| <!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="Measurements of the circular frequency of oscillations of a body of mass m, fixed in the middle of a tensioned string with length 2 l, gave the value \omega. Find the force of tension of the string."> | | <meta name="description" content="Measurements of the circular frequency of oscillations of a body of mass m, fixed in the middle of a tensioned string with length 2 l, gave the value \omega. Find the force of tension of the string."> |
| <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="Measurements of the circular frequency of oscillations of a body of mass m, fixed in the middle of a tensioned string with length 2 l, gave the value \omega. Find the force of tension of the string."> | | <meta property="og:title" content="Measurements of the circular frequency of oscillations of a body of mass m, fixed in the middle of a tensioned string with length 2 l, gave the value \omega. Find the force of tension of the string."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="Measurements of the circular frequency of oscillations of a body of mass m, fixed in the middle of a tensioned string with length 2 l, gave the value \omega. Find the force of tension of the string."> | | <meta property="og:description" content="Measurements of the circular frequency of oscillations of a body of mass m, fixed in the middle of a tensioned string with length 2 l, gave the value \omega. Find the force of tension of the string."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>Measurements of the circular frequency of oscillations of a body of mass m, fixed in the middle of a tensioned string with length 2 l, gave the value \omega. Find the force of tension of the string.</title> | | <title>Measurements of the circular frequency of oscillations of a body of mass m, fixed in the middle of a tensioned string with length 2 l, gave the value \omega. Find the force of tension of the string.</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> |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $3.2.10.$ Measurements of the circular frequency of oscillations of a body of mass $m$, fixed in the middle of a tensioned string with length $2 l$, gave the value $\omega$. Find the force of tension of the string. | | $3.2.10.$ Measurements of the circular frequency of oscillations of a body of mass $m$, fixed in the middle of a tensioned string with length $2 l$, gave the value $\omega$. Find the force of tension of the string. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| The total external force acting on the system | | The total external force acting on the system |
| $$ F=2T\sin\varphi $$ | | $$ F=2T\sin\varphi $$ |
| Newton's second law | | Newton's second law |
| $$ m\ddot{x}(t)+2T\sin\varphi=0 $$ | | $$ m\ddot{x}(t)+2T\sin\varphi=0 $$ |
| Let's use the approximation for small angles $(\varphi \ll 1)$: | | Let's use the approximation for small angles $(\varphi \ll 1)$: |
| $$ \sin\varphi\approx\varphi=\frac{x}{l} $$ | | $$ \sin\varphi\approx\varphi=\frac{x}{l} $$ |
| Harmonic oscillation equation | | Harmonic oscillation equation |
| $$ \ddot{x}(t)+\frac{2T}{ml}x(t)=0 $$ | | $$ \ddot{x}(t)+\frac{2T}{ml}x(t)=0 $$ |
| Where does the period of oscillation come from? | | Where does the period of oscillation come from? |
| $$ \omega^{*2}=\frac{2T}{ml}\Rightarrow \boxed{T=\frac{ml\omega^{*2}}{2}} $$ | | $$ \omega^{*2}=\frac{2T}{ml}\Rightarrow \boxed{T=\frac{ml\omega^{*2}}{2}} $$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Dzikan Mikita<br> | | Dzikan Mikita<br> |
| Aliaksandr Kanashenka | | Aliaksandr Kanashenka |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$T=\frac{ml\omega^{*2}}{2}$$ | | $$T=\frac{ml\omega^{*2}}{2}$$ |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> |
| </p> | | </p> |
| <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> | | <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> | | </p> |
| </footer> | | </footer> |
| </body> | | </body> |
| | | |
| </html> | | </html> |
