| <!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="What force acts in the cross-section of a homogeneous rod of length l at a distance x from the end to which the force F is applied along the rod?"> | | <meta name="description" content="What force acts in the cross-section of a homogeneous rod of length l at a distance x from the end to which the force F is applied along the rod?"> |
| <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="What force acts in the cross-section of a homogeneous rod of length l at a distance x from the end to which the force F is applied along the rod?"> | | <meta property="og:title" content="What force acts in the cross-section of a homogeneous rod of length l at a distance x from the end to which the force F is applied along the rod?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="What force acts in the cross-section of a homogeneous rod of length l at a distance x from the end to which the force F is applied along the rod?"> | | <meta property="og:description" content="What force acts in the cross-section of a homogeneous rod of length l at a distance x from the end to which the force F is applied along the rod?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>What force acts in the cross-section of a homogeneous rod of length l at a distance x from the end to which the force F is applied along the rod?</title> | | <title>What force acts in the cross-section of a homogeneous rod of length l at a distance x from the end to which the force F is applied along the rod?</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> |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.1.5.$ What force acts in the cross-section of a homogeneous rod of length $l$ at a distance $x$ from the end to which the force $F$ is applied along the rod? | | $2.1.5.$ What force acts in the cross-section of a homogeneous rod of length $l$ at a distance $x$ from the end to which the force $F$ is applied along the rod? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.5/statement.png" | | <img src="https://savchenkosolutions.com/2/2.1.5/statement.png" |
| loading="lazy" width="280" /> | | loading="lazy" width="280" /> |
| <figcaption> | | <figcaption> |
| For problem $2.1.5$ | | For problem $2.1.5$ |
| </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/2/2.1.5/draw.png" | | <img src="https://savchenkosolutions.com/2/2.1.5/draw.png" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Forces acting on a small area | | Forces acting on a small area |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| If we consider a rod of mass $m$ as a single whole, then it will move with acceleration | | If we consider a rod of mass $m$ as a single whole, then it will move with acceleration |
| $$ a = \frac{F}{m} $$ | | $$ a = \frac{F}{m} $$ |
| Since the rod is inextensible, the acceleration of all its parts is the same and equal to $a$ | | Since the rod is inextensible, the acceleration of all its parts is the same and equal to $a$ |
| </p> | | </p> |
| <p> | | <p> |
| Let's consider a small section of the rod of length $dx$ and mass $dm$. Since the rod is homogeneous | | Let's consider a small section of the rod of length $dx$ and mass $dm$. Since the rod is homogeneous |
| $$ dm = m \frac{dx}{l} $$ | | $$ dm = m \frac{dx}{l} $$ |
| Let's write down Newton's second law for this section. | | Let's write down Newton's second law for this section. |
| $$ a\, dm = F(x+dx) - F(x)\;(1) $$ | | $$ a\, dm = F(x+dx) - F(x)\;(1) $$ |
| Where $F(x+dx)$ and $F(x)$ are the interaction forces with neighbors | | Where $F(x+dx)$ and $F(x)$ are the interaction forces with neighbors |
| </p> | | </p> |
| <p> | | <p> |
| Let's integrate expression $(1)$ along the horizontal coordinate $x$: | | Let's integrate expression $(1)$ along the horizontal coordinate $x$: |
| $$ \int_x^l am\frac{dx}{l} = \int_x^l dF $$ | | $$ \int_x^l am\frac{dx}{l} = \int_x^l dF $$ |
| | | |
| $$ ma\frac{l-x}{l} $$ | | $$ ma\frac{l-x}{l} $$ |
| | | |
| $$ F(x) = \fbox{$F(1-\frac{x}{l})$} $$ | | $$ F(x) = \fbox{$F(1-\frac{x}{l})$} $$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$T = F(1 − x/l)$$ | | $$T = F(1 − x/l)$$ |
| </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> |
