| <!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 connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> | | <meta name="description" content="The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> |
| <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 connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> | | <meta property="og:title" content="The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> |
| <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 connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> | | <meta property="og:description" content="The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls.</title> | | <title>The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls.</title> |
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | | <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> |
| <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> | | <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> |
| <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> | | <script src="https://savchenkosolutions.com/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;"> |
| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| </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> |
| <img src="https://savchenkosolutions.com/2/2.1.16/statement.png" | | <img src="https://savchenkosolutions.com/2/2.1.16/statement.png" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| For problem $2.1.16$ | | For problem $2.1.16$ |
| </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/2/2.1.16/sol.png" | | <img src="https://savchenkosolutions.com/2/2.1.16/sol.png" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Series spring system | | Series spring system |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p> | | <p> |
| 1. When springs are connected in series, their deformation will be different with the same acting force, this circumstance allows us to determine the overall stiffness of the springs as follows | | 1. When springs are connected in series, their deformation will be different with the same acting force, this circumstance allows us to determine the overall stiffness of the springs as follows |
| $$ \Delta x_0 = \Delta x_1 + \Delta x_2 = \frac{F}{k_1}+\frac{F}{k_2} = \frac{F}{k_0} $$ | | $$ \Delta x_0 = \Delta x_1 + \Delta x_2 = \frac{F}{k_1}+\frac{F}{k_2} = \frac{F}{k_0} $$ |
| | | |
| $$ \boxed{k_0 = \frac{k_1 k_2}{k_1 + k_2}} $$ | | $$ \boxed{k_0 = \frac{k_1 k_2}{k_1 + k_2}} $$ |
| 2. The combined effect on the mass of the springs when the mass is at rest will be equal to the applied force $F$ | | 2. The combined effect on the mass of the springs when the mass is at rest will be equal to the applied force $F$ |
| $$ k_0 \Delta x_0 = F $$ | | $$ k_0 \Delta x_0 = F $$ |
| | | |
| $$ \boxed{ \Delta x_0 = \frac{F(k_1 + k_2)}{k_1 k_2}} $$ | | $$ \boxed{ \Delta x_0 = \frac{F(k_1 + k_2)}{k_1 k_2}} $$ |
| </p> | | </p> |
| <h3> | | <h3> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$x = F(k_1 + k_2)/(k_1k_2)$$ | | $$x = F(k_1 + k_2)/(k_1k_2)$$ |
| </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> |
