| <!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="solutions, savchenko, physics problems, olympiad physics, physics book"> | | <meta name="keywords" content="solutions, savchenko, physics problems, olympiad physics, physics book"> |
| <meta name="description" content="A website with solutions to physics problems from Savchenko Textbook"> | | <meta name="description" content="A website with solutions to physics problems from Savchenko Textbook"> |
| <meta property="og:title" content="Savchenko Solutions"> | | <meta property="og:title" content="Savchenko Solutions"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="A website with solutions to physics problems from Savchenko Textbook"> | | <meta property="og:description" content="A website with solutions to physics problems from Savchenko Textbook"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>Savchenko Solutions</title> | | <title>Savchenko Solutions</title> |
| <link rel="stylesheet" href="https://savchenko-physics.github.io/css/css-latex/style.css"> | | <link rel="stylesheet" href="https://savchenko-physics.github.io/css/css-latex/style.css"> |
| <link rel="icon" href="https://savchenko-physics.github.io/img/logo.png" type="image/png"> | | <link rel="icon" href="https://savchenko-physics.github.io/img/logo.png" type="image/png"> |
| <script src="https://savchenko-physics.github.io/js/jquery-1.10.1.min.js"></script> | | <script src="https://savchenko-physics.github.io/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 Physics Textbook</h2> | | <h2>Solutions of Savchenko Physics Textbook</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="../#2.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../#2.2">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.2.39.$ The tube of radius $r$ is filled with a porous substance of density $\rho_0$. The piston, which acts on a constant force $F$, moving in the pipe, compacts the substance to a density $\rho$. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is $\rho$, and to the left-$\rho_0$? At the initial moment, this boundary coincides with the surface of the piston. | | $2.2.39.$ The tube of radius $r$ is filled with a porous substance of density $\rho_0$. The piston, which acts on a constant force $F$, moving in the pipe, compacts the substance to a density $\rho$. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is $\rho$, and to the left-$\rho_0$? At the initial moment, this boundary coincides with the surface of the piston. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw.png" | | <img src="draw.png" |
| loading="lazy" width="280" /> | | loading="lazy" width="280" /> |
| <figcaption> | | <figcaption> |
| Geometry of the problem | | Geometry of the problem |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| From the drawing | | From the drawing |
| $$\Delta L=L_0-L$$ | | $$\Delta L=L_0-L$$ |
| During the time $\Delta t$ the piston with velocity $v$ has moved by the amount $\Delta l$ | | During the time $\Delta t$ the piston with velocity $v$ has moved by the amount $\Delta l$ |
| $$\Delta L=v\Delta t$$ | | $$\Delta L=v\Delta t$$ |
| Given a constant cross-sectional area $S=\pi r^2$, the expression from above can be rewritten as | | Given a constant cross-sectional area $S=\pi r^2$, the expression from above can be rewritten as |
| $$\frac{m}{\rho_0S}-\frac{m}{\rho S}=v\Delta t$$ | | $$\frac{m}{\rho_0S}-\frac{m}{\rho S}=v\Delta t$$ |
| $$\frac{m}{\Delta t}=Sv\frac{\rho\rho_0}{\rho-\rho_0}$$ | | $$\frac{m}{\Delta t}=Sv\frac{\rho\rho_0}{\rho-\rho_0}$$ |
| Because the mass remains constant ($m=\text{const}$): | | Because the mass remains constant ($m=\text{const}$): |
| $$m=\rho_0SL_0=\rho SL$$ | | $$m=\rho_0SL_0=\rho SL$$ |
| From where | | From where |
| $$\frac{\rho}{\rho_0}=\frac{L}{L_0}$$ | | $$\frac{\rho}{\rho_0}=\frac{L}{L_0}$$ |
| Representing the force by a small change in momentum | | Representing the force by a small change in momentum |
| $$F_\text{average}=\frac{mv_\text{system}}{\Delta t}$$ | | $$F_\text{average}=\frac{mv_\text{system}}{\Delta t}$$ |
| Substituting the value of $\frac{m}{\Delta t}$: | | Substituting the value of $\frac{m}{\Delta t}$: |
| $$F_\text{average}=Sv\frac{\rho\rho_0}{\rho-\rho_0}v_\text{system}$$ | | $$F_\text{average}=Sv\frac{\rho\rho_0}{\rho-\rho_0}v_\text{system}$$ |
| $$v_\text{system}=\frac{0+v}{2}=\frac{v}{2}$$ | | $$v_\text{system}=\frac{0+v}{2}=\frac{v}{2}$$ |
| $$F_\text{average}=\frac{0+F}{2}=\frac{F}{2}$$ | | $$F_\text{average}=\frac{0+F}{2}=\frac{F}{2}$$ |
| Substitute into the formula for $F_\text{average}$: | | Substitute into the formula for $F_\text{average}$: |
| $$\frac{F}{2}=Sv\frac{\rho\rho_0}{\rho-\rho_0}\frac{v}{2}$$ | | $$\frac{F}{2}=Sv\frac{\rho\rho_0}{\rho-\rho_0}\frac{v}{2}$$ |
| From where | | From where |
| $$\boxed{v=\sqrt{\frac{F(\rho-\rho_0)}{\pi r^2\rho\rho_0}}}$$ | | $$\boxed{v=\sqrt{\frac{F(\rho-\rho_0)}{\pi r^2\rho\rho_0}}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
