Added 2.3.49
en/2.3.49.md
+108 −0
| @@ -0,0 +1,108 @@ | |||
| + | <!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="On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? "> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? "> | ||
| + | <meta property="og:image" content="img/logo.png"> | ||
| + | <meta property="og:description" content="On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? "> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? </title> | ||
| + | <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"> | ||
| + | <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 type="text/x-mathjax-config"> | ||
| + | MathJax.Hub.Config({ | ||
| + | extensions: ['tex2jax.js'], | ||
| + | jax: ['input/TeX', 'output/HTML-CSS'], | ||
| + | tex2jax: { | ||
| + | inlineMath: [['$', '$'], ['$', '$']], | ||
| + | processEscapes: true, | ||
| + | processClass: 'tex2jax', | ||
| + | ignoreClass: 'html' | ||
| + | }, | ||
| + | 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="../../#2.3">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.3.49.$ On the horizontal table is a body of mass $m_0$. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass $m_0$ from its place, if the coefficient of friction of the body against the table is $\mu$? | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="2.3.49.png" | ||
| + | loading="lazy" width="200" /> | ||
| + | <figcaption> | ||
| + | For problem $2.3.49$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="2.3.49_1.png" | ||
| + | loading="lazy" width="300" /> | ||
| + | <figcaption> | ||
| + | Forces acting on the system | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | State of equilibrium for body $m_0$ | ||
| + | $$N = m_0g$$ | ||
| + | $$F_\text{fr} = \mu N = \mu m_0 g$$ | ||
| + | Conservation of energy for the attached weight | ||
| + | $$\frac{k\Delta x^2}{2} = mg(l+\Delta x) - mgl$$ | ||
| + | From where | ||
| + | $$\frac{k\Delta x}{2} = mg$$ | ||
| + | Considering | ||
| + | $$T=k\Delta x \Rightarrow \mu m_0 g = 2mg$$ | ||
| + | So, we get the desired mass | ||
| + | $$\boxed{m=\frac{\mu m_0}{2}}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$m=\frac{\mu m_0}{2}$$ | ||
| + | </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> aliaksandr@savchenkosolutions.com <br></small> | ||
| + | </p> | ||
| + | </footer> | ||
| + | </body> | ||
| + | |||
| + | </html> | ||
| @@ -0,0 +1,108 @@ | |||
| <!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="On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? "> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? "> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? "> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? </title> | |||
| <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"> | |||
| <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 type="text/x-mathjax-config"> | |||
| MathJax.Hub.Config({ | |||
| extensions: ['tex2jax.js'], | |||
| jax: ['input/TeX', 'output/HTML-CSS'], | |||
| tex2jax: { | |||
| inlineMath: [['$', '$'], ['$', '$']], | |||
| processEscapes: true, | |||
| processClass: 'tex2jax', | |||
| ignoreClass: 'html' | |||
| }, | |||
| 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="../../#2.3">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.3.49.$ On the horizontal table is a body of mass $m_0$. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass $m_0$ from its place, if the coefficient of friction of the body against the table is $\mu$? | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.3.49.png" | |||
| loading="lazy" width="200" /> | |||
| <figcaption> | |||
| For problem $2.3.49$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.3.49_1.png" | |||
| loading="lazy" width="300" /> | |||
| <figcaption> | |||
| Forces acting on the system | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| State of equilibrium for body $m_0$ | |||
| $$N = m_0g$$ | |||
| $$F_\text{fr} = \mu N = \mu m_0 g$$ | |||
| Conservation of energy for the attached weight | |||
| $$\frac{k\Delta x^2}{2} = mg(l+\Delta x) - mgl$$ | |||
| From where | |||
| $$\frac{k\Delta x}{2} = mg$$ | |||
| Considering | |||
| $$T=k\Delta x \Rightarrow \mu m_0 g = 2mg$$ | |||
| So, we get the desired mass | |||
| $$\boxed{m=\frac{\mu m_0}{2}}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$m=\frac{\mu m_0}{2}$$ | |||
| </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> aliaksandr@savchenkosolutions.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||