| <!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="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?"> | | <meta name="description" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?"> |
| <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="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?"> | | <meta property="og:title" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?"> | | <meta property="og:description" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?</title> | | <title>A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?</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"> |
| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | | <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> |
| </div> | | </div> |
| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
| $3.3.3$ | | $3.3.3$ |
| A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations? | | A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| For a harmonic motion, variable position depends sinusoidally on time. In this case, we suppose that load is at equilibrium position at $t = 0$. So, sine function is more adequate to this situation without phase angle or null phase angle. So, | | For a harmonic motion, variable position depends sinusoidally on time. In this case, we suppose that load is at equilibrium position at $t = 0$. So, sine function is more adequate to this situation without phase angle or null phase angle. So, |
| $$x(t) = A~\sin{~\omega t}$$ | | $$x(t) = A~\sin{~\omega t}$$ |
| where $A$ is the amplitude such that $A = 1~{\rm{cm}}$, because it is the maximum distance. Let's suppose that $x_0 = 0.5~{\rm{cm}}$ is achieved at $t=t_1$ and $A$ is achieved at $t = t_2$. Then, | | where $A$ is the amplitude such that $A = 1~{\rm{cm}}$, because it is the maximum distance. Let's suppose that $x_0 = 0.5~{\rm{cm}}$ is achieved at $t=t_1$ and $A$ is achieved at $t = t_2$. Then, |
| $$x_0 = A~\sin{~\omega t_1} \;(1)$$ | | $$x_0 = A~\sin{~\omega t_1} \;(1)$$ |
| $$A = A~\sin{~\omega t_2}$$ or | | $$A = A~\sin{~\omega t_2}$$ or |
| $$\sin{~\omega t_2} = 1$$ | | $$\sin{~\omega t_2} = 1$$ |
| Hence, $\omega t_2 = \frac{\pi}{2} + 2k\pi$ with $k\in\mathbb{Z}$, for $k=0$, $t_2 = \frac{\pi}{2\omega}$. As $\omega = \frac{2\pi}{T}$, so $t_2 = \frac{T}{4}$. Since $\Delta t = t_2 - t_1 = 0.01~{\rm{s}}$, | | Hence, $\omega t_2 = \frac{\pi}{2} + 2k\pi$ with $k\in\mathbb{Z}$, for $k=0$, $t_2 = \frac{\pi}{2\omega}$. As $\omega = \frac{2\pi}{T}$, so $t_2 = \frac{T}{4}$. Since $\Delta t = t_2 - t_1 = 0.01~{\rm{s}}$, |
| $$t_1 = \frac{T}{4} - \Delta t \;(2)$$ | | $$t_1 = \frac{T}{4} - \Delta t \;(2)$$ |
| Putting (2) into (1) and separating $T$, it is obtained | | Putting (2) into (1) and separating $T$, it is obtained |
| $$T = \frac{4\Delta t}{1-\frac{2}{\pi}\arcsin{\frac{x_0}{A}}}$$ | | $$T = \frac{4\Delta t}{1-\frac{2}{\pi}\arcsin{\frac{x_0}{A}}}$$ |
| </p> | | </p> |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$T = 0.06 {\rm{s}}$$ | | $$T = 0.06 {\rm{s}}$$ |
| </p> | | </p> |
| | | |
| | | |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| BSc. Luis Daniel Fernández Quintana<br> | | BSc. Luis Daniel Fernández Quintana<br> |
| Physics Department (FCNE)<br> | | Physics Department (FCNE)<br> |
| Universidad de Oriente, Cuba<br> | | Universidad de Oriente, Cuba<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> |
