\upsilon^2)}{\sqrt{L^2-\upsilon^2t^2}} $$ | | $$ u=-\frac{\frac{1}{2}\cdot 2t\cdot (-\upsilon^2)}{\sqrt{L^2-\upsilon^2t^2}} $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ u=\frac{t\cdot \upsilon^2}{\sqrt{L^2-\upsilon^2t^2}} $$ | | $$ u=\frac{t\cdot \upsilon^2}{\sqrt{L^2-\upsilon^2t^2}} $$ |
| </p> | | </p> |
| <p> </p> | | <p> </p> |
| <p>If you connect the center of the rod with the vertex of the right angle, then this segment will be the median of the right triangle and will be equal to half the length of the hypotenuse. It turns out that the center of the rod moves along a circle with the center at point $O$.</p> | | <p>If you connect the center of the rod with the vertex of the right angle, then this segment will be the median of the right triangle and will be equal to half the length of the hypotenuse. It turns out that the center of the rod moves along a circle with the center at point $O$.</p> |
| <p> </p> | | <p> </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$v_A = \frac{v^2t}{\sqrt{L^2 - v^2t^2}}$$ | | $$v_A = \frac{v^2t}{\sqrt{L^2 - v^2t^2}}$$ |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> |
| </p> | | </p> |
| <p> | | <p> |
| <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> |
| </p> | | </p> |
| </footer> | | </footer> |
| </body> | | </body> |
| | | |
| </html> | | </html> |
