| <!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 should be applied to the upper weight of mass m_1 so that the lower weight of mass m_2, connected to the upper one by a spring, breaks off from the floor after this force ceases?"> | | <meta name="description" content="What force should be applied to the upper weight of mass m_1 so that the lower weight of mass m_2, connected to the upper one by a spring, breaks off from the floor after this force ceases?"> |
| <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 should be applied to the upper weight of mass m_1 so that the lower weight of mass m_2, connected to the upper one by a spring, breaks off from the floor after this force ceases?"> | | <meta property="og:title" content="What force should be applied to the upper weight of mass m_1 so that the lower weight of mass m_2, connected to the upper one by a spring, breaks off from the floor after this force ceases?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="What force should be applied to the upper weight of mass m_1 so that the lower weight of mass m_2, connected to the upper one by a spring, breaks off from the floor after this force ceases?"> | | <meta property="og:description" content="What force should be applied to the upper weight of mass m_1 so that the lower weight of mass m_2, connected to the upper one by a spring, breaks off from the floor after this force ceases?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>What force should be applied to the upper weight of mass m_1 so that the lower weight of mass m_2, connected to the upper one by a spring, breaks off from the floor after this force ceases?</title> | | <title>What force should be applied to the upper weight of mass m_1 so that the lower weight of mass m_2, connected to the upper one by a spring, breaks off from the floor after this force ceases?</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="../../#2.3">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.3">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.3.46.$ What force should be applied to the upper weight of mass $m_1$ so that the lower weight of mass $m_2$, connected to the upper one by a spring, breaks off from the floor after this force ceases? | | $2.3.46.$ What force should be applied to the upper weight of mass $m_1$ so that the lower weight of mass $m_2$, connected to the upper one by a spring, breaks off from the floor after this force ceases? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.3.46.png" | | <img src="2.3.46.png" |
| loading="lazy" width="170" /> | | loading="lazy" width="170" /> |
| <figcaption> | | <figcaption> |
| For problem $2.3.46$ | | For problem $2.3.46$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.3.46_1.png" | | <img src="2.3.46_1.png" |
| loading="lazy" width="300" /> | | loading="lazy" width="300" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the system | | Forces acting on the system |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| Forces acting on the upper body in the force-less situation ($F=0$) | | Forces acting on the upper body in the force-less situation ($F=0$) |
| In the case where no external force $F$ is applied, we have the equation: | | In the case where no external force $F$ is applied, we have the equation: |
| $$N+k\Delta x'=m_2g $$ | | $$N+k\Delta x'=m_2g $$ |
| As $N=0$: | | As $N=0$: |
| $$\Delta x' = \frac{m_2g}{k}$$ | | $$\Delta x' = \frac{m_2g}{k}$$ |
| For the case where there is a force $F$ | | For the case where there is a force $F$ |
| $$\Delta x = \frac{F+m_1g}{k}$$ | | $$\Delta x = \frac{F+m_1g}{k}$$ |
| Law of energy preservation | | Law of energy preservation |
| $$m_1g\left(l-\Delta x\right)+\frac{k\Delta x^2}{2} = m_1g\left(l+\Delta x'\right)+\frac{k{\Delta x'}^2}{2}$$ | | $$m_1g\left(l-\Delta x\right)+\frac{k\Delta x^2}{2} = m_1g\left(l+\Delta x'\right)+\frac{k{\Delta x'}^2}{2}$$ |
| $$-m_1g\left(\Delta x + \Delta x'\right)=\frac{k}{2}\left({\Delta x'}^2-{\Delta x}^2\right)$$ | | $$-m_1g\left(\Delta x + \Delta x'\right)=\frac{k}{2}\left({\Delta x'}^2-{\Delta x}^2\right)$$ |
| The expression $ {\Delta x'}^2 - {\Delta x}^2 $ can be rewritten using the difference of squares: | | The expression $ {\Delta x'}^2 - {\Delta x}^2 $ can be rewritten using the difference of squares: |
| $$2m_1g=k\Delta x - k\Delta x'$$ | | $$2m_1g=k\Delta x - k\Delta x'$$ |
| Substituting the previously obtained values | | Substituting the previously obtained values |
| $$2m_1g=F+m_1g-m_2g$$ | | $$2m_1g=F+m_1g-m_2g$$ |
| From where the final result could be obtained | | From where the final result could be obtained |
| $$\boxed{F=m_1g+m_2g}$$ | | $$\boxed{F=m_1g+m_2g}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$F=(m_1+m_2)g$$ | | $$F=(m_1+m_2)g$$ |
| </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> |
