| <!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="Potential energy of interaction of a particle with a fixed point source U = V (\frac{l^2}{r^2} - \frac{2l}{r}), where r is the distance between the particle and the source, V and l are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to E?"> | | <meta name="description" content="Potential energy of interaction of a particle with a fixed point source U = V (\frac{l^2}{r^2} - \frac{2l}{r}), where r is the distance between the particle and the source, V and l are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to E?"> |
| <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="Potential energy of interaction of a particle with a fixed point source U = V (\frac{l^2}{r^2} - \frac{2l}{r}), where r is the distance between the particle and the source, V and l are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to E?"> | | <meta property="og:title" content="Potential energy of interaction of a particle with a fixed point source U = V (\frac{l^2}{r^2} - \frac{2l}{r}), where r is the distance between the particle and the source, V and l are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to E?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="Potential energy of interaction of a particle with a fixed point source U = V (\frac{l^2}{r^2} - \frac{2l}{r}), where r is the distance between the particle and the source, V and l are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to E?"> | | <meta property="og:description" content="Potential energy of interaction of a particle with a fixed point source U = V (\frac{l^2}{r^2} - \frac{2l}{r}), where r is the distance between the particle and the source, V and l are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to E?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>Potential energy of interaction of a particle with a fixed point source U = V (\frac{l^2}{r^2} - \frac{2l}{r}), where r is the distance between the particle and the source, V and l are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to E?</title> | | <title>Potential energy of interaction of a particle with a fixed point source U = V (\frac{l^2}{r^2} - \frac{2l}{r}), where r is the distance between the particle and the source, V and l are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to E?</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> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.3.40.$ Potential energy of interaction of a particle with a fixed point source $U = V (\frac{l^2}{r^2} - \frac{2l}{r})$, where $r$ is the distance between the particle and the source, $V$ and $l$ are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to $E$? | | $2.3.40.$ Potential energy of interaction of a particle with a fixed point source $U = V (\frac{l^2}{r^2} - \frac{2l}{r})$, where $r$ is the distance between the particle and the source, $V$ and $l$ are positive constants having the dimension of energy and distance, respectively. In what region does the rectilinear motion of a particle occur, if the total energy of the system is equal to $E$? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.3.40.png" | | <img src="2.3.40.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| Let us consider the case where the system (the second particle) starts moving at a distance $r$ from the first particle.</p><p> | | Let us consider the case where the system (the second particle) starts moving at a distance $r$ from the first particle.</p><p> |
| Since this is the beginning of the movement, the kinetic energy of the first and second particles tends to zero. | | Since this is the beginning of the movement, the kinetic energy of the first and second particles tends to zero. |
| $$E_{k1}\to 0;\quad E_{k2}\to 0$$ | | $$E_{k1}\to 0;\quad E_{k2}\to 0$$ |
| From where the total energy would be equal to the potential energy | | From where the total energy would be equal to the potential energy |
| $$E=E_p$$ | | $$E=E_p$$ |
| Substituting expression from the statement | | Substituting expression from the statement |
| $$\frac{E}{V}=\frac{l^2}{r^2}-\frac{2l}{r}$$ | | $$\frac{E}{V}=\frac{l^2}{r^2}-\frac{2l}{r}$$ |
| Transforming it into a quadratic equation | | Transforming it into a quadratic equation |
| $$Er^2+2Vlr-Vl^2=0$$ | | $$Er^2+2Vlr-Vl^2=0$$ |
| Let's solve it | | Let's solve it |
| $$r_{1,2}=l\frac VE\left(-1\pm\sqrt{1+E/V}\right)$$ | | $$r_{1,2}=l\frac VE\left(-1\pm\sqrt{1+E/V}\right)$$ |
| Since the distance is not less than zero, we choose a positive sign for $E>0$ | | Since the distance is not less than zero, we choose a positive sign for $E>0$ |
| $$r\geq l\frac VE\left(\sqrt{1+E/V}-1\right)$$ | | $$r\geq l\frac VE\left(\sqrt{1+E/V}-1\right)$$ |
| For $E<0$ it will be in the range between $r_1$ and $r_2$ | | For $E<0$ it will be in the range between $r_1$ and $r_2$ |
| $$l\frac VE\left(-1-\sqrt{1+E/V}\right)\leq r\leq l\frac VE\left(-1+\sqrt{1+E/V}\right)$$ | | $$l\frac VE\left(-1-\sqrt{1+E/V}\right)\leq r\leq l\frac VE\left(-1+\sqrt{1+E/V}\right)$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| With $E>0$ area of motion $r\geqslant l\frac VE\left(-1+\sqrt{1+E/V}\right)$</p><p> | | With $E>0$ area of motion $r\geqslant l\frac VE\left(-1+\sqrt{1+E/V}\right)$</p><p> |
| if $E<0$ area of motion, $r$ between $r_{1,2}=l\frac VE\left(-1\pm\sqrt{1+E/V}\right)$ | | if $E<0$ area of motion, $r$ between $r_{1,2}=l\frac VE\left(-1\pm\sqrt{1+E/V}\right)$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
| </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> |
