Translated 2.1.34-2.1.47
en/2.1.35.md
+117 −0
| @@ -0,0 +1,117 @@ | |||
| + | <!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 mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="The mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u."> | ||
| + | <meta property="og:image" content="img/logo.png"> | ||
| + | <meta property="og:description" content="The mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u."> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>The mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u.</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;"> | ||
| + | <div id = "logo"> | ||
| + | <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | ||
| + | </div> | ||
| + | <p class="author"> | ||
| + | Solutions of Savchenko Problems in Physics <br> | ||
| + | <i><b>knowledge must be free</b></i> | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.1.35^*.$ The mass of the balloon together with the rope dragging along the ground is $m$; the buoyant force acting on the balloon is $F$; the coefficient of friction of the rope on the ground is $\mu $. The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: $f = \alpha v^2$. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of $u$. | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.35/statement.png" | ||
| + | loading="lazy" width="250" /> | ||
| + | <figcaption> | ||
| + | For problem $2.1.35^*$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.35/sol.png" id="hot-airbaloon" | ||
| + | loading="lazy" width="350" /> | ||
| + | <figcaption> | ||
| + | Forces acting on a balloon | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | <p> | ||
| + | There are five forces acting on the balloon <a href="#hot-airbaloon">(see fig.)</a>: gravity $\vec{F}_{т} = M \vec{g}$, buoyancy $\vec{F}$, air resistance $\vec{F}_{c}$, Earth reaction force $\vec{N}$ and Earth friction force $\vec{F}_{пт}$. | ||
| + | </p> | ||
| + | <p> | ||
| + | Let $\vec{v}^{ \prime}$ denote the velocity of the balloon relative to the Earth. Then | ||
| + | $$ \vec{F}_{c} = - \alpha ( \vec{v}^{ \prime} - \vec{u}) $$ | ||
| + | From the condition that the balloon moves uniformly in the horizontal direction, it follows | ||
| + | $$ | \vec{F}_{c} | - | \vec{F}_{тр} | = 0 $$ | ||
| + | |||
| + | $$ | \vec{F}| + | \vec{N}| - M | \vec{g} | = 0 $$ | ||
| + | Besides, | ||
| + | $$ | \vec{F}_{тр} | = \mu | \vec{N} | $$ | ||
| + | Taking into account that | ||
| + | $$ | \vec{F}_{c}| = - \alpha ( | \vec{v}^{ \prime} | - | \vec{u} | ) $$ | ||
| + | from the last three equations we get | ||
| + | $$ | \vec{v}^{ \prime} | = | \vec{u} | - \frac{ \mu}{ \alpha} (M | \vec{g} | - | \vec{F} | ) $$ | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $v = u − \sqrt{(\mu /α)(mg − F)}$ at $αu^2 \geq \mu (mg − F)$, otherwise $v = 0$ | ||
| + | </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> | ||
| @@ -0,0 +1,117 @@ | |||
| <!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 mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="The mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="The mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>The mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u.</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;"> | |||
| <div id = "logo"> | |||
| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | |||
| </div> | |||
| <p class="author"> | |||
| Solutions of Savchenko Problems in Physics <br> | |||
| <i><b>knowledge must be free</b></i> | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.1.35^*.$ The mass of the balloon together with the rope dragging along the ground is $m$; the buoyant force acting on the balloon is $F$; the coefficient of friction of the rope on the ground is $\mu $. The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: $f = \alpha v^2$. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of $u$. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.35/statement.png" | |||
| loading="lazy" width="250" /> | |||
| <figcaption> | |||
| For problem $2.1.35^*$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.35/sol.png" id="hot-airbaloon" | |||
| loading="lazy" width="350" /> | |||
| <figcaption> | |||
| Forces acting on a balloon | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| <p> | |||
| There are five forces acting on the balloon <a href="#hot-airbaloon">(see fig.)</a>: gravity $\vec{F}_{т} = M \vec{g}$, buoyancy $\vec{F}$, air resistance $\vec{F}_{c}$, Earth reaction force $\vec{N}$ and Earth friction force $\vec{F}_{пт}$. | |||
| </p> | |||
| <p> | |||
| Let $\vec{v}^{ \prime}$ denote the velocity of the balloon relative to the Earth. Then | |||
| $$ \vec{F}_{c} = - \alpha ( \vec{v}^{ \prime} - \vec{u}) $$ | |||
| From the condition that the balloon moves uniformly in the horizontal direction, it follows | |||
| $$ | \vec{F}_{c} | - | \vec{F}_{тр} | = 0 $$ | |||
| $$ | \vec{F}| + | \vec{N}| - M | \vec{g} | = 0 $$ | |||
| Besides, | |||
| $$ | \vec{F}_{тр} | = \mu | \vec{N} | $$ | |||
| Taking into account that | |||
| $$ | \vec{F}_{c}| = - \alpha ( | \vec{v}^{ \prime} | - | \vec{u} | ) $$ | |||
| from the last three equations we get | |||
| $$ | \vec{v}^{ \prime} | = | \vec{u} | - \frac{ \mu}{ \alpha} (M | \vec{g} | - | \vec{F} | ) $$ | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $v = u − \sqrt{(\mu /α)(mg − F)}$ at $αu^2 \geq \mu (mg − F)$, otherwise $v = 0$ | |||
| </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> | |||