Translated 3.2.1-3.2.17
en/3.2.3.md
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| + | <meta name="description" content="A weight is oscillating vertically on a rubber cord. How many times will the period of vertical oscillations of the weight change if it is suspended on the same cord folded in half?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
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| + | <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="../../#3.2">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $3.2.3.$ A weight is oscillating vertically on a rubber cord. How many times will the period of vertical oscillations of the weight change if it is suspended on the same cord folded in half? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | We find the period of oscillations on a standard cord as the period of oscillations of a spring pendulum | ||
| + | $$ T_1=2\pi\sqrt{\frac{m}{k}} $$ | ||
| + | When we fold the cord, we change its length and width and, accordingly, its stiffness $k'$ changes | ||
| + | $$ T_2=2\pi\sqrt{\frac{m}{k'}} $$ | ||
| + | Since the spring is divided into two and the length of each half is $l/2$ and the stiffness of each of them is $k_1=2k$; by adding the two halves in parallel we obtain a parallel connection | ||
| + | $$ k'=2k_1=4k $$ | ||
| + | We substitute the oscillation period of the “new” cord into the expression for $T_2$ | ||
| + | $$ T_2=2\pi\sqrt{\frac{m}{4k}}=\pi\sqrt{\frac{m}{k}} $$ | ||
| + | From where we obtain the ratio of the periods of oscillations | ||
| + | $$ \boxed{\frac{T_1}{T_2}=2} $$ | ||
| + | |||
| + | |||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Dzikan Mikita<br> | ||
| + | Aliaksandr Kanashenka | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | The period will be halved | ||
| + | </p> | ||
| + | |||
| + | |||
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| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
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| + | <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> | ||
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| <!DOCTYPE html> | |||
| <html lang="en"> | |||
| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
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| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <meta name="description" content="A weight is oscillating vertically on a rubber cord. How many times will the period of vertical oscillations of the weight change if it is suspended on the same cord folded in half?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A weight is oscillating vertically on a rubber cord. How many times will the period of vertical oscillations of the weight change if it is suspended on the same cord folded in half?"> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="A weight is oscillating vertically on a rubber cord. How many times will the period of vertical oscillations of the weight change if it is suspended on the same cord folded in half?"> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>A weight is oscillating vertically on a rubber cord. How many times will the period of vertical oscillations of the weight change if it is suspended on the same cord folded in half?</title> | |||
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | |||
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| </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="../../#3.2">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $3.2.3.$ A weight is oscillating vertically on a rubber cord. How many times will the period of vertical oscillations of the weight change if it is suspended on the same cord folded in half? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| We find the period of oscillations on a standard cord as the period of oscillations of a spring pendulum | |||
| $$ T_1=2\pi\sqrt{\frac{m}{k}} $$ | |||
| When we fold the cord, we change its length and width and, accordingly, its stiffness $k'$ changes | |||
| $$ T_2=2\pi\sqrt{\frac{m}{k'}} $$ | |||
| Since the spring is divided into two and the length of each half is $l/2$ and the stiffness of each of them is $k_1=2k$; by adding the two halves in parallel we obtain a parallel connection | |||
| $$ k'=2k_1=4k $$ | |||
| We substitute the oscillation period of the “new” cord into the expression for $T_2$ | |||
| $$ T_2=2\pi\sqrt{\frac{m}{4k}}=\pi\sqrt{\frac{m}{k}} $$ | |||
| From where we obtain the ratio of the periods of oscillations | |||
| $$ \boxed{\frac{T_1}{T_2}=2} $$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Dzikan Mikita<br> | |||
| Aliaksandr Kanashenka | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| The period will be halved | |||
| </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> | |||