Translated 2.1.9-2.1.20
en/2.1.13.md
+118 −0
| @@ -0,0 +1,118 @@ | |||
| + | <!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 painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to 400 H. Cradle weight 12 kg, painter weight 72 kg. What is the acceleration of the cradle?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="The painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to 400 H. Cradle weight 12 kg, painter weight 72 kg. What is the acceleration of the cradle?"> | ||
| + | <meta property="og:image" content="img/logo.png"> | ||
| + | <meta property="og:description" content="The painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to 400 H. Cradle weight 12 kg, painter weight 72 kg. What is the acceleration of the cradle?"> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>The painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to 400 H. Cradle weight 12 kg, painter weight 72 kg. What is the acceleration of the cradle?</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="../../#2.1">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.1.13.$ The painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to $400$ H. Cradle weight $12$ kg, painter weight $72$ kg. What is the acceleration of the cradle? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.13/draw.png" | ||
| + | loading="lazy" width="270" /> | ||
| + | <figcaption> | ||
| + | Forces acting on the system | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <p> | ||
| + | Let's write the equilibrium condition for the two lower balls on the vertical and horizontal axes | ||
| + | $$ | ||
| + | \left\{\begin{matrix} | ||
| + | mg = F_1 \sin \alpha \\ | ||
| + | F_x = F_1 \cos \alpha & | ||
| + | \end{matrix}\right. $$ | ||
| + | where $P$ is the painter's pressure force on the chair. </p> | ||
| + | And for the upper ball | ||
| + | $$ T = mg + 2F_1 \sin \alpha $$ | ||
| + | |||
| + | $$ T = 3mg $$ | ||
| + | Accordingly, when the thread burns out, the upper ball will be acted upon downwards by a force of $T=3mg$. From Newton's second law, we find its initial acceleration as | ||
| + | $$ a = \frac{T}{m} = 3g $$ | ||
| + | The lower balls will be acted upon in the horizontal direction by the force $F_x$, which will be compensated by the force $F_1 \cos \alpha$, and the force of gravity $mg$ — $F_1 \sin \alpha$ | ||
| + | <p> | ||
| + | <p> | ||
| + | Thus, the lower balls will be in zero gravity | ||
| + | $$ a=0 $$ | ||
| + | NO: Something similar happens when a Slinky falls | ||
| + | </p> | ||
| + | <br> | ||
| + | |||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.13/slinki.gif" | ||
| + | loading="lazy" width="270" /> | ||
| + | <figcaption> | ||
| + | Falling Slinky | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | I recommend an interesting problem about Slinky <a href="https://s3.eu-central-1.amazonaws.com/physprob.com/files/ipho/2019_Israel_p1.pdf" target="_blank">IPhO 2019 "Springs and Slinky"</a> | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | The acceleration of the upper ball is $3g$, and the acceleration of the lower balls is zero. | ||
| + | </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 +1,118 @@ | |||
| <!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 painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to 400 H. Cradle weight 12 kg, painter weight 72 kg. What is the acceleration of the cradle?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="The painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to 400 H. Cradle weight 12 kg, painter weight 72 kg. What is the acceleration of the cradle?"> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="The painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to 400 H. Cradle weight 12 kg, painter weight 72 kg. What is the acceleration of the cradle?"> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>The painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to 400 H. Cradle weight 12 kg, painter weight 72 kg. What is the acceleration of the cradle?</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="../../#2.1">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.1.13.$ The painter works in a suspended cradle. He needed to get up quickly. He begins to pull the rope with such force that the force of his pressure on the floor of the cradle is reduced to $400$ H. Cradle weight $12$ kg, painter weight $72$ kg. What is the acceleration of the cradle? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.13/draw.png" | |||
| loading="lazy" width="270" /> | |||
| <figcaption> | |||
| Forces acting on the system | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| Let's write the equilibrium condition for the two lower balls on the vertical and horizontal axes | |||
| $$ | |||
| \left\{\begin{matrix} | |||
| mg = F_1 \sin \alpha \\ | |||
| F_x = F_1 \cos \alpha & | |||
| \end{matrix}\right. $$ | |||
| where $P$ is the painter's pressure force on the chair. </p> | |||
| And for the upper ball | |||
| $$ T = mg + 2F_1 \sin \alpha $$ | |||
| $$ T = 3mg $$ | |||
| Accordingly, when the thread burns out, the upper ball will be acted upon downwards by a force of $T=3mg$. From Newton's second law, we find its initial acceleration as | |||
| $$ a = \frac{T}{m} = 3g $$ | |||
| The lower balls will be acted upon in the horizontal direction by the force $F_x$, which will be compensated by the force $F_1 \cos \alpha$, and the force of gravity $mg$ — $F_1 \sin \alpha$ | |||
| <p> | |||
| <p> | |||
| Thus, the lower balls will be in zero gravity | |||
| $$ a=0 $$ | |||
| NO: Something similar happens when a Slinky falls | |||
| </p> | |||
| <br> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.13/slinki.gif" | |||
| loading="lazy" width="270" /> | |||
| <figcaption> | |||
| Falling Slinky | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| I recommend an interesting problem about Slinky <a href="https://s3.eu-central-1.amazonaws.com/physprob.com/files/ipho/2019_Israel_p1.pdf" target="_blank">IPhO 2019 "Springs and Slinky"</a> | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| The acceleration of the upper ball is $3g$, and the acceleration of the lower balls is zero. | |||
| </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> | |||