| <!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 body of mass m is attached to two consecutive springs of stiffness k_1 and k_2. A constant force F is applied to the free end of the spring chain. What is the total elongation of the springs, if the vibrations have already stopped?"> | | <meta name="description" content="The body of mass m is attached to two consecutive springs of stiffness k_1 and k_2. A constant force F is applied to the free end of the spring chain. What is the total elongation of the springs, if the vibrations have already stopped?"> |
| <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 body of mass m is attached to two consecutive springs of stiffness k_1 and k_2. A constant force F is applied to the free end of the spring chain. What is the total elongation of the springs, if the vibrations have already stopped?"> | | <meta property="og:title" content="The body of mass m is attached to two consecutive springs of stiffness k_1 and k_2. A constant force F is applied to the free end of the spring chain. What is the total elongation of the springs, if the vibrations have already stopped?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="The body of mass m is attached to two consecutive springs of stiffness k_1 and k_2. A constant force F is applied to the free end of the spring chain. What is the total elongation of the springs, if the vibrations have already stopped?"> | | <meta property="og:description" content="The body of mass m is attached to two consecutive springs of stiffness k_1 and k_2. A constant force F is applied to the free end of the spring chain. What is the total elongation of the springs, if the vibrations have already stopped?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>The body of mass m is attached to two consecutive springs of stiffness k_1 and k_2. A constant force F is applied to the free end of the spring chain. What is the total elongation of the springs, if the vibrations have already stopped?</title> | | <title>The body of mass m is attached to two consecutive springs of stiffness k_1 and k_2. A constant force F is applied to the free end of the spring chain. What is the total elongation of the springs, if the vibrations have already stopped?</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> |
| $2.1.17.$ A light magnet with a hook on a vertical steel plate remains stationary as long as the weight suspended from it does not exceed $m_0$ in mass. What is the magnetic force if the coefficient of friction of the magnet on steel is equal to $\mu$? With what acceleration does the magnetic suspension slide, if the mass of the load is $m > m_0$? | | $2.1.17.$ A light magnet with a hook on a vertical steel plate remains stationary as long as the weight suspended from it does not exceed $m_0$ in mass. What is the magnetic force if the coefficient of friction of the magnet on steel is equal to $\mu$? With what acceleration does the magnetic suspension slide, if the mass of the load is $m > m_0$? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.17/statement.png" | | <img src="https://savchenkosolutions.com/2/2.1.17/statement.png" |
| loading="lazy" width="120" /> | | loading="lazy" width="120" /> |
| <figcaption> | | <figcaption> |
| For problem $2.1.17$ | | For problem $2.1.17$ |
| </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/2/2.1.17/sol.png" | | <img src="https://savchenkosolutions.com/2/2.1.17/sol.png" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| Forces acting on a magnet | | Forces acting on a magnet |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p> | | <p> |
| 1. The body can be considered as free if the connections are replaced by their reactions. The friction force in this case is caused by the action of the magnetic force, i.e. | | 1. The body can be considered as free if the connections are replaced by their reactions. The friction force in this case is caused by the action of the magnetic force, i.e. |
| $$ F_{fr} = \mu F_m $$ | | $$ F_{fr} = \mu F_m $$ |
| 2. The condition of mass equilibrium, in this case, will take place when the magnitudes of gravity and friction are equal. | | 2. The condition of mass equilibrium, in this case, will take place when the magnitudes of gravity and friction are equal. |
| $$ m_0 g = \mu F_m\Rightarrow \boxed{F_m = \frac{gm_0}{\mu}} $$ | | $$ m_0 g = \mu F_m\Rightarrow \boxed{F_m = \frac{gm_0}{\mu}} $$ |
| 3. When the body starts moving at $m > m_0$, then the equation of Newton's second law becomes valid, which in projection onto the $y$ axis will be written as follows | | 3. When the body starts moving at $m > m_0$, then the equation of Newton's second law becomes valid, which in projection onto the $y$ axis will be written as follows |
| $$ mg - m_0 g = ma $$ | | $$ mg - m_0 g = ma $$ |
| | | |
| $$ \boxed{a = g \frac{m - m_0}{m}} $$ | | $$ \boxed{a = g \frac{m - m_0}{m}} $$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$F = m_0g/\mu ;$$ | | $$F = m_0g/\mu ;$$ |
| $$a = g(m − m_0)/m$$ | | $$a = g(m − m_0)/m$$ |
| </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> |
