| <!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="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="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="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="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: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: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 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"> | | <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> | | <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="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="../../#2.3">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.3">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.3.49.$ 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 $\mu$? | | $2.3.49.$ 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> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.3.49.png" | | <img src="2.3.49.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem $2.3.49$ | | For problem $2.3.49$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.3.49_1.png" | | <img src="2.3.49_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> |
| | | |
| State of equilibrium for body $m_0$ | | State of equilibrium for body $m_0$ |
| $$N = m_0g$$ | | $$N = m_0g$$ |
| $$F_\text{fr} = \mu N = \mu m_0 g$$ | | $$F_\text{fr} = \mu N = \mu m_0 g$$ |
| Conservation of energy for the attached weight | | Conservation of energy for the attached weight |
| $$\frac{k\Delta x^2}{2} = mg(l+\Delta x) - mgl$$ | | $$\frac{k\Delta x^2}{2} = mg(l+\Delta x) - mgl$$ |
| From where | | From where |
| $$\frac{k\Delta x}{2} = mg$$ | | $$\frac{k\Delta x}{2} = mg$$ |
| Considering | | Considering |
| $$T=k\Delta x \Rightarrow \mu m_0 g = 2mg$$ | | $$T=k\Delta x \Rightarrow \mu m_0 g = 2mg$$ |
| So, we get the desired mass | | So, we get the desired mass |
| $$\boxed{m=\frac{\mu m_0}{2}}$$ | | $$\boxed{m=\frac{\mu m_0}{2}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
