| <!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="Two beads of mass m each, connected to each other by a spring of stiffness k, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is l. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the x-direction after the beads jump off the rod? What will be the greatest compression strain of the spring?"> | | <meta name="description" content="Two beads of mass m each, connected to each other by a spring of stiffness k, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is l. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the x-direction after the beads jump off the rod? What will be the greatest compression strain of the spring?"> |
| <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="Two beads of mass m each, connected to each other by a spring of stiffness k, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is l. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the x-direction after the beads jump off the rod? What will be the greatest compression strain of the spring?"> | | <meta property="og:title" content="Two beads of mass m each, connected to each other by a spring of stiffness k, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is l. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the x-direction after the beads jump off the rod? What will be the greatest compression strain of the spring?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="Two beads of mass m each, connected to each other by a spring of stiffness k, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is l. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the x-direction after the beads jump off the rod? What will be the greatest compression strain of the spring?"> | | <meta property="og:description" content="Two beads of mass m each, connected to each other by a spring of stiffness k, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is l. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the x-direction after the beads jump off the rod? What will be the greatest compression strain of the spring?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>Two beads of mass m each, connected to each other by a spring of stiffness k, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is l. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the x-direction after the beads jump off the rod? What will be the greatest compression strain of the spring?</title> | | <title>Two beads of mass m each, connected to each other by a spring of stiffness k, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is l. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the x-direction after the beads jump off the rod? What will be the greatest compression strain of the spring?</title> |
| <link rel="stylesheet" href="../../css/css-latex/style.css"> | | <link rel="stylesheet" href="../../css/css-latex/style.css"> |
| <link rel="icon" href="../../img/logo.png" type="image/png"> | | <link rel="icon" href="../../img/logo.png" type="image/png"> |
| <script src="../../js/jquery-1.10.1.min.js"></script> | | <script src="../../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;"> |
| <div id = "logo"> | | <div id = "logo"> |
| <h3 id="back-link"><a href="../../#2.4">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.4">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.4.16.$ Two beads of mass $m$ each, connected to each other by a spring of stiffness $k$, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is $l$. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the $x$-direction after the beads jump off the rod? What will be the greatest compression strain of the spring? | | $2.4.16.$ Two beads of mass $m$ each, connected to each other by a spring of stiffness $k$, are held on smooth rods rigidly fixed in the wall. The spring is stretched and its length is $l$. The distance between the free ends of the rods is equal to the length of the undeformed spring. The beads are released. How fast will the spring move in the $x$-direction after the beads jump off the rod? What will be the greatest compression strain of the spring? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.4.16.png" | | <img src="2.4.16.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem $2.4.16$ | | For problem $2.4.16$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Preservation of energy at the initial stage | | Preservation of energy at the initial stage |
| $$\frac{k\left(l-l_0\right)^2}{2} = 2\frac{mv^2}{2}$$ | | $$\frac{k\left(l-l_0\right)^2}{2} = 2\frac{mv^2}{2}$$ |
| From where | | From where |
| $$ v = \sqrt{\frac{k}{2m}}\left(l-l_0\right)$$ | | $$ v = \sqrt{\frac{k}{2m}}\left(l-l_0\right)$$ |
| The greatest compression will be when it has nowhere to compress $(v_y=0)$ | | The greatest compression will be when it has nowhere to compress $(v_y=0)$ |
| Alternative for horizontal projection | | Alternative for horizontal projection |
| $$v_x=v\cos\alpha;v_x=\text{const}$$ | | $$v_x=v\cos\alpha;v_x=\text{const}$$ |
| After substituting | | After substituting |
| $$ \boxed{v_x = \sqrt{\frac{k}{2m}}\left(l-l_0\right)\cos\alpha} $$ | | $$ \boxed{v_x = \sqrt{\frac{k}{2m}}\left(l-l_0\right)\cos\alpha} $$ |
| Conservation of energy with $x_\text{max}$ deformation | | Conservation of energy with $x_\text{max}$ deformation |
| $$\frac{k\left(l-l_0\right)^2}{2} = 2\frac{mv_x^2}{2} + \frac{kx_\text{max}^2}{2}$$ | | $$\frac{k\left(l-l_0\right)^2}{2} = 2\frac{mv_x^2}{2} + \frac{kx_\text{max}^2}{2}$$ |
| From where | | From where |
| $$x_\text{max}^2=\left(l-l_0\right)^2\left(1-\cos^2\alpha\right)$$ | | $$x_\text{max}^2=\left(l-l_0\right)^2\left(1-\cos^2\alpha\right)$$ |
| After using pythagorean trigonometric identity: | | After using pythagorean trigonometric identity: |
| $$\boxed{x_\text{max} = \left(l-l_0\right) \sin \alpha}$$ | | $$\boxed{x_\text{max} = \left(l-l_0\right) \sin \alpha}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\begin{aligned}v_x=(l-l_0)\sqrt{k/(2m)} \cos\alpha; x=(l-l_0)\sin\alpha.\end{aligned}$$ | | $$\begin{aligned}v_x=(l-l_0)\sqrt{k/(2m)} \cos\alpha; x=(l-l_0)\sin\alpha.\end{aligned}$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
| </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> |
