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en/2.1.17.md
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| + | <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="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?"> | ||
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| + | <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?"> | ||
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| + | <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> | ||
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| + | </head> | ||
| + | <body style=""> | ||
| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.1.17.$ 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? | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.17/statement.png" | ||
| + | loading="lazy" width="120" /> | ||
| + | <figcaption> | ||
| + | For problem $2.1.17$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.17/sol.png" | ||
| + | loading="lazy" width="250" /> | ||
| + | <figcaption> | ||
| + | Forces acting on a magnet | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <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. | ||
| + | $$ F_{fr} = \mu F_m $$ | ||
| + | 2. The condition of mass equilibrium, in this case, will take place when the magnitudes of gravity and friction are equal. | ||
| + | $$ m_0 g = \mu F_m\Rightarrow \boxed{F_m = \frac{gm_0}{\mu}} $$ | ||
| + | 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 | ||
| + | $$ mg - m_0 g = ma $$ | ||
| + | |||
| + | $$ \boxed{a = g \frac{m - m_0}{m}} $$ | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$F = m_0g/\mu ;$$ | ||
| + | $$a = g(m − m_0)/m$$ | ||
| + | </p> | ||
| + | |||
| + | |||
| + | <footer class="row container"> | ||
| + | <br> | ||
| + | <p> | ||
| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
| + | </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> | ||
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| + | </footer> | ||
| + | </body> | ||
| + | |||
| + | </html> | ||
| @@ -0,0 +1,105 @@ | |||
| <!DOCTYPE html> | |||
| <html lang="en"> | |||
| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
| <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="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="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: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 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> | |||
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | |||
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| <script type="text/x-mathjax-config"> | |||
| MathJax.Hub.Config({ | |||
| extensions: ['tex2jax.js'], | |||
| jax: ['input/TeX', 'output/HTML-CSS'], | |||
| tex2jax: { | |||
| inlineMath: [['$', '$'], ['$', '$']], | |||
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| }, | |||
| showProcessingMessages: false, | |||
| messageStyle: 'none' | |||
| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.1.17.$ 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? | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.17/statement.png" | |||
| loading="lazy" width="120" /> | |||
| <figcaption> | |||
| For problem $2.1.17$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.17/sol.png" | |||
| loading="lazy" width="250" /> | |||
| <figcaption> | |||
| Forces acting on a magnet | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <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. | |||
| $$ F_{fr} = \mu F_m $$ | |||
| 2. The condition of mass equilibrium, in this case, will take place when the magnitudes of gravity and friction are equal. | |||
| $$ m_0 g = \mu F_m\Rightarrow \boxed{F_m = \frac{gm_0}{\mu}} $$ | |||
| 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 | |||
| $$ mg - m_0 g = ma $$ | |||
| $$ \boxed{a = g \frac{m - m_0}{m}} $$ | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$F = m_0g/\mu ;$$ | |||
| $$a = g(m − m_0)/m$$ | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </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> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||